# Topological Vector Spaces: a non-standard approach with monads and galaxies.

Niels Charlier, Hans Vernaeve

###### Abstract

By generalizing the overspill principle towards directed sets, a new and
extensive formalism is developed for monads and galaxies in non-standard
enlargements. It is shown that monads and galaxies can be manipulated using
order-preserving and order-reversing set-to-set maps, and that set properties
associated with these maps can be extended not only to internal sets but to
all monads and galaxies. An abstract theory of Intersections of Galaxies is
introduced. These concepts are applied to basic topology as well (locally
convex) topological vector spaces and their properties. Local properties and
completeness can be defined and characterized effortlessly. Duality theory is
studied in this framework, allowing in particular to formulate brief and
insightful proofs for the theorems of Mackey-Arens and Grothendieck
completeness without any technicalities.

## Preliminaries

Several proposed non-standard theories of topological vector spaces can be
found in literature (Stroyan and Luxemburg (1976); Henson and Moore (1972);
Young (1972)). In this paper we introduce a new approach with new notations
and more generalized definitions. The reader is only assumed to be familiar
with the basic principles of non-standard analysis, in particular with
saturated models.

##  1 The general theory of monads and galaxies

###  1.1 Directedness and cofinality

######  Definition 1.1.

A preordered set is a pair $(J,\leq)$ where $J$ is non-empty set and $\leq$ a
reflexive, transitive relation on $J$. If the equivalence relation defined by
$\leq\wedge\geq$ coincides with $=$, it is a (partial) order. If for any
$a,b\in J:\exists c\in J:c\geq a,b$, the (pre-)order is directed. If for any
$a,b\in J$, $a\leq b\vee a\geq b$, the (pre-)order is total. The tails; heads;
finite and infinite points of $(J,\leq)$ are ($\iota\in[{}^{\star}]{J}$):

$\begin{split}[{}^{\star}]{J}_{\leq\iota}:=\\{j\in[{}^{\star}]{J}:j\leq\iota\\}~{}~{}~{}&~{}~{}~{}[{}^{\star}]{J}_{\geq\iota}:=\\{j\in[{}^{\star}]{J}:j\geq\iota\\}\\\
{}^{\star}{J}_{\mathfrak{f}}:=\\{j\in{}^{\star}{J}:\exists\iota\in{}^{\sigma}{J}:j\leq\iota\\}\supseteq{}^{\sigma}{J};~{}~{}~{}&~{}~{}~{}{}^{\star}{J}_{\infty}:=\\{\omega\in{}^{\star}{J}:\forall
j\in{}^{\sigma}{J}:\omega\geq j\\}.\end{split}$

If the greatest elements $J_{\top}:=\\{j\in{J}:\forall k\in J:j\geq
k\\}=\emptyset$ then
${}^{\star}{J}_{\infty}\cap{}^{\star}{J}_{\mathfrak{f}}=\emptyset$ and both
sets are external. Otherwise
${}^{\star}{J}={}^{\star}{J}_{\mathfrak{f}}\supseteq{}^{\star}{J}_{\infty}={}^{\star}{J_{\top}}$,

######  Proposition 1.2.

Assume we are using a $\kappa$-saturated model. Given a pre-ordered set
$(J,\leq)$ with $|J|\leq\kappa$ Then $J$ is directed iff
${}^{\star}{J}_{\infty}\neq\emptyset$.

###### Proof.

Since ${}^{\star}{J}_{\infty}=\cap_{j\in J}{}^{\star}{J}_{\geq j}$, saturation
implies that ${}^{\star}{J}_{\infty}\neq\emptyset$ iff $\\{{}^{\star}{J}_{\geq
j}\\}_{j\in J}$ has the finite intersection property, which is equivalent to
directedness. ∎

######  Proposition 1.3.

(Finite overspill) Assume we are using a $\kappa$-saturated model. Given a
directed set $J$ with $|J|\leq\kappa$ and an internal
$K\subseteq{}^{\star}{J}$ such that ${}^{\star}{J}_{\infty}\subseteq K$. Then
there exists an $\iota\in J$ such that ${}^{\star}{J}_{\geq\iota}\subseteq K$.

###### Proof.

Since $\cap_{j\in J}{}^{\star}{J}_{\geq j}\subseteq K$, i.e. $\cap_{j\in
J}{}^{\star}{J}_{\geq j}\setminus K=\emptyset$, saturation implies
$\\{{}^{\star}{J}_{\geq j}\setminus K\\}_{j\in J}$ does not have the finite
intersection property, which due to directedness implies the existence of a
$j\in J$ such that ${}^{\star}{J}_{\geq j}\setminus K=\emptyset$, i.e.
${}^{\star}{J}_{\geq\iota}\subseteq K$. ∎

######  Corrolary 1.4.

(Infinite overspill) Assume we are using a $\kappa$-saturated model. Given a
directed set $J$ with $|J|\leq\kappa$ and an internal
$K\subseteq{}^{\star}{J}$ such that ${}^{\star}{J}_{\mathfrak{f}}\subseteq K$.
Then there exists an $\omega\in{}^{\star}{J}_{\infty}$ such that
${}^{\star}{J}_{\leq\omega}\subseteq K$.

###### Proof.

Consider the internal set
$L=\\{\iota\in{}^{\star}{J}:{}^{\star}{J}_{\leq\iota}\not\subseteq K\\}$.
Suppose ${}^{\star}{J}_{\infty}\subseteq L$, then there exists a $j\in J$ such
that ${}^{\star}{J}_{\geq\iota}\subseteq L$, i.e.
${}^{\star}{J}_{\leq\iota}\not\subseteq K$, contradicting the assumption. Then
there exists an $\omega\in{}^{\star}{J}_{\infty}\setminus L$, i.e.
${}^{\star}{J}_{\leq\omega}\subseteq K$. ∎

######  Definition 1.5.

Given preordered sets $(J,\leq)$ and $(K,\leq)$. Then the product-order
$(J,\leq)\times(K,\leq):=(J\times K,\leq)$ where given $j,j^{\prime}\in
J;k,k^{\prime}\in K$: $(j,k)\leq(j^{\prime},k^{\prime})\iff j\leq
j^{\prime}\wedge k\leq k^{\prime}$.

######  Definition 1.6.

Given preordered sets $(J,\leq)$ and $(K,\leq)$ and a mapping
$\psi:J\rightarrow K$. Then $\psi$ is order preserving if for any
$j,j^{\prime}\in J:j\geq j^{\prime}\Rightarrow\psi(j)\geq\psi(j^{\prime})$. It
is order embedding if for any $j,j^{\prime}\in J:j\geq
j^{\prime}\iff\psi(j)\geq\psi(j^{\prime})$. A order embedding bijection is
called an order isomorphism. It is order reversing if for any $j,j^{\prime}\in
J:j\geq j^{\prime}\Rightarrow\psi(j)\leq\psi(j^{\prime})$.

######  Definition 1.7.

Given a preordered set $(X,\geq)$. Then $\mathcal{P}_{[\kappa]}(X,\leq)$ are
the preordered subsets with induced pre-order [with cardinality $\leq\kappa$]
, $\vec{\mathcal{P}}_{[\kappa]}(X,\leq)$ denotes those induced pre-ordered
subsets that are directed.

######  Definition 1.8.

Given a set $X$. The ordered families with cardinality $\leq\kappa$ in $X$ are

$\begin{gathered}\mathcal{O}_{\kappa}(X):=\\{(a_{j})_{j\in J}\textnormal{ in
}X;(J,\leq)\textnormal{ is a pre-
order},|J|\leq\kappa\\}/{\equiv}\end{gathered}$

with the equivalence relation ($\equiv_{\theta}$ denoting an order isomorphism
$\theta$):

$(a_{j})_{j\in J}\equiv(b_{j})_{j\in K}\iff J\equiv_{\theta}K\textnormal{ and
}\forall j\in J:a_{j}=b_{\theta(j)}.$

We denote $\vec{\mathcal{O}}_{\kappa}(X)$ for those ordered families with a
directed index (nets). We denote $\mathcal{O}_{\kappa}(X,\leq)$ for those
ordered families where the mapping $j\rightarrow x_{j}$ is order preserving.
We implicitly identify $\mathcal{P}_{\kappa}(X,\leq)$ with the subset of
$\mathcal{O}_{\kappa}(X,\leq)$ for which the representing mappings are order
embedding, by the injection $V\rightarrow(a_{x})_{x\in V,\leq};a_{x}=x$.

######  Notation 1.9.

Given a pre-ordered set $(X,\leq)$;
${}^{\star}\mathcal{O}(X[,\leq]):={{}^{\star}\mathcal{O}_{\kappa}(X[,\leq])}$,
where $\kappa$ is a globally defined cardinality for which our non-standard
model is saturated. We denote ${}^{\star}\mathcal{O}_{\sigma[\kappa]}(X,\leq)$
for internal ordered families that have a representant with a standard index.

######  Definition 1.10.

Given
$A=(a_{j})_{j\in{}^{\star}{J}}\in{}^{\star}\mathcal{O}_{\sigma}(X,\leq)$. The
point set; restricted point set; tails; heads; finite-indexed, infinite-
indexed and standard-indexed point sets of $A$ are
($K\subseteq{}^{\star}{J},\iota\in{}^{\star}{J})$:

$\begin{gathered}A_{\\{\\}}:=\\{a_{j}\\}_{j\in{}^{\star}{J}};~{}~{}~{}~{}A_{K}:=\\{a_{j}\\}_{j\in
K};~{}~{}~{}~{}A_{\geq\iota}:=\\{a_{j}\\}_{j\in{}^{\star}{J}_{\geq}};~{}~{}~{}~{}~{}A_{\leq\iota}:=\\{a_{j}\\}_{j\in{}^{\star}{J}_{\leq\iota}};\\\
A_{\mathfrak{f}}:=\\{a_{j}\\}_{j\in{}^{\star}{J}_{\mathfrak{f}}};~{}~{}~{}~{}~{}A_{\infty}:=\\{a_{j}\\}_{j\in{}^{\star}{J}_{\infty}};~{}~{}~{}~{}~{}A_{\sigma}:=\\{a_{j}\\}_{j\in{}^{\sigma}{J}}.\end{gathered}$

We may also combine these notations, for instance $A_{\infty,K}=A_{\infty}\cap
A_{K}$. In the case $({}^{\star}{V},\leq)\in{}^{\sigma}\mathcal{P}(X,\leq)$,
the notations are non-ambiguous; ${}^{\star}{V}_{\sigma}={}^{\sigma}{V}$ and
${}^{\star}{V}_{\\{\\}}={}^{\star}{V}$.

######  Definition 1.11.

Given a pre-ordered set $(X,\leq)$. Given two sets $A\subseteq{}^{\star}{X}$
and $B\subseteq{}^{\star}{X}$. Then $A$ is cofinal in $B$ iff for each $b\in
B$ there exists a $a\in A$ such that $a\geq b$. $A$ is coinitial in $B$ iff
for each $b\in B$ there exists $a\in A$ such that $a\leq b$. In the specific
case where $A$ is also a subset of $B$ we denote this as
$A\subseteq_{\mathrm{cof}}B$ for a cofinal subset and
$A\subseteq_{\mathrm{coi}}B$ for a coinitial subset.

######  Proposition 1.12.

Given a pre-ordered set $(X,\leq)$,
$(a_{j})_{j\in{}^{\star}{J}},(b_{j})_{j\in{}^{\star}{J^{\prime}}}\in{}^{\star}\mathcal{O}_{\sigma}{(X,\leq)}$
and internal $K\subseteq{}^{\star}{J}$,
$K^{\prime}\subseteq{}^{\star}{J^{\prime}}$. The following are equivalent:

  1. (i)

$A_{\infty,K}$ coinitial in $B_{\infty,K^{\prime}}$.

  2. (ii)

$A_{\mathfrak{f},K}$ cofinal in $B_{\mathfrak{f},K^{\prime}}$ (or if $K=J$,
$K^{\prime}=J^{\prime}$: $A_{\sigma}$ cofinal in $B_{\sigma}$).

###### Proof.

(i) $\Rightarrow$ (ii): For any $k\in{}^{\star}{J}^{\prime}_{\mathfrak{f}}\cap
K^{\prime}$,
${}^{\star}{J}_{\infty}\subseteq\\{j\in{}^{\star}{J}:\exists\iota\in
K:b_{k}\leq a_{\iota}\leq a_{j}\\}$, so that because of overspill there exists
a $\iota\in{}^{\star}{J}_{\mathfrak{f}}\cap K$ such that $a_{\iota}\geq
b_{k}$.

(ii) $\Rightarrow$ (i): Given $\nu\in{}^{\star}{J}^{\prime}_{\infty}\cap
K^{\prime}$,
${}^{\star}{J}_{\mathfrak{f}}\subseteq\\{j\in{}^{\star}{J}:\exists\iota\in
K:a_{j}\leq a_{\iota}\leq b_{\nu}\\}$, so that because of overspill there
exists a $\omega\in J$ such that $a_{\omega}\leq b_{\nu}$. ∎

######  Corrolary 1.13.

Given $(X,\leq)$ directed and internal $A,B\subseteq X$. The following are
equivalent:

  1. (i)

$A\cap{}^{\star}{X}_{\infty}$ cofinal in $B\cap{}^{\star}{X}_{\infty}$;

  2. (ii)

$A\cap{}^{\star}{X}_{\mathfrak{f}}$ coinitial in
$B\cap{}^{\star}{X}_{\mathfrak{f}}$.

If $A\subseteq B$ then
$A\cap{}^{\star}{X}_{\infty}\subseteq_{\mathrm{coi}}B\cap{}^{\star}{X}_{\infty}\iff
A\cap{}^{\star}{X}_{\mathfrak{f}}\subseteq_{\mathrm{cof}}B\cap{}^{\star}{X}_{\mathfrak{f}}$.

######  Corrolary 1.14.

Given a pre-ordered set $(X,\leq)$ and directed $V,W\subseteq X$, the
following are equivalent:

  1. (i)

$V$ cofinal in $W$.

  2. (ii)

${}^{\star}{V}_{\infty}$ coinitial in ${}^{\star}{W}_{\infty}$;

If $V\subseteq W$ this is equivalent to
${}^{\star}{V}_{\infty}\subseteq{}^{\star}{W}_{\infty}$ and
${}^{\star}{V}_{\infty}={}^{\star}{V}\cap{}^{\star}{W}_{\infty}$.

###### Proof.

For the last statement, notice that
${}^{\star}{V}\cap{}^{\star}{W}_{\infty}\subseteq{}^{\star}{V}_{\infty}$
always, hence ${}^{\star}{V}_{\infty}\subseteq{}^{\star}{W}_{\infty}$ implies
${}^{\star}{V}_{\infty}={}^{\star}{V}\cap{}^{\star}{W}_{\infty}$. Now assume
this to be the case and suppose there exists a $b\in W$ such that $\forall
a\in V:a\not\geq b$. Because of the transfer principle
${}^{\star}{V}\cap{}^{\star}{X}_{\infty}=\emptyset$, a contradiction because
${}^{\star}{V}_{\infty}$ cannot be empty. Hence $V\subseteq_{\mathrm{cof}}W$
and thus
${}^{\star}{V}_{\infty}\subseteq_{[\mathrm{coi}]}{}^{\star}{W}_{\infty}$. ∎

###  1.2 Monads and Galaxies

From now on we will work with a set called $E$. For $V\subseteq E$,
respectively $A\subseteq{}^{\star}{E}$ we denote
${V}^{\mathfrak{c}}:=E\setminus V$, respectively
${A}^{\mathfrak{c}}:={}^{\star}{E}\setminus A$. For
$\mathcal{U}\in{}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\subseteq)}$. The
directedness property is equivalent to

$\forall U_{1},\dots,U_{n}\in\mathcal{U}_{\sigma}:\exists
U^{\prime}\in\mathcal{U}_{\sigma}:U^{\prime}\subseteq U_{1}\cap\dots
U_{n}\textnormal{, resp. }U^{\prime}\supseteq U_{1}\cup\dots U_{n}.$

######  Definition 1.15.

Given
$\mathcal{U}=(U_{j})_{j\in{}^{\star}{J}}\in{}^{\star}{\mathcal{O}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively ${}^{\star}{\mathcal{O}_{\sigma}(\mathcal{P}(E),\subseteq)}$.
Then the $\cap$-closure, resp. $\cup$-closure $\vec{\mathcal{U}}\in$
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$, resp.
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\subseteq)}$ is

$\vec{\mathcal{U}}:=(V_{j})_{S\in{}^{\star}\mathcal{P}_{\mathrm{fin}}(J),\subseteq}~{}~{}~{}V_{j_{1},\dots,j_{n}}:=\bigcap_{i=1,\dots,n}U_{j_{i}},\textnormal{resp.}\bigcup_{i=1,\dots,n}U_{j_{i}}.$

Note that $\cap\vec{\mathcal{U}}_{\sigma}=\cap\mathcal{U}_{\sigma}$.

######  Definition 1.16.

Given $\mathcal{U}\in{\mathcal{O}(\mathcal{P}(E),\supseteq)}$, respectively
${\mathcal{O}(\mathcal{P}(E),\subseteq)}$. Then the downward closure
${}^{\downarrow}{\mathcal{U}}\in$ ${\mathcal{P}(\mathcal{P}(E))}$ is

${}^{\downarrow}{\mathcal{U}}:=\\{V\in\mathcal{P}(E):\exists
U\in\mathcal{U}:V\supseteq U\textnormal{, resp. }V\subseteq U\\}.$

For, $\mathcal{U}\in{{}^{\star}\mathcal{O}(\mathcal{P}(E),\supseteq)}$ the
internal downward closure ${}^{\downarrow}{\mathcal{U}}$ is defined by
transferring. Given
$\mathcal{U}=(U_{j})_{j\in{}^{\star}{J}}\in{}^{\star}{\mathcal{O}(\mathcal{P}(E),\supseteq)}$,
respectively ${}^{\star}{\mathcal{O}(\mathcal{P}(E),\subseteq)}$. Then the
external downward closure
${}^{\mathrel{\rotatebox{-90.0}{$\rightsquigarrow$}}}{\mathcal{U}}\in$
$\mathcal{P}(\mathcal{P}({}^{\star}{E}))$ is

${}^{\mathrel{\rotatebox{-90.0}{$\rightsquigarrow$}}}{\mathcal{U}}:=\\{V\in\mathcal{P}({}^{\star}{E}):U\in\mathcal{U}:V\supseteq
U\textnormal{, resp. }V\subseteq U\\}.$

######  Proposition 1.17.

(Cauchy principle) Given
$\mathcal{U}\in{}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively ${}^{\star}{\mathcal{O}_{\sigma}(\mathcal{P}(E)),\subseteq)}$.
Then

$\begin{split}\forall
A\in{}^{\star}{\mathcal{P}}(E):~{}~{}~{}&\cup\mathcal{U}_{\infty}\subseteq
A\iff\exists U\in U_{\sigma}:U\subseteq A;\\\
resp.~{}~{}&A\subseteq\cap\mathcal{U}_{\infty}\iff\exists U\in
U_{\sigma}:A\subseteq U.\end{split}$

###### Proof.

This follows from finite overspill. ∎

######  Theorem 1.18.

(Nucleus principle) Given
$\mathcal{U}\in{}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively
${}^{\star}{\mathcal{O}_{\sigma}(\mathcal{P}(E)),\supseteq)}$. Then

$\begin{split}\forall
A\in{}^{\star}{\mathcal{P}}(E):~{}~{}~{}&A\subseteq\cap\mathcal{U}_{\sigma}\iff\exists
U\in\mathcal{U}_{\infty}:A\subseteq U;\\\
resp.~{}~{}&\cup\mathcal{U}_{\sigma}\subseteq A\iff\exists
U\in\mathcal{U}_{\infty}:U\subseteq A.\end{split}$

###### Proof.

This follows from infinite overspill. ∎

######  Proposition 1.19.

Given
$\mathcal{U},\mathcal{V}\in{}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively ${}^{\star}{\mathcal{O}_{\sigma}(\mathcal{P}(E)),\subseteq)}$.
The following statements are equivalent:

  1. (i)

$\cap\mathcal{U}_{\sigma}\subseteq\cap\mathcal{V}_{\sigma}$, resp.
$\cup\mathcal{U}_{\sigma}\supseteq\cup\mathcal{V}_{\sigma}$;

  2. (ii)

$\cup\mathcal{U}_{\infty}\subseteq\cup\mathcal{V}_{\infty}$, resp.
$\cap\mathcal{U}_{\infty}\supseteq\cap\mathcal{V}_{\infty}$;

  3. (iii)

$\mathcal{U}_{\sigma}$ cofinal in $\mathcal{V}_{\sigma}$;

  4. (iv)

$\mathcal{V}_{\infty}$ coinitial in $\mathcal{U}_{\infty}$.

###### Proof.

The equivalence (iii) $\iff$ (iv) follows from proposition 1.12. The
implications (iii) $\Rightarrow$ (i) and (iv) $\Rightarrow$ (ii) are clear.
The implications (ii) $\Rightarrow$ (iii) and (i) $\Rightarrow$ (iv) follow
from the Cauchy, resp. nucleus principle. ∎

######  Proposition 1.20.

Given
$\mathcal{U}\in{}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E)),\subseteq)}$.

$\cap\mathcal{U}_{\sigma}=\cap\mathcal{U}_{\mathfrak{f}}=\cup\mathcal{U}_{\infty}\textnormal{,
resp. }\cup\mathcal{U}_{\sigma}=\cup\mathcal{U}_{f}=\cap\mathcal{U}_{\infty}.$

###### Proof.

The inclusion $\cap\mathcal{U}_{\sigma}\subseteq\cup\mathcal{U}_{\infty}$
follows from the nucleus principle;
$\cap\mathcal{U}_{\infty}\subseteq\cup\mathcal{U}_{\sigma}$ from the Cauchy
principe (each time applied to a singleton). The other inclusions are clear. ∎

######  Notation 1.21.

Proposition 1.19 defines a pre-order on
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E),\supseteq)}$,
respectively
${}^{\star}{\vec{\mathcal{O}}_{\sigma}(\mathcal{P}(E)),\subseteq)}$, denoted
as $\succeq$. The equivalence relationship defined by $\preceq\land\succeq$ is
denoted as $\sim$.

######  Definition 1.22.

The $({}^{\filleddiamond})$monads and $({}^{\filleddiamond})$galaxies [of
cofinality $\leq\kappa$] in $E$ are

$\begin{split}\mathfrak{M}_{[\kappa]}({}^{\star}{E}):=\vec{\mathcal{O}}_{[\kappa]}(\mathcal{P}(E),\supseteq)/{\sim};~{}~{}~{}&~{}~{}~{}~{}{}^{\filleddiamond}\mathfrak{M}_{[\kappa]}({}^{\star}{E}):={}^{\star}{\vec{\mathcal{O}}_{\sigma[\kappa]}(\mathcal{P}(E),\supseteq)}/{\sim};\\\
\mathfrak{G}_{[\kappa]}({}^{\star}{E}):=\vec{\mathcal{O}}_{[\kappa]}(\mathcal{P}(E),\subseteq)/{\sim};~{}~{}~{}&~{}~{}~{}~{}{}^{\filleddiamond}\mathfrak{G}_{[\kappa]}({}^{\star}{E}):={}^{\star}{\vec{\mathcal{O}}_{\sigma[\kappa]}(\mathcal{P}(E),\subseteq)}/{\sim};\end{split}$

We identify the left-side sets as subsets of the right-side sets. A
representant of a $[{}^{\filleddiamond}]$monad or
$[{}^{\filleddiamond}]$galaxy is called a $[{}^{\filleddiamond}]$base. Since
subsets are (partially) ordered, we may assume w.l.o.g. that the index of a
$[{}^{\filleddiamond}]$base is ordered when this is convenient. A
$[{}^{\filleddiamond}]$subbase is a
$\mathcal{U}\in[{}^{\star}]\mathcal{O}_{[\sigma]}(\mathcal{P}(E),\subseteq/\supseteq)$
such that $\vec{\mathcal{U}}$ is a base. The pre-order $\succeq$ induces an
order on the quotients, where the left (resp. right) side of the order
relation is called a finer (resp. coarser)
$[{}^{\filleddiamond}]$monad/galaxy. Consider the mappings:

$\begin{split}{}^{\filleddiamond}\mathfrak{M}({{}^{\star}{E}})\rightarrow\mathcal{P}({}^{\star}{E}):\mathcal{U}\rightarrow\cup\mathcal{U}_{\infty}=\cap\mathcal{U}_{\sigma};\\\
{}^{\filleddiamond}\mathfrak{G}({}^{\star}{E})\rightarrow\mathcal{P}({}^{\star}{E}):\mathcal{U}\rightarrow\cap\mathcal{U}_{\infty}=\cup\mathcal{U}_{\sigma}.\end{split}$

Because of proposition 1.19 these are in fact injections, and we will
consistently use them as implicit identifications. In this identification the
‘is finer than’ order corresponds with $\subseteq$ for monads and $\supseteq$
for galaxies. Then we have:

$\begin{gathered}\mathfrak{M}_{1}({}^{\star}{E})(=\mathfrak{M}_{2}({}^{\star}{E})=\dots)=\mathfrak{G}_{1}({}^{\star}{E})(=\mathfrak{G}_{2}({}^{\star}{E})=\dots)={}^{\sigma}{\mathcal{P}}(E);\\\
{}^{\filleddiamond}\mathfrak{M}_{1}({}^{\star}{E})(={}^{\filleddiamond}\mathfrak{M}_{2}({}^{\star}{E})=\dots)={}^{\filleddiamond}\mathfrak{G}_{1}({}^{\star}{E})(={}^{\filleddiamond}\mathfrak{G}_{2}({}^{\star}{E})=\dots)={}^{\star}{\mathcal{P}}(E).\end{gathered}$

However, all other monads and galaxies are identified as external sets. We
must thus be careful: $\mathfrak{M}({E})$ and $\mathfrak{G}({E})$ may be
regarded as standard sets, but the implicit identification of their elements
as sets is external.

######  Proposition 1.23.

$\mathfrak{M}({}^{\star}{E})\cap{}^{\filleddiamond}\mathfrak{M}_{\kappa}({}^{\star}{E})=\mathfrak{M}_{\kappa}({}^{\star}{E})$;
$\mathfrak{G}({}^{\star}{E})\cap{}^{\filleddiamond}\mathfrak{G}_{\kappa}({}^{\star}{E})=\mathfrak{G}_{\kappa}({}^{\star}{E})$.

###### Proof.

Given
$\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})\cap{}^{\filleddiamond}\mathfrak{M}_{\kappa}({}^{\star}{E})$.
Suppose $\mathcal{U}$ is a standard base for $\mathfrak{m}$ and
$(U_{j})_{j\in{}^{\star}{J}}$ is a ${}^{\filleddiamond}$base with
$|J|\leq\kappa$. Then for any $j\in{}^{\sigma}{J}$, let $V_{j}$ be a set in
$\mathcal{U}$ such that ${}^{\star}{V_{j}}\subseteq U_{j}$. Then
$(V_{j})_{j\in J}$ is a standard base for $\mathfrak{m}$. For galaxies the
proof is analogue. ∎

######  Example 1.24.

Standard copies of sets are galaxies with finite sets as a base. In
particular, ${}^{\sigma}{E}$ is the galaxy that has all finite sets of $E$ as
its base. For any pre-ordered set $X$, ${}^{\star}{X}_{\mathfrak{f}}$ is a
galaxy and ${}^{\star}{X}_{\infty}$ is a monad.

######  Definition 1.25.

We call $\emptyset$, resp. ${}^{\star}{E}$ the trivial monad, resp. galaxy. A
${}^{\filleddiamond}$monad or ${}^{\filleddiamond}$galaxy is proper if it is
not trivial. The cauchy principle implies that a monad., resp. galaxy with a
given (sub)base is proper iff the (sub)base has the (non-empty) finite
intersection, resp. (non-covering) finite union property.

######  Definition 1.26.

Given spaces $E_{1},\dots,E_{n},F$ and an internal mapping
$\psi:{}^{\star}\mathcal{P}(E_{1})\times\dots\times{}^{\star}\mathcal{P}(E_{n})\rightarrow{}^{\star}\mathcal{P}(F)$
that is either order preserving in each argument or order reversing in each
argument. We define, given bases $(U_{j_{1}})_{j_{1}\in{}^{\star}{J_{1}}}$ in
$E_{1},\dots,(U_{j_{n}})_{j_{n}\in{}^{\star}{J_{n}}}$ in $E_{n}$ (using the
product order):

$\begin{gathered}{}^{\dagger}\psi((U_{j_{1}})_{j_{1}\in{}^{\star}{J_{1}}},\dots,(U_{j_{n}})_{j_{n}\in{}^{\star}{J_{n}}}):=(\psi(U_{j_{1}},\dots
U_{j_{n}}))_{(j_{1},\dots,j_{n})\in{}^{\star}{(J_{1}\times\dots\times
J_{n})}}.\end{gathered}$

Now note that order preservation or reversion and proposition 1.19 imply that
given ${}^{\filleddiamond}$bases
$\mathcal{U}_{1},\dots\mathcal{U}_{n},\mathcal{V}_{1},\dots\mathcal{V}_{n}$ we
have $\mathcal{U}_{1}\sim
V_{1},\dots,\mathcal{U}_{n}\sim\mathcal{V}_{n}\Rightarrow{}^{\dagger}\psi(\mathcal{U}_{1},\dots,\mathcal{U}_{n})\sim{}^{\dagger}\psi(\mathcal{V}_{1},\dots,\mathcal{V}_{n}$).
Hence we may interpret the definition above of ${}^{\dagger}\psi$ as a mapping
of monads and galaxies that is independent of the representant. If the mapping
is order preserving, ${}^{\filleddiamond}{\mathfrak{M}}_{\kappa}(E)^{n}$ is
mapped to ${}^{\filleddiamond}{\mathfrak{M}}_{\kappa}(E)$ and
${}^{\filleddiamond}{\mathfrak{G}}_{\kappa}(E)^{n}$ is mapped to
${}^{\filleddiamond}{\mathfrak{G}}_{\kappa}(E)$. If the mapping is order
reversing, ${}^{\filleddiamond}{\mathfrak{M}}_{\kappa}(E)^{n}$ is mapped to
${}^{\filleddiamond}{\mathfrak{G}}_{\kappa}(E)$ and
${}^{\filleddiamond}{\mathfrak{G}}_{\kappa}(E)^{n}$ is mapped to
${}^{\filleddiamond}{\mathfrak{M}}_{\kappa}(E)$. If $\phi$ is a standard map,
then standard monads and galaxies are mapped onto standard monads and
galaxies.

######  Proposition 1.27.

Given spaces $E_{1},\dots,E_{n},F$ and a mapping
$\psi:\mathcal{P}({}^{\star}{E_{1}})\times\dots\times\mathcal{P}({}^{\star}{E_{n}})\rightarrow\mathcal{P}({}^{\star}{F})$
that is order preserving or reversing, such that $\psi$ restricted to
${}^{\star}(\mathcal{P}(E_{1})\times\dots\times\mathcal{P}(E_{n}))$ is
internal. For any
$\mathfrak{m}_{1}\in{}^{\filleddiamond}{\mathfrak{M}}(E_{1}),\dots,\mathfrak{m}_{n}\in{}^{\filleddiamond}{\mathfrak{M}}(E_{n})$
or
$\mathfrak{g}_{1}\in{}^{\filleddiamond}{\mathfrak{G}}(E_{1}),\dots,\mathfrak{g}_{n}\in{}^{\filleddiamond}{\mathfrak{G}}(E_{n})$,
we have (applying $\dagger$ to the internal restriction):

$\psi(\mathfrak{m}_{1},\dots\mathfrak{m}_{n})={}^{\dagger}\psi(\mathfrak{m}_{1},\dots\mathfrak{m}_{n})~{}~{}~{}~{}~{}~{}\psi(\mathfrak{g}_{1},\dots,\mathfrak{g}_{n})={}^{\dagger}\psi(\mathfrak{g}_{1},\dots,\mathfrak{g}_{n}).$

###### Proof.

Assume a single argument mapping. Given monad $\mathfrak{m}$ with base
$\mathcal{U}$ (the proof for a galaxy is analogue). If $\psi$ is order-
preserving we have

$\begin{split}{}^{\dagger}\psi(\mathfrak{m})&=\bigcap\psi(\mathcal{U}_{\sigma})\supseteq\psi(\bigcap\mathcal{U}_{\sigma})=\psi(\mathfrak{m})=\psi(\bigcup\mathcal{U}_{\infty})\supseteq\bigcup\psi(\mathcal{U}_{\infty})={}^{\dagger}\psi(\mathfrak{m}).\end{split}$

while if $\psi$ is order-reversing

$\begin{split}{}^{\dagger}\psi(\mathfrak{m})&=\bigcup\psi(\mathcal{U}_{\sigma})\supseteq\psi(\bigcup\mathcal{U}_{\sigma})=\psi(\mathfrak{m})=\psi(\bigcup\mathcal{U}_{\infty})\supseteq\bigcap\psi(\mathcal{U}_{\infty})={}^{\dagger}\psi(\mathfrak{m}).\end{split}$

For multiple arguments, first fixate all but one argument on an arbitrary
internal set; then use recursion for multiple monads or galaxies. ∎

######  Corrolary 1.28.

Given spaces $E_{1},\dots,E_{n},F$ and an internal mapping
$\psi:{}^{\star}{E_{1}}\times\dots\times{}^{\star}{E_{n}}\rightarrow{}^{\star}{F}$.
For any
$\mathfrak{m}_{1}\in{}^{\filleddiamond}{\mathfrak{M}}(E),\dots,\mathfrak{m}_{n}\in{}^{\filleddiamond}{\mathfrak{M}}(E_{n})$
or
$\mathfrak{g}_{1}\in{}^{\filleddiamond}{\mathfrak{G}}(E_{1}),\dots,\mathfrak{g}_{n}\in{}^{\filleddiamond}{\mathfrak{G}}(E_{n})$
we have

$\psi(\mathfrak{m}_{1},\dots\mathfrak{m}_{n})={}^{\dagger}\psi(\mathfrak{m}_{1},\dots\mathfrak{m}_{n})~{}~{}~{}~{}~{}~{}\psi(\mathfrak{g}_{1},\dots,\mathfrak{g}_{n})={}^{\dagger}\psi(\mathfrak{g}_{1},\dots,\mathfrak{g}_{n})$

where we have interpreted $\psi$ as a
$\mathcal{P}({}^{\star}{E_{1}})\times\dots\times\mathcal{P}({}^{\star}{E_{n}})\rightarrow\mathcal{P}({}^{\star}{G})$
mapping that is internal for internal sets and order-preserving.

######  Corrolary 1.29.

${}^{\filleddiamond}{\mathfrak{M}}(E)$ and
${}^{\filleddiamond}{\mathfrak{G}}(E)$ are each closed for finite union and
intersection.111In fact, one can prove that $\mathfrak{M}({}^{\star}{E})$,
resp. $\mathfrak{G}({}^{\star}{E})$ are closed for infinite intersections,
resp. unions; but this is not the case for unions of monads and intersections
of galaxies. The complement of a ${}^{\filleddiamond}$monad, respectively
${}^{\filleddiamond}$galaxy is a ${}^{\filleddiamond}$galaxy, respectively
${}^{\filleddiamond}$monad.

###### Proof.

Consider the mappings ($A,B\subseteq{}^{\star}{E}$): $(A,B)\rightarrow A\cap
B$, $(A,B)\rightarrow A\cup B$ and $A\rightarrow{A}^{\mathfrak{c}}$. The first
two are order preserving in each argument; the latter is order reversing and
each is internal for internal sets; hence their $\dagger$-extensions coincides
with the general set operators. ∎

######  Definition 1.30.

Given an internal order-preserving mapping
$\psi:{}^{\star}\mathcal{P}(E)\rightarrow{}^{\star}\mathcal{P}(E)$. Then the
set property as defined by the set

$\mathcal{Q}_{\psi}=\\{V\in{}^{\star}\mathcal{P}(E):\psi(V)=V\\}.$

is extendable to a $\dagger$-property as follows:

${}^{\dagger}\mathcal{Q}_{\psi}=\\{\mathfrak{a}\in{}^{\filleddiamond}{\mathfrak{M}}(E)\cup{}^{\filleddiamond}{\mathfrak{G}}(E):{}^{\dagger}\psi(\mathfrak{a})=\mathfrak{a}\\}.$

This is equivalent to $\mathfrak{a}$ having a ${}^{\filleddiamond}$base in
$\mathcal{A}_{\psi}$. If $\psi$ is defined on external sets, then the property
is also defined on external sets by

${}^{\ostar}{\mathcal{Q}}_{\psi}=\\{V\in\mathcal{P}({}^{\star}{E}):\psi(V)=V\\}.$

in which case ${}^{\dagger}\mathcal{A}_{\psi}$ is precisely the restriction of
this set to monads and galaxies. In this case we may leave off the $\dagger$
symbol.

######  Lemma 1.31.

Given an order-preserving map
$\psi:\mathcal{P}({}^{\star}{E})\rightarrow\mathcal{P}({}^{\star}{E})$ such
that $\psi(A)\supseteq A$ (respectively $\psi(A)\subseteq A$) for any
$A\subseteq{}^{\star}{E}$ and associated set property
${}^{\ostar}{\mathcal{Q}}_{\psi}$. Then ${}^{\ostar}{\mathcal{Q}}_{\psi}$ is
closed for intersection (respectively union).

###### Proof.

Given a family of sets $\\{U_{j}\\}_{j\in
J}\subseteq{}^{\ostar}{\mathcal{Q}}_{\psi}$ and $U$ its intersection. Then

$\psi(U)=\psi\big{(}\bigcap_{j\in J}U_{j}\big{)}\subseteq\bigcap_{j\in
J}\psi(U_{j})=\bigcap_{j\in J}U_{j}=U.$

If the assumption was $\psi(U)\supseteq U$, the result follows. The other case
is analogue. ∎

######  Remark 1.32.

The order-preserving extension of an internal mapping is unique on monads and
galaxies, however this is not the case for other external sets. In fact, we
will later discover an example where two order-preserving mappings coincide
entirely on standard sets, internal sets, monads and galaxies, but not on all
external sets.

######  Proposition 1.33.

(Combined Cauchy Principle) Given
$\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$ and
$\mathfrak{g}\in{}^{\filleddiamond}\mathfrak{G}({}^{\star}{E})$. Then

$\mathfrak{m}\subseteq\mathfrak{g}\iff\exists
A\subseteq{}^{\star}{\mathcal{P}(E)}:\mathfrak{m}\subseteq
A\subseteq\mathfrak{g}.$

###### Proof.

The inclusion is equivalent to $\mathfrak{m}\cap{\mathfrak{g}}^{\mathfrak{c}}$
being a trivial monad; i.e. each base of this monad containing the empty set.
Given bases $\mathcal{U}$ for $\mathfrak{m}$ and $\mathcal{V}$ for
$\mathfrak{g}$, the statement is also equivalent with the existence of a
$U\in\mathcal{U}$ and $V\in\mathcal{V}$ such that $U\subseteq V$. ∎

Given
$\mathscr{A}=(a_{j})_{j\in{}^{\star}{J}}\in{}^{\star}\vec{\mathcal{O}}(E)$
with $|J|\leq\kappa$. Then $(\mathscr{A}_{\geq j})_{j\in{}^{\star}{J}}$ is a
base (of tails), hence
$\mathscr{A}_{\infty}\in{}^{\filleddiamond}\mathfrak{M}_{\kappa}({}^{\star}{E})$.

######  Definition 1.34.

The $\kappa$-thin $[{}^{\filleddiamond}]$monads are the infinite point sets of
$[{}^{\star}]$nets with a standard index of cardinality $\leq\kappa$. We
denote them as
$[{}^{\filleddiamond}]\mathfrak{M}^{\kappa}({}^{\star}{E})\subseteq[{}^{\filleddiamond}]\mathfrak{M}_{\kappa}({}^{\star}{E})$.
In particular, we call $\aleph_{0}$-thin $[{}^{\filleddiamond}]$monads
sequential, they are associated with $[{}^{\star}]$sequences.

The following propositions show that nets are fine representatives of monads
and sequences are fine representatives of monads with countable cofinality.

######  Proposition 1.35.

Given $\mathfrak{m}\in\mathfrak{M}_{\kappa}({}^{\star}{E})$, there exists a
cardinality $\lambda\geq\kappa$ such that
$\mathfrak{m}\in\mathfrak{M}^{\lambda}({}^{\star}{E})$.

###### Proof.

Let $\mathcal{U}$ be a base for $\mathfrak{m}$. Let
$J=\\{(U,e):U\in\mathcal{U},e\in U\\}$ with
$(U,e)\leq(U^{\prime},e^{\prime})\iff U\leq U^{\prime}$ and consider the net
$A=(a_{j})_{j\in{}^{\star}{J}};a_{(U,e)}:=e$. Then $A_{\infty}=\mathfrak{m}$.
∎

######  Proposition 1.36.

Given
$\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}_{\kappa}({}^{\star}{E})$,
there exists an
$\mathfrak{m}^{\prime}\in{}^{\filleddiamond}\mathfrak{M}^{\kappa}(\mathfrak{m})$.

###### Proof.

Let $\mathcal{U}$ be a ${}^{\filleddiamond}$base for $\mathfrak{m}$. Consider
a net $\mathscr{A}=(a_{U})_{U\in\mathcal{U}}$ where $a_{U}\subseteq U$ is
arbitrarily chosen for each $U$. Then
$\mathscr{A}_{\infty}\subseteq\mathfrak{m}$. ∎

######  Corrolary 1.37.

Any monad of countable cofinality contains a sequential finer monad.

###  1.3 Properties of standard monads and galaxies

######  Definition 1.38.

A ring of sets $\Sigma$ over $E$ is a family of sets $\Sigma$ in $E$ such that
$\emptyset\in\Sigma,E\in\Sigma$, closed for finite intersection and union. The
discrete ring is the ring of all subsets of $E$.

######  Definition 1.39.

Given a ring $\Sigma$ and a set $A\in{}^{\star}{E}$. Then the filter and monad
over $\Sigma$ generated by $A$ are

$\begin{split}\mathrm{fil_{\Sigma}~{}}{A}&:=\\{U\in\Sigma,A\subseteq{}^{\star}{U}\\};\\\
\boldsymbol{\mu}_{\Sigma}(A)&:=\bigcap{}^{\sigma}(\mathrm{fil_{\Sigma}~{}}{A})=\bigcup{}^{\star}(\mathrm{fil_{\Sigma}~{}}{A})_{\infty}.\end{split}$

The ideal and galaxy over $\Sigma$ generated by $A$ are

$\begin{split}\mathrm{idl_{\Sigma}~{}}{A}&:=\\{U\in\Sigma,{}^{\star}{U}\subseteq
A\\};\\\
\boldsymbol{\Gamma}_{\Sigma}(A)&:=\bigcup{}^{\sigma}(\mathrm{fil_{\Sigma}~{}}{A})=\bigcap{}^{\star}(\mathrm{fil_{\Sigma}~{}}{A})_{\infty}.\end{split}$

In the absence of an explicit $\Sigma$, the discrete ring is implied. In the
case of a singleton, we may leave off the accolades.

Now notice that for $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ with base
$\mathcal{U}$, we have
$\mathrm{fil~{}}\mathfrak{m}={}^{\downarrow}{\mathcal{U}}$ and
$\boldsymbol{\mu}(\mathfrak{m})=\mathfrak{m}$, hence
$\mathfrak{m}\rightarrow\mathrm{fil~{}}\mathfrak{m}$ is a standard bijection
into the set of filters on $E$. The filter is the greatest base (for the
$\subseteq$ order) of $\mathfrak{m}$; the bases of $\mathfrak{m}$ are
precisely the cofinal subsets of its filter. All properties of (standard)
monads can be interpreted as properties of filters, relating them to standard
analysis. The case for galaxies and ideals is analogue.

######  Definition 1.40.

A non-trivial monad that does not contain any strictly finer non-trivial
monads is an ultramonad. A non-trivial galaxy that is not contained in any
strictly finer non-trivial galaxies is an ultragalaxy.

######  Proposition 1.41.

The ultramonads of $E$ are precisely
$\\{\boldsymbol{\mu}(e):e\in{}^{\star}{E}\\}$.

###### Proof.

We may construct a base for a strictly finer non-trivial monad iff there
exists a $U\subseteq E$ such that $\mathrm{fil~{}}\mathfrak{m}$ does not
contain either $U$ or its complement. First consider
$\mathfrak{m}=\boldsymbol{\mu}(e)$ for $e\in{}^{\star}{E}$, then
$U\in\mathrm{fil~{}}\mathfrak{m}$ iff $e\in{}^{\star}{U}$ but otherwise
$e\in{}^{\star}{{U}^{\mathfrak{c}}}$. For the opposite implication, suppose
$\mathfrak{m}$ contains points $e,e^{\prime}$ such that
$e^{\prime}\notin\boldsymbol{\mu}(e)$, then $\boldsymbol{\mu}(e)$ is a
strictly finer non-trivial monad of $\mathfrak{m}$. ∎

Similarly, the ultragalaxies are precisely
$\\{\boldsymbol{\Gamma}({e}^{\mathfrak{c}})={\boldsymbol{\mu}(e)}^{\mathfrak{c}}:e\in{}^{\star}{E}\\}$.

######  Definition 1.42.

The kernel of a monad $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ is

$\ker\mathfrak{m}:=\bigcap_{U\in\mathrm{fil~{}}{\mathfrak{m}}}U=\mathfrak{m}\cap{}^{\sigma}{E}.$

Note that
${}^{\star}\ker\mathfrak{m}=\cap{}^{\star}(\mathrm{fil~{}}\mathfrak{m})_{\infty}\subseteq\mathfrak{m}$.
A monad is free if $\ker\mathfrak{m}=\emptyset$. A monad is principal if
${}^{\star}\ker\mathfrak{m}=\mathfrak{m}$ (which is equivalent to finite
cofinality). An ultramonad $\boldsymbol{\mu}(e)$ is principal iff $e$ is
standard and free otherwise.

######  Definition 1.43.

A galaxy $\mathfrak{g}\in\mathfrak{M}({}^{\star}{E})$ is covering if
${}^{\sigma}{E}\subseteq\mathfrak{g}$.

Hence free monads and covering galaxies are each others complements.

######  Definition 1.44.

Given a ring $\Sigma$. A set $A\subseteq{}^{\star}{E}$ is $\Sigma$-chromatic
if $\forall a\in A:\boldsymbol{\mu}_{\Sigma}(a)\subseteq A$. Naturally, for
any $U\in\Sigma$, ${}^{\star}{U}$ is $\Sigma$-chromatic. Again the absence of
an explicit $\Sigma$ refers to the discrete ring so that a set is chromatic if
it is a union of monads.

The following is a generalization of a well-known result (Stroyan and
Luxemburg (1976), 8.2.2, p. 199):

######  Proposition 1.45.

Given a ring $\Sigma$ and
$\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$. Then

$\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})=\bigcup_{e\in\mathfrak{m}}\boldsymbol{\mu}_{\Sigma}(e)$

###### Proof.

The inclusion
$\bigcup_{e\in\mathfrak{m}}\boldsymbol{\mu}_{\Sigma}(e)\subseteq\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})$
is clear, assume then that it is strict. Take
$e\in\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})\setminus\mathfrak{m}$. Let
$\mathfrak{A}:=\\{\boldsymbol{\mu}_{\Sigma}(e^{\prime}):e^{\prime}\in\mathfrak{m}\\}\subseteq\mathfrak{M}({}^{\star}{E})$
and note that for any $\mathfrak{a}\in\mathfrak{A}$,
$\mathfrak{a}\subseteq{\\{e\\}}^{\mathfrak{c}}$. Using the Cauchy principle,
consider the galaxy $\mathfrak{g}_{e}$ with base
$\mathcal{U}=\\{U\in\Sigma:\exists\mathfrak{a}\in\mathfrak{A}:\mathfrak{a}\subseteq{}^{\star}{U};e\notin{}^{\star}{U}\\}$.
Since we have $\mathfrak{m}\subseteq\mathfrak{g}_{e}$, the combined Cauchy
principle gives us that $\mathfrak{m}\subseteq{}^{\star}{U}$ with
$U\in\mathcal{U}\subseteq\Sigma$ and $e\not\in{}^{\star}{U}$. This is a
contradiction because $e\in\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})$. Hence we
can conclude that $\mathfrak{m}=\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})$.

∎

######  Corrolary 1.46.

Any $\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$ or
$\mathfrak{g}\in{}^{\filleddiamond}\mathfrak{G}({}^{\star}{E})$ is
$\Sigma$-chromatic iff it is standard and has a base in $\Sigma$.

###### Proof.

If $\mathfrak{m}$ is standard with a base in $\Sigma$, then clearly
$\mathfrak{m}=\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})$. If otherwise
$\mathfrak{m}$ is $\Sigma$-chromatic then
$\mathfrak{m}=\bigcup_{e\in\mathfrak{m}}\boldsymbol{\mu}_{\Sigma}(e)=\boldsymbol{\mu}_{\Sigma}(\mathfrak{m})$.
If $\mathfrak{g}$ is a standard galaxy with a base in $\Sigma$, it is a union
of $\Sigma$-chromatic sets and therefore $\Sigma$-chromatic. Assume otherwise
that $\mathfrak{g}$ is $\Sigma$-chromatic. Since the complement of a chromatic
set is chromatic, the first part of the proof implies that $\mathfrak{g}$ is
standard. Take any $U\in\mathrm{idl~{}}\mathfrak{g}$, then
$\boldsymbol{\mu}_{\Sigma}({}^{\star}{U})=\bigcup_{e\in{}^{\star}{U}}\boldsymbol{\mu}_{\Sigma}(e)\subseteq\mathfrak{g}$;
hence because of the combined Cauchy principle, there exists a
$V\in\Sigma\cap\mathrm{idl~{}}\mathfrak{g}$ such that $U\subseteq V$. ∎

######  Corrolary 1.47.

Given a ring $\Sigma$ and an internal set $A$. Then $A$ is $\Sigma$-chromatic
iff $A\in{}^{\sigma}{\Sigma}$. In particular, $A$ is chromatic iff it is
standard.

######  Definition 1.48.

Given a ring $\Sigma$ and $A,B\subseteq{}^{\star}{E}$. Then $A$ is
$\Sigma$-distinguishable from $B$ iff
$B\cap\boldsymbol{\mu}_{\Sigma}(A)=\emptyset$ (equivalently due to the Cauchy
principle, there exists a $V\in\Sigma$ such that $A\subseteq{}^{\star}{V}$ and
$B\subseteq{}^{\star}{{V}^{\mathfrak{c}}}$). Then $A$ and $B$ are
$\Sigma$-separated iff
$\boldsymbol{\mu}_{\Sigma}(A)\cap\boldsymbol{\mu}_{\Sigma}(B)=\emptyset$
(equivalently due to the Cauchy principle, there exists disjunctive
$V,W\in\Sigma$ with met $A\subseteq V$ and $B\subseteq W$). We apply these
terms also to points $e,e^{\prime}\in{}^{\star}{E}$ (as singletons). Note that
a pair being mutually distinguishable is strictly weaker than being separated.

######  Notation 1.49.

Given a ring $\Sigma$, then the complements of sets in $\Sigma$ are a ring as
well, denoted as $\overline{\Sigma}$. The discrete ring is of course equal to
its complement.

######  Lemma 1.50.

Given a ring $\Sigma$, and $e,e^{\prime}\in{}^{\star}{E}$:
$e\in\boldsymbol{\mu}_{\Sigma}(e^{\prime})\iff
e^{\prime}\in\boldsymbol{\mu}_{\overline{\Sigma}}(e)$.

###### Proof.

Suppose $e\notin\boldsymbol{\mu}_{\overline{\Sigma}}(e^{\prime})$, then there
exists a $U\in\overline{\Sigma}$ with $e\in U$ and $e^{\prime}\notin U$. Then
${U}^{\mathfrak{c}}\in\Sigma$, hence
$e^{\prime}\notin\boldsymbol{\mu}_{\Sigma}(e^{\prime})$. The opposite
implication is identical. ∎

######  Lemma 1.51.

Given a ring $\Sigma$. Then $A\subseteq{}^{\star}{E}$ is $\Sigma$-chromatic
iff ${A}^{\mathfrak{c}}$ is $\overline{\Sigma}$-chromatic.

###### Proof.

Given $e\in{A}^{\mathfrak{c}}$, then
$e\notin\boldsymbol{\mu}_{\Sigma}(e^{\prime})$, then because of lemma 1.50,
$e^{\prime}\notin\boldsymbol{\mu}_{\overline{\Sigma}}(e)$ for any
$e^{\prime}\in A$, hence
$\boldsymbol{\mu}_{\overline{\Sigma}}(e)\subseteq{A}^{\mathfrak{c}}$. The
opposite implication is identical. ∎

######  Proposition 1.52.

Given a ring $\Sigma$. Then $A\subseteq{}^{\star}{E}$ is $\Sigma$-chromatic
iff for all
$e\in{A}^{\mathfrak{c}}:A\subseteq\boldsymbol{\Gamma}_{\Sigma}({e}^{\mathfrak{c}})$.

###### Proof.

Because of lemma 1.51, $A\subseteq{}^{\star}{E}$ is $\Sigma$-chromatic iff

${A}^{\mathfrak{c}}=\bigcup_{e\in
A}\boldsymbol{\mu}_{\overline{\Sigma}}(e)\iff A={\big{(}\bigcup_{e\in
A}\boldsymbol{\mu}_{\overline{\Sigma}}(e)\big{)}}^{\mathfrak{c}}=\bigcap{\boldsymbol{\mu}_{\overline{\Sigma}}(e)}^{\mathfrak{c}}=\bigcap\boldsymbol{\Gamma}_{\Sigma}({e}^{\mathfrak{c}}).$

∎

In particular, a set is chromatic iff it is exactly equal to the union of all
standard monads it contains (and not larger) iff it it is exactly equal to the
intersection of all standard galaxies it contains (and not smaller).

######  Lemma 1.53.

(Chromatic inclusion) Given two sets $A,B\subseteq{}^{\star}{E}$ where $A$ is
chromatic. Then

$A\subseteq B\iff\mathfrak{M}(A)\subseteq\mathfrak{M}(B).$

###### Proof.

The $\Rightarrow$ implication is trivial. For $\Leftarrow$, take $e\in A$.
Since $A$ is chromatic $\boldsymbol{\mu}(e)\subseteq A$. Then
$\boldsymbol{\mu}(e)\in\mathfrak{M}(A)\subseteq\mathfrak{M}(B)$, hence,
$\boldsymbol{\mu}(e)\subseteq B$. ∎

######  Proposition 1.54.

Given $\mathfrak{m}\in\mathfrak{M}^{\aleph_{0}}({}^{\star}{E})$,
$Q\subseteq{}^{\star}{E}$ such that ${}^{\sigma}{E}\subseteq Q$. Then
$\mathfrak{m}\subseteq Q\iff\mathfrak{m}\subseteq\boldsymbol{\Gamma}(Q)$.

###### Proof.

Let $\mathscr{A}=(a_{n})_{n\in\mathbb{N}}$ be a sequence such that
$\mathfrak{m}={}^{\star}{\mathscr{A}}_{\infty}$. Assume $\mathfrak{m}\subseteq
Q$ (the other implication is trivial). Since
${}^{\star}{\mathscr{A}}_{\sigma}={}^{\sigma}\\{a_{n}\\}_{n\in\mathbb{N}}\subseteq{}^{\sigma}{E}\subseteq
Q$,
${}^{\star}{\mathscr{A}}_{\\{\\}}={}^{\star}{\mathscr{A}}_{\sigma}\cup\mathfrak{m}\subseteq
Q$. But then ${}^{\star}{\mathscr{A}}_{\\{\\}}\subseteq\boldsymbol{\Gamma}(Q)$
hence $\mathfrak{m}\subseteq\boldsymbol{\Gamma}(Q)$ ∎

######  Corrolary 1.55.

Given $\mathfrak{m}\in\mathfrak{M}_{\aleph_{0}}(E)$, $Q\subseteq{}^{\star}{E}$
such that ${}^{\sigma}{E}\subseteq Q$. If $\mathfrak{m}\subseteq Q$, then
$\mathfrak{m}\cap\boldsymbol{\Gamma}(A)\neq\emptyset$.

###### Proof.

Combine with corollary 1.37. ∎

###  1.4 I.o.G.-properties

Given a directed set $J$ and a family $(\mathfrak{g}_{j})_{j\in J}$ in
$\mathfrak{G}({}^{\star}{E})$ that is order-preserving for $\supseteq$, each
having a base $\mathcal{U}_{j}$. Define:

$\mathfrak{h}:=\bigcap_{j\in{}^{\sigma}{J}}\mathfrak{g}_{j};~{}~{}~{}\mathcal{I}:=\mathrm{idl~{}}\mathfrak{h};~{}~{}~{}\mathfrak{g}:=\boldsymbol{\Gamma}(\mathfrak{h})=\bigcup{}^{\sigma}\big{(}\bigcap_{j\in
J}\mathcal{U}_{j}\big{)}.$

We call $\mathfrak{h}$ an Intersection of Galaxies (I.o.G). Now it is clear
that for a standard set $V\subseteq E$, we have a single set property
characterized by

$V\in\mathcal{I}\iff{}^{\star}{V}\subseteq\mathfrak{h}\iff{}^{\star}{V}\subseteq\mathfrak{g}\iff
V\in\bigcap_{j\in J}\mathrm{idl~{}}\mathfrak{g}_{j}.$

We will consider three ways of extending this property to all (including
external) subsets of $E$. The $\star$-property is naturally defined by
${}^{\ostar}\mathcal{I}:={}^{\mathrel{\rotatebox{-90.0}{$\rightsquigarrow$}}}{{}^{\star}}\mathcal{I}$.
The $\mathfrak{c}$-extension (chromatic property) is
${}^{\mathfrak{c}}\mathcal{I}:=\mathcal{P}(\mathfrak{h})$. Finally, the
$\mathfrak{s}$-extension (standard property) is:

${}^{\mathfrak{s}}\mathcal{I}:=\\{A\subseteq{}^{\star}{E}:\forall
j\in{}^{\sigma}{J}:A\in{}^{\mathrel{\rotatebox{-90.0}{$\rightsquigarrow$}}}{{}^{\star}}\mathcal{U}_{j}\\}.$

Each of them is (externally) downward closed.

######  Proposition 1.56.

Given $\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({E})$ with
${}^{\filleddiamond}$base $\mathcal{V}$, we have

$\mathfrak{m}\in{}^{\mathfrak{c}}\mathcal{I}~{}(\mathfrak{m}\subseteq\mathfrak{h})\iff\mathfrak{m}\in{}^{\mathfrak{s}}\mathcal{I}\iff\forall
j\in J:\exists V\in\mathcal{V}_{\sigma},U\in{}^{\sigma}\mathcal{I}:V\subseteq
U.$

###### Proof.

The first equivalence follows from fixating $j\in{}^{\sigma}{J}$ and applying
the combined Cauchy principle on $\mathfrak{m}\subseteq\mathfrak{g}_{j}$. The
second equivalence follows from fixating $j\in{}^{\sigma}{J}$ and applying the
Cauchy principle on $\mathfrak{m}\subseteq U$ for a certain
$U\in{}^{\star}\mathcal{U}_{j}$. ∎

######  Corrolary 1.57.

Given any set $A\subseteq{}^{\star}{E}$, then
$A\in{}^{\mathfrak{c}}\mathcal{I}~{}(A\subseteq\mathfrak{h})$ iff any internal
set it contains is in ${}^{\mathfrak{s}}\mathcal{I}$. If $A$ is internal, then
$A\in{}^{\mathfrak{c}}\mathcal{I}(A\subseteq\mathfrak{h})\iff
A\in{}^{\mathfrak{s}}\mathcal{I}$.

Given $\mathfrak{g}^{\prime}\in{}^{\filleddiamond}\mathfrak{G}({}^{\star}{E})$
with base $\mathcal{U}$, then
$\mathfrak{g}^{\prime}\in{}^{\mathfrak{c}}\mathcal{I}~{}(\mathfrak{g}^{\prime}\subseteq\mathfrak{h})\iff\mathcal{U}\subseteq{}^{\mathfrak{s}}\mathcal{I}$.

######  Proposition 1.58.

Given $\mathfrak{g}^{\prime}\in\mathfrak{G}({}^{\star}{E})$, we have

$\mathfrak{g}^{\prime}\in{}^{\ostar}\mathcal{I}\iff\mathfrak{g}^{\prime}\in{}^{\mathfrak{c}}\mathcal{I}~{}(\mathfrak{g}^{\prime}\subseteq\mathfrak{h})\iff\mathfrak{g}^{\prime}\subseteq\mathfrak{g}\iff\forall
V\in\mathrm{idl~{}}\mathfrak{g}^{\prime}:V\in\mathcal{I}.$

###### Proof.

We will prove the equivalence of the first and last statement. Assume
$\mathfrak{g}^{\prime}\in{}^{\ostar}\mathcal{I}$ and take
$V\in\mathrm{idl~{}}\mathfrak{g}^{\prime}$, then
${}^{\star}{V}\in{}^{\ostar}\mathcal{I}$, in fact in ${}^{\star}\mathcal{I}$
since it is internal which transfers to $V\in\mathcal{I}$. Assume otherwise
the last statement is true. Then take any
$N\in{}^{\star}(\mathrm{idl~{}}\mathfrak{g}^{\prime})_{\infty}$, because of
the transfer principle $N\in{}^{\star}\mathcal{I}$, but then
$\mathfrak{g}\in{}^{\ostar}\mathcal{I}$ because it is downward closed. The
second equivalence is trivial; the third equivalence is because of the Cauchy
principle. ∎

######  Proposition 1.59.

Given $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$, then the following are
equivalent:

  1. (i)

$\mathfrak{m}\subseteq\mathfrak{g}$;

  2. (ii)

$\mathfrak{m}\in{}^{\ostar}\mathcal{I}$;

  3. (iii)

$\mathrm{fil~{}}\mathfrak{m}$ contains a set in $\mathcal{I}$;

  4. (iv)

$\mathfrak{m}$ has a base in $\mathcal{I}$.

###### Proof.

(i) $\iff$ (iii): This follows from the combined Cauchy principle.

(ii) $\Rightarrow$ (iii): If $\mathfrak{m}\in{}^{\ostar}\mathcal{I}$, then
each $U\in{}^{\star}(\mathrm{fil~{}}\mathfrak{m})_{\infty}$ is in
${}^{\ostar}\mathcal{I}$, in fact ${}^{\star}\mathcal{I}$ since they are
internal; then because of the transfer principle there exists a
$U\in\mathrm{fil~{}}\mathfrak{m}\cap\mathcal{I}$.

(iii) $\Rightarrow$ (iv): If $\mathcal{U}$ is a base of $\mathfrak{m}$ and
$V\in\mathcal{I}\cap\mathrm{fil~{}}\mathfrak{m}$, then $\\{U\cap
V:U\in\mathcal{U}\\}$ is a base of $\mathfrak{m}$ in $\mathcal{I}$.

(iv) $\Rightarrow$ (ii): Trivial. ∎

In this case the distinction between $\star$-properties and
$\mathfrak{s}/\mathfrak{c}$-properties are not merely non-standard tools, but
have particular relevance in the standard world. For monads these carry over
to two distinctive properties of filters, as will become clearer in concrete
examples.

######  Lemma 1.60.

Given a ring $\Sigma$. If for all $j\in J:\mathfrak{g}_{j}$ is
$\Sigma$-chromatic, then $\mathfrak{h}$ is $\Sigma$-chromatic.

###### Proof.

Since the $\Sigma$-chromatic sets are closed for intersection. ∎

###  1.5 Topology and convergence

For any ring $\Sigma$, clearly if $U\in\Sigma$, then $\forall e\in
U:\boldsymbol{\mu}_{\Sigma}(e)\subseteq{}^{\star}{U}$. Suppose the opposite
implication was also true, i.e. for any $U\subseteq E$:

$U\in\Sigma\iff{}^{\star}{U}=\bigcup_{e\in
U}\boldsymbol{\mu}_{\Sigma}(e)\iff\forall e\in U:\exists
V\in\mathrm{fil_{\Sigma}~{}}e:V\subseteq U$

where we used the Cauchy principle for the second equivalence (note that this
is strictly stronger than $\Sigma$-chromatic). It can be seen that the above
statement is equivalent to $\Sigma$ being closed for the infinite union.

We will now work with a topological space $(E,\tau)$, i.e. $\tau$ is a ring
that is closed for the infinite union, i.e. the open sets. The
$\overline{\tau}$ is closed for infinite intersection, i.e. the closed sets.

######  Definition 1.61.

For $e\in E$, the topological monad of $e$ is
$\mathbf{m}_{[\tau]}(e):=\boldsymbol{\mu}_{\tau}(e)$, $\tau$ is left off when
the topology is clear from the context. Then
$\mathrm{nbh}_{[\tau]}(e):=\mathrm{fil~{}}\mathbf{m}_{[\tau]}(e)$ is the
filter of neighborhoods of $e$. Any base of $\mathbf{m}(e)$ is a base of
neighborhoods of $e$.

A set $U\subseteq E$ is thus open iff for any $e\in U$;
$\mathbf{m}(e)\subseteq{}^{\star}{E}$, equivalently it contains a neighborhood
of $e$. It is closed if it contains any $e$ such that
$\mathbf{m}(e)\cap{}^{\star}{U}\neq\emptyset$. The interior of a set is its
largest open subset and the closure of a set is its smallest closed superset.
The $\star$-extensions of the open and closed properties are extended to
external sets by letting ${}^{\ostar}\tau$ be the closure of ${}^{\star}\tau$
for infinite unions; equivalently the ⋆closure of any
$A\subseteq{}^{\star}{E}$ is

$\overline{A}^{\tau}:=\\{e\in{}^{\star}{E}:\forall
V\in{}^{\star}\mathrm{nbh}_{\tau}(e):A\cap V\neq\emptyset\\}.$

Since ⋆interior and ⋆closure are order preserving mappings, they have
$\dagger$-extensions that coincide with the $\star$-extensions on monads and
galaxies. Hence, a [${}^{\filleddiamond}$]monad or
[${}^{\filleddiamond}$]galaxy is ⋆closed iff it has
[${}^{\filleddiamond}$]base of [⋆]closed sets and ⋆open iff it has
[${}^{\filleddiamond}$]base of [⋆]open sets. This implies that in fact for any
$e\in{}^{\star}{E}$:
$\overline{\boldsymbol{\mu}(e)}^{\tau}=\boldsymbol{\mu}_{\overline{\tau}}(e)$.
In the case of standard monads and galaxies, because of corollary 1.46 ⋆open
coincides with being $\tau$-chromatic and ⋆closed coincides with being
$\overline{\tau}$-chromatic.

######  Definition 1.62.

A monad $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ converges towards a point
$e\in E$, denoted $\mathfrak{m}\xrightarrow{\tau}e$ if
$\mathfrak{m}\subseteq\mathbf{m}(e)$, i.e. each neighborhood of $e$ contains
$\mathfrak{m}$.

The accumulation points of $\mathfrak{m}$ are

$\nabla_{\tau}{\mathfrak{m}}:=\ker\overline{\mathfrak{m}}^{\tau}.$

######  Proposition 1.63.

Given $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$:

$\begin{split}\nabla_{\tau}\mathfrak{m}&=\\{e\in
E:\mathbf{m}(e)\cap\mathfrak{m}\neq\emptyset\\}=\\{e\in
E:\exists\mathfrak{m}^{\prime}\in\mathfrak{M}(\mathfrak{m}):\mathfrak{m}^{\prime}\xrightarrow{\tau}e\\}.\end{split}$

###### Proof.

Assume $e\in\nabla_{\tau}\mathfrak{m}$. Take any
$U\in{}^{\star}\mathrm{nbh}_{\infty}(e)$; since $e\in\overline{\mathfrak{m}}$
there exists an $e^{\prime}\in U\cap\mathfrak{m}$, hence
$\boldsymbol{\mu}(e^{\prime})\subseteq\mathbf{m}(e)\cap\mathfrak{m}$. Now
assume otherwise $e\in E$ and there exists an
$e^{\prime}\in\mathbf{m}(e)\cap\mathfrak{m}$. Now recall that
$\overline{\mathfrak{m}}$ has a base $\mathcal{U}$ of closed sets. Since for
each $U\in\mathcal{U}$ we have $\mathbf{m}(e)\cap{}^{\star}{U}\ni e^{\prime}$,
the closure implies $e\in U$. Therefore $e\in\overline{\mathfrak{m}}$. ∎

######  Corrolary 1.64.

For $e\in{}^{\star}{E}$, $\nabla_{\tau}\boldsymbol{\mu}{(e)}=\\{e^{\prime}\in
E:\boldsymbol{\mu}(e)\xrightarrow{\tau}e^{\prime}\\}$.

######  Corrolary 1.65.

For $V\subseteq E$, $\overline{V}^{\tau}=\nabla_{\tau}{{}^{\star}{V}}$.

######  Corrolary 1.66.

A standard galaxy is ⋆closed iff it contain its accumulation points.

######  Lemma 1.67.

Given $A\subseteq{}^{\star}{E}$. Then $\boldsymbol{\mu}_{\tau}(A)$ is ⋆closed
iff for any closed $V\subseteq E$ such that $A\cap{}^{\star}{V}=\emptyset$,
$A$ and ${}^{\star}{V}$ are $\tau$-separated.

###### Proof.

Assume $\boldsymbol{\mu}_{\tau}(A)$ is ⋆closed and take a closed $V\subseteq
E$. Then ${\boldsymbol{\mu}_{\tau}(A)}^{\mathfrak{c}}$ is an ⋆open galaxy that
contains ${}^{\star}{V}$, since it is $\tau$-chromatic it must also contain
$\boldsymbol{\mu}_{\tau}({}^{\star}{V})$; hence
$\boldsymbol{\mu}_{\tau}({}^{\star}{V})\cap\boldsymbol{\mu}_{\tau}(A)=\emptyset$.
Assume otherwise that for any closed $V$, $A$ and ${}^{\star}{V}$ are
$\tau$-separated and take an open $U$ such that $A\subseteq{}^{\star}{U}$.
Then ${{}^{\star}{U}}^{\mathfrak{c}}$ is closed and thus contained in an open
set $W$ that is disjunctive from $\boldsymbol{\mu}_{\tau}(A)$, i.e.
$\boldsymbol{\mu}_{\tau}(A)\subseteq{W}^{\mathfrak{c}}\subseteq{}^{\star}{U}$.
Hence $\boldsymbol{\mu}_{\tau}(A)$ has a closed base. ∎

A topology is T1 if for any $e\in E$, $\\{e\\}$ is closed (i.e.
$\ker\mathbf{m}(e)=\\{e\\}$); equivalently any standard point is
$\tau$-distinguishable from any other standard point. A topology is Hausdorff
(T2) if each pair of standard points is $\tau$-separated; equivalently there
exist no monads that converge to more than one point. A topology is regular if
$\mathbf{m}(e)$ is ⋆closed for any $e\in E$; equivalently (lemma 1.67) each
singleton and closed set are $\tau$-separated. Note that a topology is
Hausdorff regular (T3) iff it is (T1) and regular.

If $V\subseteq E$, then the induced topology of $V$ is determined by the
monads $\forall e\in
V:\mathbf{m}_{V}(e):=\mathbf{m}_{E,\tau}(e)\cap{}^{\star}{V}$.

######  Definition 1.68.

The near-standard points of ${}^{\star}{E}$ are

$\mathrm{ns}_{\tau}({}^{\star}{E}):=\bigcup_{e\in
E}\mathbf{m}_{\tau}(e)=\\{e\in{}^{\star}{E}:\nabla_{\tau}{\boldsymbol{\mu}(e)}\neq\emptyset\\}.$

Since for any $e\in E$,
$e^{\prime}\in\mathbf{m}(e):\boldsymbol{\mu}_{\Sigma}(e^{\prime})\subseteq\mathbf{m}(e)$,
$\mathrm{ns}({}^{\star}{E})$ is $\tau$-chromatic. If the topology is Hausdorff
then for each $e\in\mathrm{ns}_{\tau}({}^{\star}{E})$, we define
$\mathrm{st}_{\tau}~{}e$ to be the unique element in
$\nabla_{\tau}{\boldsymbol{\mu}(e)}$, i.e.
$\boldsymbol{\mu}(e)\xrightarrow{\tau}\mathrm{st~{}}e$.

Hence, if the space is Hausdorff, we have for $V\subseteq{}^{\star}{E}$

$\overline{V}^{\tau}=\mathrm{st}_{\tau}~{}{}^{\star}{V}.$

######  Lemma 1.69.

Suppose $E$ is Hausdorff. Given
$\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$ and
$\mathfrak{g}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$. Then
$\mathrm{st~{}}\mathfrak{m}$ and $\mathrm{st~{}}\mathfrak{g}$ are closed.

###### Proof.

Suppose $e\notin\mathrm{st~{}}\mathfrak{m}$, then
$\mathbf{m}(e)\subseteq{\mathfrak{m}}^{\mathfrak{c}}$. Because of the combined
principle of Cauchy, there is an open neighborhood $U$ of $e$ such that
$\mathfrak{m}\cap{}^{\star}{U}=\emptyset$. Hence
$\mathrm{st~{}}\mathfrak{m}\subseteq\mathrm{st~{}}{}^{\star}({U}^{\mathfrak{c}})=E\setminus
U$ since its closed, hence
$\mathbf{m}(e)\cap\mathrm{st~{}}\mathfrak{m}=\emptyset$. Now let $\mathcal{U}$
be a base for $\mathfrak{g}$. Then
$\mathrm{st~{}}\mathfrak{g}=\cap_{U\in\mathcal{U}_{\infty}}\mathrm{st~{}}\mathcal{U}$,
an intersection of closed sets is closed. ∎

######  Proposition 1.70.

$\mathrm{ns}({}^{\star}{E})$ is the I.o.G. of ⋆open covering galaxies of
${}^{\star}{E}$.

###### Proof.

It is clear that any ⋆open covering galaxy must contain
$\mathrm{ns}({}^{\star}{E})$. Suppose then that there exists an element
$e\in{}^{\star}{E}$ contained in every ⋆open covering galaxy, then since
${\overline{\boldsymbol{\mu}(e)}}^{\mathfrak{c}}$ is open it cannot be
covering, hence there exists an $e^{\prime}\in\nabla\boldsymbol{\mu}(e)$. ∎

######  Proposition 1.71.

Given a set $V\subseteq E$. The following are equivalent:

  1. (i)

${}^{\star}{V}\subseteq\mathrm{ns}_{\tau}({}^{\star}{E})$;

  2. (ii)

$\forall\mathfrak{m}\in\mathfrak{M}({}^{\star}{V}),\mathfrak{m}\neq\emptyset:\nabla_{\tau}\mathfrak{m}\neq\emptyset$.

  3. (iii)

Each open cover of $E$ contains a finite subcover of $V$.

###### Proof.

(i) $\iff$ (ii): From the definition of $\mathrm{ns}({}^{\star}{E})$ and
proposition 1.63 follows that
$\mathfrak{m}\cap\mathrm{ns}_{\tau}({}^{\star}{E})\neq\emptyset\iff\nabla_{\tau}\mathfrak{m}\neq\emptyset$.

(ii) $\iff$ (iii): Every open cover $\mathcal{U}$ is the subbase of an ⋆open
covering galaxy $\mathfrak{g}$. By proposition 1.70,
${}^{\star}{V}\subseteq\mathfrak{g}$, hence by the Cauchy principle there must
be finite sets in $\mathcal{U}$ that cover $V$. ∎

######  Definition 1.72.

A set $V\subseteq E$ that satisfies the equivalent conditions of proposition
1.71 is called relatively compact.222Since relative compactness is the
associated set property of an I.o.G., we can use section 1.4 to define
⋆relative-compact, srelative-compact and crelative-compact on the basis of
open covering galaxies and automatically get the implications for monads and
galaxies. However, the approach with convergence and accumulation points is
natural and sufficient. $E$ is a compact space if
${}^{\star}{E}\subseteq\mathrm{ns}({}^{\star}{E})$. A set $V\subseteq E$ is
compact if ${}^{\star}{V}\subseteq\mathrm{ns}_{V}{({}^{\star}{V})}$, i.e. the
near-standard points for the space $V$ with the induced topology of $E$.

A set that that is both closed and relatively compact, is always compact since
in that case $\mathrm{ns}_{V}({}^{\star}{V})=\mathrm{ns}_{E}({}^{\star}{V})$.
If the topology is Hausdorff, a compact set set is always closed since for
$e\in\mathrm{ns}_{E}({}^{\star}{V})$: $\nabla\boldsymbol{\mu}(e)\cap
V\neq\emptyset\iff\mathrm{st~{}}e\in V$. This however does not imply that the
closure of a relatively compact set is (relatively) compact.

######  Lemma 1.73.

If $E$ is regular, then any $\tau$-chromatic
$A\subseteq\mathrm{ns}({}^{\star}{E})$, $\boldsymbol{\mu}_{\tau}(A)$ is
⋆closed.

###### Proof.

Take $e\in\overline{A}$. Since $\boldsymbol{\mu}_{\tau}(e)$ is ⋆open, there
exists an $e^{\prime}\in\boldsymbol{\mu}_{\tau}(e)\cap A$. Because of lemma
1.50 $e\in\overline{\boldsymbol{\mu}}(e^{\prime})$. Since
$A\subseteq\mathrm{ns}({}^{\star}{E})$ there exists a $\hat{e}\in A$ such that
$e^{\prime}\in\mathbf{m}(\hat{e})\subseteq\boldsymbol{\mu}_{\tau}(A)$. Since
$\mathbf{m}(\hat{e})$ is ⋆closed,
$\overline{\boldsymbol{\mu}}(e^{\prime})\subseteq\mathbf{m}(\hat{e})$, then
since $A$ is $\tau$-chromatic $e\in A$. ∎

######  Corrolary 1.74.

If $E$ is regular, the closure of a relatively compact set is compact.

###### Proof.

The lemma implies that $\mathrm{ns}({}^{\star}{E})$ itself is ⋆closed, hence
the ⋆closure of any subset of $\mathrm{ns}({}^{\star}{E})$ must still be in
$\mathrm{ns}({}^{\star}{E})$. ∎

######  Corrolary 1.75.

If $E$ is regular, any disjunctive pair of a relatively compact set and a
closed set is $\tau$-separated.

###### Proof.

Combine with lemma 1.73. ∎

######  Definition 1.76.

The compact points of $E$ are

$\mathrm{comp}_{\tau}({}^{\star}{E}):=\boldsymbol{\Gamma}(\mathrm{ns}_{\tau}({}^{\star}{E})),$

i.e. the galaxy of relatively compact sets. If $E$ is regular, it is also the
galaxy of compact sets and $\mathrm{comp}_{\tau}({}^{\star}{E})$ is ⋆closed.

######  Proposition 1.77.

(Local property) Given a galaxy $\mathfrak{g}$. Then the following statements
are equivalent:

  1. (i)

$\mathrm{ns}({}^{\star}{E})\subseteq\mathfrak{g}$

  2. (ii)

every convergent monad $\mathfrak{m}$ has a base
$\mathcal{U}\subseteq\mathrm{idl~{}}\mathfrak{g}$.

  3. (iii)

every point has a neighborhood in $\mathcal{U}$;

  4. (iv)

$\mathfrak{g}$ is $\tau$-chromatic;

###### Proof.

(i) $\Rightarrow$ (ii): Given a base $\mathcal{V}$, because of the combined
Cauchy principle it contains a set in $U\in\mathrm{idl~{}}\mathfrak{g}$, then
consider the base $\\{V\cap U:V\in\mathcal{V}\\}$;

(ii) $\Rightarrow$ (iii) $\Rightarrow$ (iv): Trivial;

(i) $\iff$ (iii): $\tau$-chromatic is equivalent to ⋆open for galaxies, so
that this follows from proposition 1.70. ∎

######  Example 1.78.

$E$ is locally compact iff each near-standard point is compact iff each point
has a compact neighborhood.

######  Definition 1.79.

Given a topological space $F$ and a function
$\psi:{}^{\star}{E}\rightarrow{}^{\star}{F}$. Then $\psi$ is $\psi$ is
ccontinuous if for any $e\in E$,
$\psi(\mathbf{m}(e))\subseteq\mathbf{m}(\psi(e))$; $\psi$ is scontinuous if
for any $e\in{}^{\sigma}{E}$ and $V\in{}^{\sigma}\mathrm{nbh}(\psi(e))$, there
exists a $U\in{}^{\sigma}\mathrm{nbh}(\psi(e))$ such that
$\psi(U)\subseteq\psi(V)$. Finally, $\psi$ is ⋆continuous if for any
$e\in{}^{\star}{E}$ and $V\in{}^{\star}\mathrm{nbh}(\psi(e))$, there exists a
$U\in{}^{\star}\mathrm{nbh}(\psi(e))$ such that $\psi(U)\subseteq\psi(V)$.

The Cauchy principle implies that an internal function $\psi$ is scontinuous
iff it is ccontinuous. For standard functions all three properties coincide
with continuity.

##  2 Topological Vector Spaces

###  2.1 Linear monads

We work with a $\mathbb{K}$-vector space $(E,+,\cdot)$, where $\mathbb{K}$ is
either $\mathbb{R}$ or $\mathbb{C}$. We have the usual topology on
$\mathbb{K}$, which means that:

$\begin{gathered}\forall\alpha\in\mathbb{K}:\mathbf{m}_{\mathbb{K}}(\alpha)=\\{\beta\in{}^{\star}{\mathbb{K}}:\forall
n\in\mathbb{N}:|\alpha-\beta|<1/n\\};\\\
\mathrm{ns}({}^{\star}{\mathbb{K}})={}^{\star}\mathbb{K}_{\mathfrak{f}}=\\{\alpha\in{}^{\star}{\mathbb{K}}:\exists
n\in{}^{\sigma}\mathbb{N}:|\alpha|\leq n\\}.\end{gathered}$

Given $\mathfrak{m}\in{}^{\star}{E}$. Then for any
$\mathfrak{m}^{\prime}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$,
$\mathfrak{m}^{\prime}+\mathfrak{m}$ is a ${}^{\filleddiamond}$monad in ⋆E and
for any
$\mathfrak{m}^{\prime}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{\mathbb{K}})$,
$\mathfrak{m}^{\prime}\cdot\mathfrak{m}$ is a ${}^{\filleddiamond}$monad in
$E$ (due to corollary 1.28).

The balanced and convex hulls of sets have non-standard equivalents applicable
to external sets. Given $A\subseteq{}^{\star}{E}$, the ⋆balanced hull is

$\mathrm{bal}(A):=\bigcup_{\begin{subarray}{c}\alpha\in{}^{\star}{\mathbb{K}}\\\
|\alpha|\leq 1\end{subarray}}\alpha A.$

In particular, sets that are closed for the
${}^{\star}\mathbb{K}_{\mathfrak{f}}$-scalar multiplication are ⋆balanced. In
fact, a set is closed for ${}^{\star}\mathbb{K}_{\mathfrak{f}}$-scalar
multiplication iff it is closed for $\mathbb{K}$-scalar multiplication and is
⋆balanced. The ⋆convex hull of a subset $A\subseteq{}^{\star}{E}$ is

$\mathrm{co}(A):=\big{\\{}\sum_{i=1}^{\omega}a_{i}x_{i}:(x_{i})_{i\leq\omega}\textnormal{
internal in }A,(a_{i})_{i\leq\omega}\textnormal{ internal in
}{}^{\star}{[0,1]},\sum_{i=1}^{\omega}a_{i}=1\big{\\}}.$

The convex and balanced hull is then

$\mathrm{cobal}(A):=\mathrm{co}(\mathrm{bal}(A))=\big{\\{}\sum_{i=1}^{\omega}a_{i}x_{i}:(x_{i})\textnormal{
int. in }A,(a_{i})\textnormal{ int. in
}{}^{\star}{\mathbb{K}},\sum_{i=1}^{\omega}|a_{i}|\leq 1\big{\\}}$

and the ⋆closed, ⋆convex, ⋆balanced hull
$\overline{\mathrm{cobal}}(A):=\overline{\mathrm{cobal}(A)}$.

These properties coincide with the ⋆-extension of their standard equivalent,
so that for a standard $V$ we have
${}^{\star}({\mathrm{bal}}(V))=\mathrm{bal}({}^{\star}{V})$ and
${}^{\star}({\mathrm{co}}(V))=\mathrm{co}({}^{\star}{V})$. Furthermore, the
two maps define set properties: $A$ is ⋆balanced iff $A=\mathrm{bal}(A)$ and
⋆convex iff $A=\mathrm{co}(A)$. Note that because of the transfer principle,
an internal set $A$ is convex precisely iff for any two $e,e^{\prime}\in A$
and $\alpha\in{}^{\star}[0,1]$: $\alpha e+(1-\alpha)e\in A$, however this
condition is not sufficient for external sets. Then, since the maps are order-
preserving the maps and associated properties have $\dagger$-extensions that
coincide with the ⋆-extensions on ${}^{\filleddiamond}$monads and
${}^{\filleddiamond}$galaxies. Hence, a $[{}^{\filleddiamond}]$monad is
⋆balanced iff it has a base existing out of $[{}^{\star}]$balanced sets and
⋆convex iff it has a base existing out of $[{}^{\star}]$convex sets.

######  Definition 2.1.

Given two sets $A\subseteq{}^{\star}{E}$ and $B\subseteq{}^{\star}{E}$. Then

$\begin{split}&B\textbf{ ${}^{\mathfrak{s}}$absorbs }A\textnormal{ if
}\exists\alpha_{0}\in{}^{\sigma}\mathbb{R}_{+}:\forall\alpha\in{}^{\star}\mathbb{K},|\alpha|\geq\alpha_{0}:A\subseteq\alpha
B;\\\ &B\textbf{ ${}^{\star}$absorbs }A\textnormal{ if
}\exists\alpha_{0}\in{}^{\star}\mathbb{R}_{+}:\forall\alpha\in{}^{\star}\mathbb{K},|\alpha|\geq\alpha_{0}:A\subseteq\alpha
B;\\\ &B\textbf{ ${}^{\mathfrak{c}}$absorbs }A\textnormal{ if
}\mathbf{m}_{\mathbb{K}}(0)\cdot A\subseteq B.\end{split}$

When $A$ is a singleton, we refer to it as a point. We call a set sabsorbing
if it sabsorbs all points of ${}^{\sigma}{E}$; cabsorbing if it cabsorbs
${}^{\sigma}{E}$; ⋆absorbing if it ⋆absorbs all points of ${}^{\star}{E}$.

######  Proposition 2.2.

Given $\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$ and
$\mathfrak{g}\in{}^{\filleddiamond}\mathfrak{G}({}^{\star}{E})$. Then
$\mathfrak{m}$ cabsorbs $\mathfrak{g}$ iff any internal set containing
$\mathfrak{m}$ sabsorbs any internal set contained in $\mathfrak{g}$.

###### Proof.

Given an internal $A\subseteq\mathfrak{g}$,
$\mathfrak{m}_{A}:=\mathbf{m}_{\mathbb{K}}(0)\cdot A$ is a
${}^{\filleddiamond}$monad with base $(\\{\alpha A:|\alpha|\leq
1/n\\})_{n\in\mathbb{N}}$. For any internal set $B$, because of the Cauchy
principle $\mathfrak{m}_{A}\subseteq B$ is equivalent with $\exists n\in
N:\forall\alpha\in\mathbb{K},|\alpha|>n:\alpha A\subseteq B$. ∎

######  Corrolary 2.3.

Given internal sets $A,B\subseteq{}^{\star}{E}$. Then $A$ sabsorbs $B$ iff $A$
cabsorbs $B$.

######  Corrolary 2.4.

Given $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ and
$\mathfrak{g}\in\mathfrak{G}({}^{\star}{E})$. Then $\mathfrak{m}$ cabsorbs
$\mathfrak{g}$ iff $\mathfrak{m}$ ⋆absorbs any set in
$\mathrm{idl~{}}\mathfrak{g}$.

###### Proof.

Given $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ and
$V\in\mathrm{idl~{}}\mathfrak{g}$. If
$\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\star}{V}\subseteq\mathfrak{m}$, any set
in $\mathrm{fil~{}}\mathfrak{m}$ absorbs $V$. Transfer implies any set in
${}^{\star}(\mathrm{fil~{}}\mathfrak{m})_{\infty}$ ⋆absorbs ${}^{\star}{V}$,
hence $\mathfrak{m}$ ⋆absorbs ${}^{\star}{V}$. If otherwise $\mathfrak{m}$
⋆absorbs ${}^{\star}{V}$, any set in $\mathrm{fil~{}}\mathfrak{m}$ ⋆absorbs
${}^{\star}{V}$, byt transfer it absorbs $V$ and therefore it cabsorbs
${}^{\sigma}{V}$. ∎

A topology $\tau$ is compatible with the vector structure if addition and
multiplication are continuous for this topology. In this case $(E,\tau)$ is a
topological vector space (TVS).

######  Proposition 2.5.

Addition is continuous for $\tau$ iff

  1. 1.

$\mathbf{m}_{\tau}(e)=\mathbf{m}_{\tau}(0)+e$ for each $e\in E$;

  2. 2.

$\mathbf{m}_{\tau}(0)$ is closed for addition.

###### Proof.

Addition is continuous iff
$\mathbf{m}(e_{1})+\mathbf{m}(e_{2})\subseteq\mathbf{m}(e_{1}+e_{2})$. Assume
that is true, than (2) follows from the case $e_{1}=-e_{2}=0$. If $e_{1}=0$ en
$e_{2}=e$ then $\mathbf{m}(0)+e\subseteq\mathbf{m}(e)$, but if $e_{1}=e_{2}=e$
then $\mathbf{m}(e)-e\subseteq\mathbf{m}(0)$, proving (1). Assuming otherwise
(1) and (2) are true, then
$\mathbf{m}(e_{1})+\mathbf{m}(e_{2})=e_{1}+e_{2}+\mathbf{m}(0)=\mathbf{m}(e_{1}+e_{2})$.
∎

######  Definition 2.6.

If addition on $E$ is continuous, we define for any $e\in{}^{\star}{E}$:

$\mathbf{m}_{\tau}(e):=\mathbf{m}_{\tau}(0)+e.$

Note that for a non-standard $e$, $\mathbf{m}(e)$ is a
${}^{\filleddiamond}$monad but not necessarily a standard monad. As these are
equivalence classes, we denote for $e,e^{\prime}\in{}^{\star}{E}$:

$e\approx_{[\tau]}e^{\prime}\iff e-e^{\prime}\in\mathbf{m}(0).$

######  Proposition 2.7.

Suppose addition on $E_{\tau}$ is continuous. Then multiplication is
continuous iff

  1. 1.

$\mathbf{m}_{\tau}(0)$ is closed for
${}^{\star}\mathbb{K}_{\mathfrak{f}}$-scalar multiplication, i.e. it is
⋆balanced and closed for ${}^{\sigma}{\mathbb{K}}$-scalar multiplication.

  2. 2.

$\mathbf{m}_{\tau}(0)$ is cabsorbing, i.e.
$\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\sigma}{E}\subseteq\mathbf{m}_{\tau}(0)$.

###### Proof.

Multiplication is continuous iff
$\mathbf{m}_{\mathbb{K}}(\alpha)\cdot\mathbf{m}(e)\subseteq\mathbf{m}(\alpha
e)$ for each $\alpha\in\mathbb{K}$ and $e\in E$. Assume that is true, then (1)
follows from the case $e=0$ and (2) from $\alpha=0$. Assuming otherwise (1)
and (2) are true, then
$\mathbf{m}_{\mathbb{K}}(\alpha)\cdot\mathbf{m}(e)=(\alpha+\mathbf{m}_{\mathbb{K}}(0))\cdot(e+\mathbf{m}(0))=\alpha
e+\mathbf{m}(0)=\mathbf{m}(\alpha e)$. ∎

######  Corrolary 2.8.

If $E_{\tau}$ is compatible with the vector structure, every neighborhood of
$0$ is absorbing and contains a balanced neighborhood of $0$.

######  Definition 2.9.

A monad $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ is linear if it is closed
and for ${}^{\star}\mathbb{K}_{\mathfrak{f}}$-multiplication and it contains
$\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\sigma}{E}$. Any linear
$\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ defines a topology $\tau$ on $E$
compatible with the topology by setting for $e\in E$:

$\mathbf{m}_{\tau}(e):=\mathfrak{m}+e.$

We will from now on assume that $E_{\tau}$ is a Hausdorff TVS, where
$\Lambda_{[\tau]}$ is a base of $\mathbf{m}_{\tau}(0)$ existing out of
absorbing, open and balanced sets. Given $\lambda\in\Lambda$ and
$\alpha\in\mathbb{K}$, $\mathbf{m}(0)\subseteq\alpha\lambda$, then applying
the Cauchy principle there exists a $\lambda^{\prime}\in\Lambda$ such that
$\lambda^{\prime}\subseteq\alpha\lambda$. Furthermore, applying the Cauchy
principle to $\mathbf{m}(0)+\mathbf{m}(0)\subseteq\lambda$ gives us that for
any $\lambda\in\Lambda$ there exists a $\lambda^{\prime}\in\Lambda$ such that
$\lambda^{\prime}+\lambda^{\prime}\subseteq\lambda$.

Note that for $V\subseteq E$,
$\overline{V}=\bigcap_{\lambda\in\Lambda}(V+\lambda)$, while for
$A\subseteq{}^{\star}{E}$,
$\overline{A}=\bigcap_{\lambda\in{}^{\star}\Lambda}(A+\lambda)$.

######  Definition 2.10.

A set $A\subseteq{}^{\star}{E}$ is $\approx_{\tau}$-saturated if
$A=A+\mathbf{m}_{\tau}(0)$.

Note that this is not equivalent to $\tau$-chromatic since for non-standard
points $e^{\prime}\in{}^{\star}{E}$, in general
$\boldsymbol{\mu}_{\tau}(e)\neq\mathbf{m}(e)$; they may be larger or smaller.

######  Lemma 2.11.

Any $\approx_{\tau}$-saturated set is ⋆open and ⋆closed.

###### Proof.

Let $A$ be a $\approx_{\tau}$-saturated set and take $e\in A$. For any
$\lambda\in{}^{\star}\Lambda_{\infty}$ we have $e+\lambda\subseteq A$, hence
$A$ is ⋆open. Now take $e\in\overline{A}$. For any
$\lambda\in{}^{\star}\Lambda_{\infty}$ we have $e^{\prime}\in A$ such that
$e-e^{\prime}\in\lambda$, but then $e\in\mathbf{m}(e^{\prime})\subseteq A$,
hence $A$ is ⋆closed. ∎

######  Corrolary 2.12.

$\mathbf{m}_{\tau}(0)$ is ⋆closed, i.e. $E$ is regular.

This means that $E_{\tau}$ has a closed base of neighborhoods of $0$, in
particular $\\{\overline{\lambda}:\lambda\in\Lambda\\}$ is such a base.
Furthermore for topological vector spaces T1=T2=T3, i.e. $E$ is Hausdorff iff
$\ker\mathbf{m}_{\tau}(0)=\\{0\\}$.

######  Lemma 2.13.

Given a closed $V\subseteq E$ and relatively compact $K\subseteq E$. Then
$K+V$ is closed.

###### Proof.

Suppose $e\in{(V+K)}^{\mathfrak{c}}$. Then $e-V\cap K=\emptyset$. Since $e-V$
is closed and $K$ relatively compact, because of regularity (corollary 1.75)
$\boldsymbol{\mu}_{\tau}(e-{}^{\star}{V})\cap\boldsymbol{\mu}_{\tau}({}^{\star}{K})=\emptyset$
and since
$\mathbf{m}(e)-{}^{\star}{V}\subseteq\boldsymbol{\mu}_{\tau}(e-{}^{\star}{V})$
this implies $\mathbf{m}(e)\cap{}^{\star}(K+V)=\emptyset$. ∎

Given $e\in\mathrm{cobal}({}^{\sigma}{E})$. Then we have internal sequences
$(e_{i})_{i\leq\nu}$ in ${}^{\sigma}{E}$ and $(\alpha_{i})_{i\leq\nu}$ in
${}^{\star}\mathbb{K}$ such that $\sum_{i=0}^{\nu}|\alpha_{i}|\leq 1$ and
$e=\sum_{i=0}^{\nu}\alpha_{i}e_{i}$. But if $\\{e_{i}\\}_{i\leq\nu}$ is an
internal subset of ${}^{\sigma}{E}$, due to the Cauchy principle it must be
standard and finite, hence we assume w.lo.g that
$\nu\in{}^{\sigma}{\mathbb{N}}$. Since ${}^{\sigma}{E}$ is the galaxy of
finite sets, $\mathrm{cobal}({}^{\sigma}{E})$ is a galaxy that has the convex,
balanced hulls of finite sets as base. Since $\mathrm{ns}({}^{\star}{E})$ is
closed for (finite) addition and ${}^{\star}\mathbb{K}_{\mathfrak{f}}$-scalar
multiplication,
$\mathrm{cobal}({}^{\sigma}{E})\subseteq\mathrm{ns}({}^{\star}{E})$. Since
$\mathrm{st~{}}\mathrm{cobal}({}^{\sigma}{E})={}^{\sigma}{E}\subseteq\mathrm{cobal}({}^{\sigma}{E})$,
the galaxy is ⋆closed, i.e. equal to
$\overline{\mathrm{cobal}}({}^{\sigma}{E})$. These observations leads us to
the following definition:

######  Definition 2.14.

The quasistandard points are the ${}^{\star}{\mathbb{K}}_{\mathfrak{f}}$-span
of the standard points, i.e.

$\mathrm{qs}({}^{\star}{E}):=\overline{\mathrm{cobal}}({}^{\sigma}{E})=\\{\sum_{i=0}^{n}\alpha_{i}e_{i}:n\in\mathbb{N},(\alpha_{i})_{i\leq
n}\textnormal{ in }{}^{\star}{\mathbb{K}}_{\mathfrak{f}},(e_{i})_{i\leq
n}\textnormal{ in }{}^{\sigma}{E}\\}.$

Note that this galaxy is independent of the topology.

###  2.2 Boundedness and precompactness

######  Definition 2.15.

The set of finite points is the following I.o.G.:

$\mathrm{Fin}_{\tau}({}^{\star}{E}):=\bigcap_{\lambda\in\Lambda_{\tau}}{}^{\star}{\mathbb{K}}_{\mathfrak{f}}\cdot\lambda=\bigcap_{\lambda\in\Lambda_{\tau}}\bigcup_{\alpha_{\lambda}\in\mathbb{R}_{+}}{}^{\star}(\alpha_{\lambda}\lambda);$

i.e. $e\in{}^{\star}{E}$ is finite iff for any $\lambda\in\Lambda$ there
exists an $\alpha\in\mathbb{R}_{+}$ such that $e\in\alpha\lambda$. The set is
$\tau$-chromatic, $\tau$-saturated, closed for addition and
${}^{\star}{\mathbb{K}}_{\mathfrak{f}}$-multiplication. The corresponding
galaxy is the set of bounded points:

$\mathrm{Bdd}_{\tau}({}^{\star}{E}):=\boldsymbol{\Gamma}(\mathrm{Fin}_{\tau}({}^{\star}{E}))=\bigcup{}^{\sigma}\\{\bigcap_{\lambda\in\lambda}\alpha_{\lambda}\lambda\\}_{(\alpha_{\lambda})_{\lambda}\in\mathbb{R}_{+}^{\Lambda}}$

i.e. $e\in{}^{\star}{E}$ is bounded iff there exists a
$(\alpha_{\lambda})_{\lambda}\in\mathbb{R}_{+}^{\Lambda}$ such that
$\forall\lambda\in{}^{\star}{\Lambda}:e\in\alpha_{\lambda}\lambda$.

######  Proposition 2.16.

A point $e\in{}^{\star}{E}$ is finite iff it is cabsorbed by
$\mathbf{m}_{\tau}(0)$.

###### Proof.

Assume $e\in\mathrm{Fin}_{\tau}({}^{\star}{E})$ and
$\epsilon\in\mathbf{m}_{\mathbb{K}}(0)$. For any $\lambda\in\Lambda$ there
exists an $\alpha_{\lambda}\in\mathbb{R}_{+}$ such that
$e\in{}^{\star}(\alpha_{\lambda}\lambda)$. Then $\epsilon
e\in\epsilon\alpha({}^{\star}\lambda)\subseteq{}^{\star}\lambda$ since
$\lambda$ is balanced and $|\epsilon\alpha_{\lambda}|<1$. Assume otherwise
$e\notin\mathrm{Fin}_{\tau}({}^{\star}{E})$. There exists a
$\lambda\in\Lambda$ such that $\forall\alpha\in\mathbb{K}$,
$e\notin{}^{\star}(\alpha\lambda)$. Because of infinite overspill there exists
an $\omega\in{}^{\star}\mathbb{K}_{\infty}$ such that
$e\notin\omega({}^{\star}\lambda)$, hence
$\omega^{-1}e\notin{}^{\star}\lambda$. ∎

The corresponding set property for $V\subseteq E$,
${}^{\star}{V}\subseteq\mathrm{Fin}({}^{\star}{E})$ is bounded, from section
1.4 we immediately get definitions for cbounded, sbounded and ⋆bounded sets:
$A\subseteq{}^{\star}{E}$ is [${}^{\mathfrak{s}}|{}^{\star}$]bounded iff for
any $\lambda\in[{}^{\sigma}|{}^{\star}]\Lambda$ there exists an
$\alpha\in[{}^{\sigma}|{}^{\star}]\mathbb{R}_{+}$ such that
$A\subseteq\alpha\lambda$. The section also gives us the implications for
${}^{\filleddiamond}$monads and ${}^{\filleddiamond}$galaxies, in particular
for a monad $\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ we distinguish:

  * •

$\mathfrak{m}\subseteq\mathrm{Fin}({}^{\star}{E})$ is cbounded, sbounded or
finite, i.e. or any $\lambda\in\Lambda$ there exists an
$\alpha\in\mathbb{R}_{+}$ and a $U$ in $\mathrm{fil~{}}\mathfrak{m}$ such that
$U\subseteq\alpha\lambda$; in light of proposition 2.16 a monad is finite iff
when multiplied with a monad in ${}^{\star}\mathbb{K}$ that converges to $0$,
the resulting monad converges to $0$;

  * •

$\mathfrak{m}\subseteq\mathrm{Bdd}({}^{\star}{E})$ is ⋆bounded or just
bounded, i.e. $\mathrm{fil~{}}\mathfrak{m}$ contains a bounded set and
therefore a base existing out of bounded sets.

Hence, a set in $E$ is bounded iff each monad it contains is finite; a set in
${}^{\star}{E}$ is cbounded iff each monad it intersects is finite.

######  Proposition 2.17.

Given $\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$, the
following statements are equivalent:

  1. (i)

For each $e\in\mathfrak{m}$; $\mathfrak{m}\leq\mathbf{m}_{\tau}(e)$;

  2. (ii)

there exists an $e\in{}^{\star}{E}$ such that
$\mathfrak{m}\leq\mathbf{m}_{\tau}(e)$;

  3. (iii)

for each $\lambda\in\Lambda_{\tau}$ there exists an internal (standard if
$\mathfrak{m}$ is standard) $U_{\lambda}\supseteq\mathfrak{m}$ such that
$U_{\lambda}-U_{\lambda}\subseteq{}^{\star}\lambda$;

###### Proof.

(i) $\Rightarrow$ (ii): Trivial.

(ii) $\Rightarrow$ (iii): Given $\lambda\in\Lambda$, take $\lambda^{\prime}$
such that $\lambda^{\prime}+\lambda^{\prime}\subseteq\lambda$. Because of the
Cauchy principle there exists an internal $U_{\lambda}\supseteq\mathfrak{m}$
such that $U_{\lambda}\subseteq e+{}^{\star}\lambda^{\prime}$. Then
$U_{\lambda}-U_{\lambda}\subseteq{}^{\star}\lambda^{\prime}+{}^{\star}\lambda^{\prime}\subseteq{}^{\star}\lambda$.

(iii) $\Rightarrow$ (i): Given $e\in\mathfrak{m}$ and take
$\lambda\in\Lambda$. Because of the assumption there exists a
$U_{\lambda}\supseteq\mathfrak{m}$ such that
$U_{\lambda}-U_{\lambda}\subseteq{}^{\star}\lambda$, since $e\in U_{\lambda}$,
$U_{\lambda}\subseteq e+{}^{\star}\lambda$. ∎

######  Definition 2.18.

A $[{}^{\filleddiamond}]$monad that satisfies the equivalent conditions of
2.17 is called $[{}^{\filleddiamond}]$Cauchy.

######  Definition 2.19.

The set of prenearstandard points is the following I.o.G.:

$\mathrm{pns}_{\tau}({}^{\star}{E}):=\bigcap_{\lambda\in\Lambda_{\tau}}{}^{\sigma}{E}+{}^{\star}\lambda=\bigcap_{\lambda\in\Lambda_{\tau}}\bigcup_{e\in{}^{\sigma}{E}}e+{}^{\star}\lambda;$

i.e. $e\in{}^{\star}{E}$ is prenearstandard iff for any $\lambda\in\Lambda$
there exists an $e^{\prime}\in E$ such that $e-e^{\prime}\in\lambda$. The set
is $\tau$-chromatic, $\tau$-saturated, closed for addition and
${}^{\star}{\mathbb{K}}_{\mathfrak{f}}$-multiplication. The corresponding
galaxy is the set of precompact points:

$\mathrm{pcomp}_{\tau}({}^{\star}{E}):=\boldsymbol{\Gamma}(\mathrm{pns}_{\tau}({}^{\star}{E}))=\bigcup{}^{\sigma}\\{\bigcap_{\lambda\in\lambda}S_{\lambda}+\lambda\\}_{(S_{\lambda})_{\lambda}\in\mathcal{P}_{\mathrm{Fin}}(E)^{\Lambda}}$

i.e. $e\in{}^{\star}{E}$ is precompact iff there exists a
$(S_{\lambda})_{\lambda}\in\mathcal{P}_{\mathrm{Fin}}(E)^{\Lambda}$ such that
$\forall\lambda\in{}^{\star}{\Lambda}:e\in({}^{\star}{S})_{\lambda}+{}^{\star}\lambda.$

######  Proposition 2.20.

Given $e\in{}^{\star}{E}$. Then $e\in\mathrm{pns}_{\tau}({}^{\star}{E})$ iff
$\mathbf{m}_{\tau}(e)\in\mathfrak{M}({}^{\star}{E})$.

###### Proof.

Assume $e\in\mathrm{pns}({}^{\star}{E})$. Then we can choose
$(e_{\lambda})_{\lambda}$ in $E$ such that for each $\lambda\in\Lambda$,
$e-e_{\lambda}\in\lambda$. Then consider the monad $\mathfrak{m}$ with base
$(e_{\lambda}+\lambda)_{\lambda\in\Lambda}$. Given $\lambda\in\Lambda$, take
$\lambda^{\prime}$ such that
$\lambda^{\prime}+\lambda^{\prime}\subseteq\lambda$. For any $e^{\prime}\in
e_{\lambda^{\prime}}+\lambda^{\prime}$,
$e-e^{\prime}=(e-e_{\lambda^{\prime}})-(e^{\prime}-e_{\lambda^{\prime}})\in\lambda^{\prime}+\lambda^{\prime}\subseteq\lambda$,
so that $e_{\lambda^{\prime}}+\lambda^{\prime}\subseteq e+\lambda$; thus
$\mathfrak{m}\subseteq\mathbf{m}(e)$. For any $e^{\prime\prime}\in
e+\lambda^{\prime}$ similarly $e_{\lambda}-e^{\prime\prime}\in\lambda$, so
that $e+\lambda^{\prime}\subseteq e_{\lambda}+\lambda$; thus also
$\mathbf{m}(e)\supseteq\mathfrak{m}$. Hence
$\mathbf{m}(e)=\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$. Assuming otherwise
$\mathbf{m}(e)\in\mathfrak{M}({}^{\star}{E})$, then $\mathbf{m}(e)$ is cauchy.
Given $\lambda$, there exists a $U\subseteq{}^{\star}{E}$ such that
$\mathbf{m}(e)\subseteq U$ and $U-U\subseteq\lambda$. Picking any
$e^{\prime}\in U$, we have $e-e^{\prime}\subseteq\lambda$. ∎

The associated set property for a set $V\subseteq
E,{}^{\star}{V}\subseteq\mathrm{pns}({}^{\star}{E})$ is precompact or totally
bounded, from section 1.4 we immediately get definitions for cprecompact,
sprecompact and ⋆precompact sets: $A\subseteq{}^{\star}{E}$ is
$[{}^{\mathfrak{s}}|{}^{\star}]$precompact iff for any
$\lambda\in[{}^{\sigma}|{}^{\star}]\Lambda$ there exists an
$S\in[{}^{\sigma}|{}^{\star}]\mathcal{P}_{\mathrm{fin}}(E)$ such that
$A\subseteq S+\lambda$). The section also gives implications for
${}^{\filleddiamond}$monads and ${}^{\filleddiamond}$galaxies while
proposition 2.20 implies that a standard monad is cprecompact iff it is a
union of Cauchy monads. Then for a monad
$\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ we distinguish:

  * •

$\mathfrak{m}\subseteq\mathrm{pns}({}^{\star}{E})$ is cprecompact, sprecompact
or totally finite, i.e. for any $\lambda\in\Lambda$ there exists a $U$ in
$\mathrm{fil~{}}\mathfrak{m}$ and a $S\in\mathcal{P}_{\mathrm{fin}}(E)$ such
that $U\subseteq S+\lambda$. Each ultramonad it contains is Cauchy, i.e. for
any $\lambda\in\Lambda$ there exists a $U$ in $\mathrm{fil~{}}\mathfrak{m}$
such that $U-U\subseteq\lambda$;

  * •

$\mathfrak{m}\subseteq\mathrm{pcomp}({}^{\star}{E})$ are ⋆precompact or just
precompact, i.e. $\mathrm{fil~{}}\mathfrak{m}$ contains a precompact set and
therefore a base existing out of precompact sets.

Hence, a set in $E$ is precompact iff each ultramonad it contains is Cauchy, a
set in ${}^{\star}{E}$ is cprecompact iff each ultramonad it intersects is
Cauchy.

######  Remark 2.21.

Note that because of proposition 1.54, sequential monads are finite iff they
are bounded and totally finite iff they are precompact. From its corollary
follows that monads with countable cofinality that are finite, respectively
totally finite always intersect with a bounded, respectively precompact set.
This is the basis of many special properties of metrizables spaces where
$\mathbf{m}_{\tau}(0)$ has countable cofinality.

######  Proposition 2.22.

(Local property for TVS) Given a covering
$\mathfrak{g}\in\mathfrak{G}(\mathrm{Fin}_{\tau}({}^{\star}{E}))$, closed for
$\mathbb{K}$-scalar multiplication. The following statements are equivalent:

  1. (i)

$\mathbf{m}(0)\subseteq\mathfrak{g}$;

  2. (ii)

$\mathrm{Fin}_{\tau}({}^{\star}{E})=\mathfrak{g}$.

  3. (iii)

$\mathfrak{g}$ is $\tau$-chromatic;

  4. (iv)

$\mathfrak{g}$ is $\approx_{\tau}$-saturated.

  5. (v)

There exists a neighborhood of $0$ in $\mathrm{idl~{}}\mathfrak{g}$;

  6. (vi)

Each finite monad has a base in $\mathrm{idl~{}}\mathfrak{g}$.

###### Proof.

(i) $\Rightarrow$ (ii) : Take $e\in\mathrm{Fin}({}^{\star}{E})$. Proposition
2.16 gives $\mathbf{m}_{\mathbb{K}}(0)\cdot
e\subseteq\mathbf{m}(0)\subseteq\mathfrak{g}$, then there exists a
$\alpha\in\mathbb{R}_{+}$ (combined principle of Cauchy) such that $\alpha
e\in\mathfrak{g}$. Since $\mathfrak{g}$ is closed for $\mathbb{K}$-scalar
multiplication, $e\in\mathfrak{g}$..

(ii) $\Rightarrow$ (iii), (iv): Trivial.

(iii), (iv) $\Rightarrow$ (i): Trivial.

(i) $\iff$ (v): The combined Cauchy principle.

(ii) $\iff$ (vi): Chromatic inclusion and the combined Cauchy principle. ∎

######  Example 2.23.

$E$ is locally bounded, i.e.
$\mathbf{m}(0)\subseteq\mathrm{Bdd}({}^{\star}{E})$ iff
$\mathrm{Fin}({}^{\star}{E})=\mathrm{Bdd}({}^{\star}{E})$, in this case there
exists a bounded $\lambda\in\Lambda$. For any $\lambda^{\prime}\in\Lambda$
there exists an $\alpha\in\mathbb{R}_{+}$ such that
$\alpha^{-1}\lambda\subseteq\lambda^{\prime}$. Hence,

$\mathbf{m}(0)=\bigcap_{\alpha\in\mathbb{R}_{+}}{}^{\star}(\alpha^{-1}\lambda)=\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\star}\lambda.$

Looking back at the I.o.G. definition of $\mathrm{Fin}({}^{\star}{E})$ with
galaxies of the form
$\bigcup_{\alpha_{\lambda}\in\mathbb{R}_{+}}{}^{\star}(\alpha_{\lambda}\lambda)$,
consider that for a bounded $\lambda$, this galaxy contains all the others so
that $\mathrm{Fin}({}^{\star}{E})$ is indeed a single galaxy.

Since ${}^{\sigma}{E}\subseteq\mathrm{pcomp}({}^{\star}{E})$ and the latter is
$\approx_{\tau}$-saturated,
$\mathrm{ns}({}^{\star}{E})\subseteq\mathrm{pns}({}^{\star}{E})$. Given any
$S\in\mathcal{P}_{\mathrm{fin}}(E)$ and $\lambda\in\Lambda$; there exists an
$\alpha\in\mathbb{R}_{+}$ such that $S+\lambda\subseteq\alpha\lambda$, hence
$\mathrm{pns}({}^{\star}{E})\subseteq\mathrm{Fin}({}^{\star}{E})$. Therefore
we get

$\begin{gathered}\mathrm{qs}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})\subseteq\mathrm{pns}({}^{\star}{E})\subseteq\mathrm{Fin}({}^{\star}{E});\\\
\mathrm{qs}({}^{\star}{E})\subseteq\mathrm{comp}({}^{\star}{E})\subseteq\mathrm{pcomp}({}^{\star}{E})\subseteq\mathrm{Bdd}({}^{\star}{E}).\\\
\end{gathered}$

Hence, closed convex balanced hulls of finite sets are compact, compact sets
are precompact and precompact sets are bounded. Convergent monads are Cauchy
and Cauchy monads are finite. We may now derive special properties of
(subspaces of) topological vector spaces from chromatic inclusions that do not
generally hold, each with their equivalent characteristics:

  * •

$\mathrm{Bdd}({}^{\star}{E})\subseteq\mathrm{pns}({}^{\star}{E})$. Bounded
sets are precompact. Bounded monads are totally finite. Bounded ultramonads
are Cauchy. Pseudo-Henson-Moore.

  * •

$\mathrm{Fin}({}^{\star}{E})\subseteq\mathrm{pns}({}^{\star}{E})$. Finite
monads are totally finite. Finite ultramonads are Cauchy. Henson-Moore.
Implies Pseudo-Henson-Moore.

  * •

$\mathrm{pcomp}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})$. Precompact
sets are compact. Precompact monads have an accumulation point. Pseudo-
complete.

  * •

$\mathrm{pns}({}^{\star}{E})\cap\mathrm{Bdd}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})$.
Bounded sets are complete.333A set $V\subseteq E$ is complete if
$\mathrm{ns}_{V}({}^{\star}{V})\subseteq\mathrm{pns}({}^{\star}{V}).$ Closed
subsets of a complete set are complete.. Bounded totally finite monads have an
accumulation point. Bounded Cauchy monads are convergent. Quasi-complete
Implies pseudo-complete.

  * •

$\mathrm{pns}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})$. Totally
finite monads have an accumulation point. Cauchy monads are convergent.
Complete. Implies quasi-complete.

  * •

$\mathrm{Bdd}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})$. Bounded sets
are compact. Bounded monads have an accumulation point. Bounded ultramonads
are convergent. Heine-Borel. Implies Quasi-complete and Pseudo-Henson-Moore.

  * •

$\mathrm{Fin}({}^{\star}{E})\subseteq\mathrm{ns}({}^{\star}{E})$. Finite
monads have an accumulation point. Finite ultramonads converge. Henson-Moore-
Complete. Implies Heine-Borel, Henson-Moore and complete.

We immediately see that a set is compact iff it is precompact and complete.

######  Proposition 2.24.

The following statements are equivalent:

  1. (i)

$\mathbf{m}_{\tau}(0)\subseteq\mathrm{comp}_{\tau}({}^{\star}{E})$, i.e. $E$
is locally compact;

  2. (ii)

$\mathbf{m}_{\tau}(0)\subseteq\mathrm{pcomp}_{\tau}({}^{\star}{E})$, i.e. $E$
is locally precompact.

  3. (iii)

$\mathbf{m}_{\tau}(0)\subseteq\mathrm{qs}({}^{\star}{E})$;

  4. (iv)

$E$ is finite dimensional;

In this case
$\mathrm{Fin}({}^{\star}{E})=\mathrm{Bdd}({}^{\star}{E})=\mathrm{pns}({}^{\star}{E})=\mathrm{pcomp}({}^{\star}{E})=\mathrm{ns}({}^{\star}{E})=\mathrm{comp}({}^{\star}{E})=\mathrm{qs}({}^{\star}{E})$.
In particular, $E$ is Henson-Moore-Complete.

###### Proof.

(i) $\Rightarrow$ (ii): Trivial.

(ii) $\Rightarrow$ (iii): Let $\lambda$ be a precompact neighborhood of $0$.
Hence, there exists an $S\in\mathcal{P}_{\mathrm{fin}}(E)$ so that recursively

$\begin{split}\lambda&\subseteq S+\frac{\lambda}{2}\subseteq
S+\frac{1}{2}(S+\frac{\lambda}{2})\subseteq\mathrm{cobal}(S)+\frac{\lambda}{4}\\\
&\subseteq....\subseteq\mathrm{cobal}(S)+\frac{\lambda}{2^{n}}~{}~{}(\forall
n\in\mathbb{N})\Rightarrow\lambda\subseteq\overline{\mathrm{cobal}}(S).\end{split}$

(iii) $\iff$ (iv): The galaxy with base
$\\{\mathrm{span}(S)\\}_{S\in\mathcal{P}_{\mathrm{fin}}({}^{\star}{E})}$
(finite dimensional subspaces) is equal to
${}^{\star}\mathbb{K}\cdot\mathrm{qs}({}^{\star}{E})$, which is equal to
${}^{\star}{E}$ if $E$ is finite-dimensional. If $E$ is not finite-
dimensional, then for any finite $S\subseteq E$ there exists an
$e\in{\mathrm{span}(S)}^{\mathfrak{c}}$, then for
$\epsilon\in\mathbf{m}_{\mathbb{K}}(0)$, we have $\epsilon e\in\mathbf{m}(0)$.
Since $\mathbf{m}(0)\nsubseteq\mathrm{span}(S)$ and $S$ was arbitrary,
$\mathbf{m}(0)\nsubseteq{}^{\star}\mathbb{K}\cdot\mathrm{qs}({}^{\star}{E})$.

(iii) $\Rightarrow$ (i): Since
$\mathrm{ns}({}^{\star}{E})=\mathrm{qs}({}^{\star}{E})$ is a galaxy, then
$\mathrm{ns}({}^{\star}{E})=\mathrm{comp}({}^{\star}{E})$. ∎

######  Corrolary 2.25.

The only topology on Hausdorff finite dimensional spaces is the Euclidean
topology.

###### Proof.

Indeed, if $E$ is finite-dimensional,
$\mathbf{m}(0)=\mathbf{m}(0)\cap\mathrm{qs}({}^{\star}{E})$, the
$\mathbf{m}_{\mathbb{K}}(0)$-span of ${}^{\sigma}{E}$, which is uniquely
defined. ∎

###  2.3 Completion

We can identify $E$ with the quotient space
$\mathrm{ns}({}^{\star}{E})/\mathbf{m}(0)$, then $\mathrm{st~{}}\cdot$ is
identified with the projection map. Consider now
$\hat{E}:=\mathrm{pns}({}^{\star}{E})/\mathbf{m}(0)$ and let
$\hat{\mathrm{st}}~{}\cdot$ be the projection map for this quotient space.
Then $E$ is identified with a subspace of $\hat{E}$ and $\mathrm{st~{}}\cdot$
is the restriction of $\hat{\mathrm{st}}~{}\cdot$ to
$\mathrm{ns}({}^{\star}{E})$. Now, consider the map

$c:\mathcal{P}(E)\rightarrow\mathcal{P}(\hat{E}):V\rightarrow\hat{\mathrm{st}}~{}{}^{\star}{V}.$

Since $c$ is order-preserving, ${}^{\star}{c}$ is $\dagger$-extendable; so let

$\mathbf{m}_{\hat{E}}(0):={}^{\dagger}{c}(\mathbf{m}_{E}(0)).$

Since projections are linear, $c$ is linear. Then it is clear that
$\mathbf{m}_{\hat{E}}(0)$ must be closed for addition and
${}^{\star}\mathbb{K}_{\mathfrak{f}}$-scalar multiplication. Furthermore,
since $c(E)=\hat{E}$, we find for any $\epsilon\in\mathbf{m}_{\mathbb{K}}(0)$
that

$\epsilon\cdot{}^{\sigma}{\hat{E}}=\epsilon\cdot{}^{\dagger}{c}({}^{\sigma}E)={}^{\dagger}{c}(\epsilon\cdot{}^{\sigma}{E})\subseteq{}^{\dagger}{c}(\mathbf{m}_{E}(0))=\mathbf{m}_{\hat{E}}(0).$

Hence, the monad is linear and defines a topology $\hat{\tau}$ compatible with
the vector structure. Since any $e\in\mathrm{pns}({}^{\star}{E})$ is
$\tau$-separated from the origin, there exists a $\lambda\in\Lambda$ that is
$\tau$-separated from $e$ which implies that $\hat{\mathrm{st}}~{}e$ is not in
$\mathbf{m}_{\hat{E}}(0)$, hence $\hat{\tau}$ is Hausdorff. Now for any
$V\subseteq E$, $c(V)\cap E=\mathrm{st~{}}{}^{\star}{V}=\overline{V}$, hence
$\mathbf{m}_{\hat{E}}(0)\cap E=\mathbf{m}_{E}(0)$, i.e. $\hat{\tau}$ induces
$\tau$ on $E$. If $e\in\mathrm{pns}({}^{\star}\hat{E})$, then for any
$\lambda\in\Lambda$,
$e\in{}^{\sigma}\hat{E}+c(\lambda)={}^{\dagger}{c}({}^{\sigma}{E}+\lambda)$,
i.e. applying the Cauchy principle to each of these galaxies, there exist
$(e_{\lambda})_{e\in\Lambda}$ in $E$ such that
$e\in\cap_{\lambda}c(e_{\lambda}+\lambda)={}^{\dagger}{c}(\mathbf{m}_{E}(e^{\prime}))=\mathbf{m}_{\hat{E}}(e^{\prime})$
for some $e^{\prime}\in\mathrm{pns}({}^{\star}{E})$. Then
$e\approx\hat{\mathrm{st}}~{}e^{\prime}$ so that
$e\in\mathrm{ns}({}^{\star}\hat{E})$. In other words, $\hat{E}$ is complete.
Then $\hat{\mathrm{st}}_{\tau}~{}\cdot$ coincides with
$\mathrm{st}_{\hat{\tau}}~{}\cdot$ on ${}^{\star}{E}$ and $c$ coincides on $E$
with closure in $\hat{E}$, but then $\overline{E}^{\hat{\tau}}=\hat{E}$.
Finally, it is clear that $\hat{E}$ is embeddable in any complete space
containing $E$, so that $\hat{E}$ is (up to an isomorphism) the unique
completion of $E$.

We can use the $\hat{\mathrm{st}}~{}\cdot$ map to generate weaker forms of
completions:
$\hat{\mathrm{st}}~{}\mathrm{Bdd}({}^{\star}{E})\cap\mathrm{pns}({}^{\star}{E})$
is the quasi-completion; $\hat{\mathrm{st}}~{}\mathrm{pcomp}({}^{\star}{E})$
the pseudo-completion.

######  Lemma 2.26.

Given an internal $A\subseteq E$ such that $A$ is sfinite. Then
$\mathrm{st~{}}A$ is finite.

###### Proof.

There exist $(\gamma_{\lambda})_{\lambda}\in\mathbb{K}^{\Lambda}$ such that
$A\subseteq\cap_{\lambda\in{}^{\sigma}\Lambda}{}^{\star}(\gamma_{\lambda}\lambda)$.
Then $\mathrm{st~{}}A\\\
\subseteq\mathrm{st~{}}\cap_{\lambda\in{}^{\sigma}\Lambda}{}^{\star}(\gamma_{\lambda}\lambda)\subseteq\cap_{\lambda\in{}^{\sigma}\Lambda}\gamma_{\lambda}(\mathrm{st~{}}{}^{\star}\lambda)=\cap_{\lambda\in\Lambda}\gamma_{\lambda}\overline{\lambda}$.
∎

######  Lemma 2.27.

Given an internal $A\subseteq E$ such that $A$ is sprecompact. Then
$\mathrm{st~{}}A$ is precompact.

###### Proof.

There exist
$(S_{\lambda})_{\lambda}\in\mathcal{P}_{\mathrm{Fin}}(E)^{\Lambda}$ such that
$A\subseteq\cap_{\lambda\in{}^{\sigma}\Lambda}S_{\lambda}+{}^{\star}\lambda$.
Then
$\mathrm{st~{}}A\subseteq\cap_{\lambda\in{}^{\sigma}\Lambda}S_{\lambda}+\mathrm{st~{}}{}^{\star}\lambda=\cap_{\lambda\in\Lambda}S_{\lambda}+\overline{\lambda}$.
∎

The two lemma’s above may be applied to
$\hat{\mathrm{st}}~{}\cdot=\mathrm{st}_{\hat{\tau}}~{}\cdot$ as well. In the
case that $A$ is sprecompact, $\hat{\mathrm{st}}~{}A$ is compact.

######  Lemma 2.28.

Given an internal $A\subseteq\mathrm{ns}({}^{\star}{E})$. Then
$\mathrm{st~{}}A=\hat{\mathrm{st}}~{}A$ is compact.

###### Proof.

Since $A$ is sprecompact, lemma 2.27 tells us that $\mathrm{st~{}}A$ is
precompact. Furthermore, $\mathrm{st~{}}A$ is closed by lemma 1.69. Since
$\mathrm{st~{}}\cdot=\hat{\mathrm{st}}~{}\cdot$ on
$\mathrm{ns}({}^{\star}{E})$, $\mathrm{st~{}}A$ is in fact closed in $\hat{E}$
and therefore complete. ∎

###  2.4 Locally convex spaces

$E$ is locally convex iff $\mathbf{m}_{\tau}(0)$ is ⋆convex. In this case we
may assume that $\Lambda$ is a base existing out of absorbing, open, balanced
and convex sets. Furthermore, we will in that case interpret $\Lambda$ as a
family of continuous semi-norms, using the Minkowski-function

$\|e\|_{\lambda}:=\inf\\{r\in\mathbb{R}_{+}:e\in r\lambda\\}.$

such that the open (respectively closed) unit ball of the seminorm is
precisely $\lambda$ (respectively $\overline{\lambda}$). Then
$e\approx_{\tau}0\iff\forall\lambda\in\Lambda_{\tau}:\|e\|_{\lambda}\approx
0$; a single norm is sufficient (i.e. the space is normable) (induced by a
norm) iff it is locally bounded.

We will assume that $E$ is locally convex in this section.

######  Proposition 2.29.

$\overline{\mathrm{cobal}}(\mathrm{Fin}_{\tau}({}^{\star}{E}))=\mathrm{Fin}_{\tau}({}^{\star}{E}).$

###### Proof.

Since
$\overline{\mathrm{cobal}}({}^{\star}\mathbb{K}_{\mathfrak{f}}\cdot\overline{\lambda})={}^{\star}\mathbb{K}_{\mathfrak{f}}\cdot\overline{\lambda}$,
this follows from lemma 1.31. ∎

######  Corrolary 2.30.

The closed convex balanced hull of a bounded set is bounded, hence
$\overline{\mathrm{cobal}}({\mathrm{Bdd}_{\tau}({}^{\star}{E})})=\mathrm{Bdd}_{\tau}({}^{\star}{E})$.

######  Proposition 2.31.

$\overline{\mathrm{cobal}}(\mathrm{pns}_{\tau}({}^{\star}{E}))=\mathrm{pns}_{\tau}({}^{\star}{E}).$

###### Proof.

Since
${}^{\sigma}{E}\subseteq\mathrm{qs}({}^{\star}{E})\subseteq\mathrm{pns}({}^{\star}{E})$,

$\mathrm{pns}({}^{\star}{E})=\bigcap_{\lambda\in\Lambda}\mathrm{qs}({}^{\star}{E})+\overline{\lambda}.$

Now
$\overline{\mathrm{cobal}}(\mathrm{qs}({}^{\star}{E})+\overline{\lambda})=\mathrm{qs}({}^{\star}{E})+\overline{\lambda}$.
Indeed, the sum of two ⋆convex, ⋆balanced sets is ⋆convex and ⋆balanced and
because of lemma 2.13 this galaxy is also ⋆closed. Hence, the result follows
from lemma 1.31. ∎

######  Corrolary 2.32.

The closed convex balanced hull of a precompact set is precompact, hence
$\overline{\mathrm{cobal}}({\mathrm{pcomp}_{\tau}({}^{\star}{E})})=\mathrm{pcomp}_{\tau}({}^{\star}{E})$.

######  Proposition 2.33.

$\overline{\mathrm{cobal}}(\mathrm{ns}_{\tau}({}^{\star}{E}))=\mathrm{cobal}(\mathrm{comp}_{\tau}({}^{\star}{E}))+\mathbf{m}(0)$.

###### Proof.

Take $e\in{}^{\star}{\mathrm{cobal}}(\mathrm{ns}({}^{\star}{E}))$. There exist
internal sequences $(e_{n})_{n\leq\omega}$ in $\mathrm{ns}({}^{\star}{E})$ en
$(\alpha_{n})_{n\leq\omega}$ in ${}^{\star}\mathbb{K}$ such that
$\sum_{i=0}^{\omega}|\alpha_{i}|\leq 1$ and
$e=\sum_{i=0}^{\omega}\alpha_{i}e_{i}$. Let
$U:=\mathrm{st~{}}\\{e_{n}\\}_{n\leq\omega}$, a compact set due to lemma 2.28.
Now $\\{e_{n}\\}_{n\leq\omega}$ is sprecompact, so that because of infinite
overspill there exists a $\lambda\in{}^{\star}\Lambda_{\infty}$ and
$S_{\lambda}\in{}^{\star}\mathcal{P}_{\mathrm{Fin}}(V)$ such that
$\\{e_{n}\\}_{n\leq\omega}\subseteq S_{\lambda}+\lambda$. For each $e_{i}$ we
find an $e^{\prime}_{i}\in S_{\lambda}\subseteq{}^{\star}{V}$ such that
$e_{i}-e^{\prime}_{i}\in\lambda$, hence $e_{i}\approx e^{\prime}_{i}$. Let
$e^{\prime}:=\sum_{i=0}^{\omega}\alpha_{i}(e^{\prime}_{i})$. Then
$e^{\prime}\in{}^{\star}\mathrm{cobal}(V)\subseteq\mathrm{cobal}(\mathrm{comp}({}^{\star}{E}))$.For
arbitrary $\lambda\in\Lambda$ we find

$\|e-e^{\prime}\|_{\lambda}\leq\sum_{i=0}^{\omega}|\alpha_{i}|\|e_{i}-e^{\prime}_{i}\|_{\lambda}\leq(\sum_{i=0}^{\omega}|\alpha_{i}|)\max\|e_{i}-e^{\prime}_{i}\|_{\lambda}\approx
0$

so that $e^{\prime}\approx e$, hence
$e\in{}^{\star}\mathrm{cobal}(V)+\mathbf{m}(0)\subseteq\mathrm{cobal}(\mathrm{comp}_{\tau}({}^{\star}{E}))+\mathbf{m}(0)$.

Now since
$\mathbf{m}(0)\subseteq\mathrm{cobal}(\mathrm{ns}_{\tau}({}^{\star}{E}))$ and
the latter is closed for addition,
$\mathrm{cobal}(\mathrm{ns}_{\tau}({}^{\star}{E}))$ is
$\approx_{\tau}$-saturated, and thus also ⋆closed; from which the result
follows. ∎

######  Corrolary 2.34.

The closed convex balanced hulls of compact sets are compact ($E$ has the
convex convex compactness property) iff $\mathrm{ns}({}^{\star}{E})$ is
⋆convex iff $\mathrm{comp}({}^{\star}{E})$ is ⋆convex.

###### Proof.

Note that $\mathrm{ns}({}^{\star}{E})$ and $\mathrm{comp}({}^{\star}{E})$ are
always ⋆balanced and ⋆closed. Then if
$\mathrm{comp}({}^{\star}{E})=\overline{\mathrm{cobal}}(\mathrm{comp}({}^{\star}{E}))$;
$\overline{\mathrm{cobal}}(\mathrm{ns}({}^{\star}{E}))=\mathrm{cobal}(\mathrm{comp}({}^{\star}{E})+\mathbf{m}(0))=\overline{\mathrm{cobal}}(\mathrm{comp}({}^{\star}{E}))+\mathbf{m}(0)=\mathrm{ns}({}^{\star}{E})$.
On the other hand, if
$\mathrm{ns}({}^{\star}{E})=\mathrm{cobal}(\mathrm{ns}({}^{\star}{E}))$,
convex hulls of compact sets are clearly compact. ∎

The convex compactness property may be understood as a weak form of
completeness. The smallest subspace of $\hat{E}$ containing $E$ that has the
convex compactness property is
$\hat{E}_{c}=\hat{\mathrm{st}}~{}\overline{\mathrm{cobal}}(\mathrm{ns}({}^{\star}{E}))$.
Then for a set $V\subseteq E$, we have
${}^{\star}{V}\subseteq\mathrm{cobal}(\mathrm{ns}({}^{\star}{E}))$ iff all of
its monads converge in $\hat{E}_{c}$. Furthermore, since
$\overline{\mathrm{cobal}}(\mathrm{comp}({}^{\star}{E}))\subseteq\mathrm{pcomp}({}^{\star}{E})$,
pseudo-completeness (and quasi-completeness) are sufficient conditions for the
convex compactness property.

In general it is not so that sets contain a greatest balanced, convex subset
(for the $\subseteq$ order). In the case of a galaxy that is closed for
addition there appears to be such a thing as a greatest balanced, convex
subgalaxy.

######  Lemma 2.35.

Given $\mathfrak{g}\in\mathfrak{G}({E})$ that is closed for addition. Given
two balanced convex sets $V,W\in\mathrm{idl~{}}\mathfrak{g}$;
$\mathrm{cobal}({}^{\star}{V}\cup{}^{\star}{W})\subseteq\mathfrak{g}$.

###### Proof.

Since $V$ and $W$ are balanced and convex, $\mathrm{cobal}(V\cup W)\subseteq
V+W$. Since $\mathfrak{g}$ is closed for addition we must have
${}^{\star}\mathrm{cobal}(V\cup
W)\subseteq{}^{\star}{V}+{}^{\star}{W}\subseteq\mathfrak{g}$. ∎

Hence, there exists a galaxy $\mathfrak{g}^{\prime}\subseteq\mathfrak{g}$ that
takes as its base all ⋆convex, ⋆balanced sets of
$\mathrm{idl~{}}\mathfrak{g}$. Then $\mathfrak{g}^{\prime}$ must contain any
⋆convex, ⋆balanced subgalaxie of $\mathfrak{g}$. If $\mathfrak{g}$ is ⋆closed,
this works for ⋆convex, ⋆balanced, ⋆closed sets. In particular for compact
points, the following definition makes sense:

######  Definition 2.36.

The convex-compact points is the galaxy of points in ${}^{\star}{E}$ which are
contained in a convex, balanced, compact subset of $E$. We denote:

$\mathrm{coco}({}^{\star}{E}):=\bigcup\\{{}^{\star}{V}:{}^{\star}{V}\subseteq\mathrm{ns}({}^{\star}{E}),\overline{\mathrm{cobal}}({V})=V\\}.$

Note that $\mathrm{coco}({}^{\star}{E})=\mathrm{comp}({}^{\star}{E})$ iff $E$
has the convex compactness property.

###  2.5 Linear maps

In this section we will work with a second topological vector space
$(G,\tau^{\prime})$ where $\mathbf{m}_{G,\tau^{\prime}}(0)$ has a base
$\mathrm{K}$ with the same assumptions as $\Lambda$.

For linear maps, continuity is determined in the origin. This means that a
linear map $\psi:{}^{\star}{E}\rightarrow{}^{\star}{G}$ is ccontinuous iff
$\psi(\mathbf{m}_{E}(0))\subseteq\mathbf{m}_{G}(0)$, as can easily be proven
using linearity. There appears to be a second characterization:

######  Proposition 2.37.

Given a linear map $\psi:{}^{\star}{E}\rightarrow{}^{\star}{G}$. Then $\psi$
is ccontinuous iff
$\psi(\mathrm{Fin}({}^{\star}{E}))\subseteq\mathrm{Fin}({}^{\star}{G})$.

###### Proof.

First assume $\psi$ is ccontinuous. Then since

$\mathbf{m}_{\mathbb{K}}(0)\cdot\psi(\mathrm{Fin}({}^{\star}{E}))=\psi(\mathbf{m}_{\mathbb{K}}(0)\cdot\mathrm{Fin}({}^{\star}{E}))\subseteq\psi(\mathbf{m}_{E}(0))\subseteq\mathbf{m}_{G}(0)$

this follows from proposition 2.16. Now assume otherwise that
$\psi(\mathrm{Fin}({}^{\star}{E}))\subseteq\mathrm{Fin}({}^{\star}{G})$. Since
$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathbf{m}_{E}(0)$ is the monad of
neighborhoods of $0$ in $E$ multiplied with neighborhoods of $0$ in
$\mathbf{m}_{\mathbb{K}}(0)$, it is in fact equal to $\mathbf{m}_{E}(0)$.
Then, using proposition 2.16 again,

$\begin{split}\psi(\mathbf{m}_{E}(0))&=\psi(\mathbf{m}_{\mathbb{K}}(0)\cdot\mathbf{m}_{E}(0))=\mathbf{m}_{\mathbb{K}}(0)\cdot\psi(\mathbf{m}_{E}(0))\\\
&\subseteq\mathbf{m}_{\mathbb{K}}(0)\cdot\psi(\mathrm{Fin}({}^{\star}{E}))\subseteq\mathbf{m}_{\mathbb{K}}(0)\cdot\mathrm{Fin}({}^{\star}{G})\subseteq\mathbf{m}_{G}(0)\end{split}$

∎

This means that ccontinuity (including scontinuity for internal maps and
continuity for standard maps) is equivalent with mapping finite monads onto
finite monads. For internal maps, this implies that bounded sets are mapped
onto bounded sets, i.e.
$\psi(\mathrm{Bdd}({}^{\star}{E}))\subseteq\mathrm{Bdd}({}^{\star}{G})$ (since
galaxies are mapped onto galaxies). The latter however does not in general
imply ccontinuity (it does so obviously when
$\mathrm{Bdd}({}^{\star}{G})=\mathrm{Fin}({}^{\star}{G})$, i.e. the space is
locally bounded).

######  Proposition 2.38.

Given a linear map $\theta:E\rightarrow G$. Then $\theta(\mathbf{m}(0))$ is a
linear monad for the subspace $\theta(E)$ of $G$.

###### Proof.

Because of corollary 1.28, $\theta(\mathbf{m}(0))$ is a monad in $G$ . Due to
linearity of $\theta$, $\theta(\mathbf{m}(0))$ is closed for addition and
$\mathbb{K}_{\mathfrak{f}}$-scalar multiplication. We have cabsorption since
$\mathbf{m}_{\mathbb{K}}(0)\cdot\theta(E)=\theta(\mathbf{m}(0)\cdot
E)\subseteq\theta(\mathbf{m}(0))$. ∎

Hence, a surjective linear map $E\rightarrow G$ defines a topology on $G$. If
$G$ is a TVS, $\theta$ is a linear isomorphism and
$\theta(\mathbf{m}_{E}(0))=\mathbf{m}_{F}(0)$, i.e. $\theta$ is a
homeomorfism, then $\theta$ is a TVS-isomorfism.

Given a subspace $H\subseteq E$, and $\pi_{H}:E\rightarrow E/H$ is the
projection to quotient space $E/H$, then $\pi_{H}(\mathbf{m}_{E}(0))$ is a
linear monad of $E/H$. Then $\mathbf{m}_{E/H}(0):=\pi_{H}(\mathbf{m}_{E}(0))$
defines a natural topology on $E/H$. Under this topology $\pi_{H}$ is open and
continuous. If $H$ is closed, then
$\pi_{H}({H}^{\mathfrak{c}})=E/H\setminus\\{0\\}$ is open and $E/H$ is
Hausdorff.

######  Definition 2.39.

A family $\\{\psi_{j}\\}_{j\in J}$ of linear maps $E\rightarrow G$ is
equicontinuous if for each $\kappa\in\mathrm{K}$ there exists a
$\lambda\in\Lambda$ such that $\psi_{j}(\lambda)\subseteq\kappa$ for all $j\in
J$.

######  Proposition 2.40.

A family $\\{\psi_{j}\\}_{j\in J}$ of linear maps $E\rightarrow G$ is
equicontinuous iff for each $j\in{}^{\star}{J}$, $\psi_{j}$ is ccontinuous
(i.e. scontinuous).

###### Proof.

Suppose that for each $j\in{}^{\star}{J}$, $\psi_{j}$ is ccontinuous. Take any
$\kappa\in\mathrm{K}$. Then

$\mathbf{m}_{E}(0)\subseteq\bigcap_{j\in{}^{\star}{J}}\psi_{j}^{-1}({}^{\star}{\kappa}).$

so that due to the Cauchy principle there exists a $\lambda\in\Lambda$ such
that ${}^{\star}{\lambda}\subseteq\psi_{j}^{-1}({}^{\star}{\kappa})$ for each
$j\in J$. Assume otherwise that the family is equicontinuous and take
$j\in{}^{\star}{J}$. There exist for each $\kappa\in\mathrm{K}$ a
$\lambda_{\kappa}\in\Lambda$ such that due to the transferring principle
$\psi_{j}(\lambda_{\kappa})\subseteq\kappa$, hence

$\psi_{j}(\mathbf{m}_{E}(0))\subseteq\psi_{j}(\bigcap_{\kappa\in\mathrm{K}}{}^{\star}{\lambda_{\kappa}})\subseteq\bigcap_{\kappa\in\mathrm{K}}\psi_{j}({}^{\star}{\lambda_{\kappa}})\subseteq\bigcap_{\kappa\in\mathrm{K}}{}^{\star}\kappa=\mathbf{m}_{G}(0).$

∎

##  3 Duality Theory

###  3.1 Dual pairs and poles

The following lemma will be assumed without proof; it is Hahn-Banach for
finite dimensions and can be proven inductively by a well known argument:

######  Lemma 3.1.

Given an finite dimensional space $E$ with euclidean norm $\|.\|$, a subspace
$H$ and a linear functional $\psi$ on $H$ with $\|\psi\|=\sup_{\|e\|\leq
1}\psi(e)\leq 1$. There exists a linear functional $\hat{\psi}$ on $E$ such
that ${\left.\kern-1.2pt\hat{\psi}\vphantom{\big{|}}\right|_{H}}=\psi$ and
$\|\hat{\psi}\|=1$.

######  Theorem 3.2.

(Hahn Banach) Given Hausdorff TVS $E$, a subspace $H$, a closed absorbing
balanced convex set $D$ and a linear functional $\psi$ on $H$ such that
$\sup_{e\in D\cap H}|\psi(e)|\leq 1$. There exists a linear functional
$\hat{\psi}$ on $E$ such that
${\left.\kern-1.2pt\hat{\psi}\vphantom{\big{|}}\right|_{H}}=\psi$ and
$\sup_{e\in D}|\psi(e)|\leq 1$.

###### Proof.

Consider the set $\mathcal{R}$ of finite dimensional subspaces of $E$;
directed for $\subseteq$. For any $R\in\mathcal{R}$, since $E$ is Hausdorff
the induced topology is Euclidean. Now take any
$R\in{}^{\star}\mathcal{R}_{\infty}$, then ${}^{\sigma}{E}\subseteq R$. Let
$D_{R}:={}^{\star}{D}\cap R$, a ⋆balanced, ⋆convex and ⋆closed set. Because of
the transfer principle, $D_{R}$ is of the form $\\{e\in R:\|e\|\leq 1\\}$ for
some ⋆Euclidean norm. Apply lemma 3.1, using the transfer principle, on $R$
with subspace $H_{R}:={}^{\star}{H}\cap R$ and
$\psi_{R}:={\left.\kern-1.2pt{}^{\star}\psi\vphantom{\big{|}}\right|_{H_{R}}}$.
Then there is a linear functional $\hat{\psi_{R}}$ on $R$ such that
$\sup_{e\in D_{R}}|\hat{\psi_{R}}(e)|\leq 1$. Since $D$ is absorbing, for any
$e\in{}^{\sigma}{E}$ there exists an $\alpha\in\mathbb{R}_{+}$ such that
$e\in{}^{\sigma}{(\alpha D)}\subseteq\alpha D_{R}$, hence
$\hat{\psi_{R}}(e)\leq\alpha\sup_{e\in D}\hat{\psi_{R}}(D_{R})\leq\alpha$.
Hence $\hat{\psi_{R}}(e)$ is finite and we can set
$\hat{\psi}(e):=\mathrm{st~{}}\hat{\psi_{R}}(e).$ ∎

######  Corrolary 3.3.

Given a Hausdorff TVS $E$, a closed absorbing balanced convex $D\subseteq E$
and a point $x\in{D}^{\mathfrak{c}}$. There exists a linear functional $\psi$
on $E$ such that $\sup_{e\in D}|\psi(e)|\leq 1$ and $|\psi(x)|>1$.

###### Proof.

Let $H=\mathbb{K}\cdot x$. Then there exists an $r\in\left]0,1\right[$ such
that $H\cap D=\\{\alpha x:|\alpha|\leq r\\}$. Let $\psi$ be the linear
functional on $H$ defined by $\psi(\alpha x):=\alpha r^{-1}$. ∎

A dual pair $(E,F)$ are two vector spaces $E$ and $F$ over a field
$\mathbb{F}$ and a bilinear mapping $\langle,\rangle:E\times
F\rightarrow\mathbb{F}$ such that both spaces are separated by each other:

$\forall e\in E\setminus\\{0\\}:\exists f\in F:\langle f,e\rangle\neq
0;~{}~{}~{}\forall f\in F\setminus\\{0\\}:\exists e\in E:\langle
f,e\rangle\neq 0.$

For instance, a vector space $E$ over $\mathbb{K}$ and its algebraic dual
$E^{*}$ form a dual pair. In fact, fixing $E$, any possible $F$ may be
identified with a subset of $E^{*}$ using the isomorphic embedding
$f\rightarrow\langle\cdot,f\rangle$ . Note that the definition of a dual pair
is symmetric, so that we can flip the roles of $E$ and $F$ in any definition
or proposition. Given a dual pair $(E,F)$, Given $V\subseteq E$, its polar is

$V^{\circ}:=\\{f\in F:\sup_{e\in V}|\langle e,f\rangle|\leq 1\\}\subseteq F.$

The bipolar of $V$ is $V^{\circ\circ}:=(V^{\circ})^{\circ}$. It can be proven
that for $U,V\subseteq E$:

  * •

$U\subseteq V\Rightarrow U^{\circ}\supseteq V^{\circ}$ hence $(U\cap
V)^{\circ}\supseteq U^{\circ}\cup V^{\circ}$;

  * •

$(U\cup V)^{\circ}=U^{\circ}\cap V^{\circ}$;

  * •

$V\subseteq V^{\circ\circ}$ hence $V^{\circ\circ\circ}=V^{\circ}$.

The non-standard extensions ${}^{\star}{E}$ and ${}^{\star}{F}$ form a dual
pair of vector spaces over ${}^{\star}{\mathbb{K}}$. Because of the transfer
principle ${}^{\star}{(V^{\circ})}=({}^{\star}{V})^{\circ}$, hence we may
write ${}^{\star}{V^{\circ}}$ unambiguously.

######  Lemma 3.4.

Given a dual pair $(E,F)$ and a set $A\subseteq{}^{\star}{E}$ closed for
${}^{\sigma}\mathbb{K}$-scalar multiplication. Then

$A^{\circ}=\\{f\in{}^{\star}{F}:|\langle e,f\rangle|\approx 0,\forall e\in
A\\}.$

###### Proof.

Take $f\in{}^{\star}{F}$. Then $f\in A^{\circ}$ iff $|\langle f,\alpha
e\rangle|\leq 1$ for any $e\in A$ and $\alpha\in\mathbb{R}_{+}$, which is
equivalent with $|\langle f,e\rangle|\leq\frac{1}{\alpha}$. Since $\alpha$ was
arbitrary, $\langle f,e\rangle\approx 0$. ∎

In particular, if $E$ is a TVS then $f\in\mathbf{m}_{E}(0)^{\circ}$ iff
$e\approx_{\tau}0\Rightarrow\langle e,f\rangle\approx 0$, i.e. $f$ is
scontinuous (identified as en element of ${}^{\star}{F}^{*}$). If $E$ is a
TVS, then $E^{\prime}$ is the subset of $E^{*}$ containing the continuous
linear functionals, the topological dual. Then we have
${}^{\sigma}{E^{\prime}}={}^{\sigma}{E}^{*}\cap\mathbf{m}_{E}(0)^{\circ}$.

######  Lemma 3.5.

Given a dual pair $(E,F)$ and $A\subseteq{}^{\star}{F}$. Then $A^{\circ}$ is
⋆balanced and ⋆convex.

###### Proof.

Let $e_{1},e_{2},\dots,e_{\omega}$ be an internal sequence in $A^{\circ}$ and
$\alpha_{1},\dots,\alpha_{\omega}$ and internal sequence
${}^{\star}{\mathbb{K}}$ such that $\sum_{i=1}^{\omega}|\alpha_{i}|\leq 1$
then we get for $f\in A$:

$|\langle
f,\sum_{i=1}^{\omega}\alpha_{i}e_{i}\rangle|\leq\sum_{i=1}^{\omega}|\alpha_{i}||\langle
f,e_{i}\rangle|\leq 1.$

∎

######  Lemma 3.6.

Suppose $E$ is a locally convex Hausdorff TVS. Given a dual pair $(E,F)$ and
$A\subseteq{}^{\star}{(F\cap{E}^{\prime})}$. Then $A^{\circ}$ is ⋆closed.

###### Proof.

Since $f$ is ⋆continuous, there is a $\lambda\in{}^{\star}\Lambda$ such that
$|\langle e^{\prime},f\rangle|\leq 1$ for any $e^{\prime}\in\lambda$. Take $e$
in the ⋆closure of $A^{\circ}$ and $f\in A$. For any
$\epsilon\in{}^{\star}\mathbb{R}_{+}$ there exists a $e_{\epsilon}\in
A^{\circ}$ such that $e-e_{\epsilon}\in\epsilon\lambda$. Then $|\langle
e,f\rangle|\leq|\langle e_{\epsilon},f\rangle|+|\langle
e-e_{\epsilon},f\rangle|\leq 1+\epsilon$. Since $\epsilon$ was arbitrary,
$e\in A^{\circ}$. ∎

######  Definition 3.7.

Given a dual pair $(E,F)$ where $E$ is a TVS. A set $V\subseteq E$ satisfies
the bipolar identity if $V^{\circ\circ}=\overline{\mathrm{cobal}}(V)$.

######  Lemma 3.8.

Suppose $E$ is a locally convex Hausdorff TVS. Given $F\subseteq E^{*}$.
Suppose any convex and balanced $V\subseteq E$ satisfies for the duality
$(E,E^{*})$

$(V^{\circ}\cap F)^{\circ}=\overline{V}.$

Then $(E,F)$ is a dual pair for which all sets in $E$ satisfy the bipolar
identity.

###### Proof.

We know $F$ is separated by $E$. Given $e\in E\setminus\\{0\\}$. There is a
$\lambda\in\Lambda$ such that $e\notin\overline{\lambda}$. The assumption
implies $e\notin(\lambda^{\circ}\cap F)^{\circ}$, i.e. there exists an
$f\in\lambda^{\circ}\cap F$ such that $\langle e,f\rangle>1$, in particular
$|\langle e,f\rangle|\neq 0$. Now in the duality $(E,F)$, given $W\subseteq E$
lemma 3.5 and order preservation imply that
$W^{\circ\circ}=\mathrm{cobal}{(W)}^{\circ\circ}$. But the assumption implies
$\mathrm{cobal}{(W)}^{\circ\circ}=\overline{\mathrm{cobal}}{(W)}$. ∎

######  Proposition 3.9.

Suppose $E$ is a locally convex Hausdorff TVS. Then $(E,E^{\prime})$ is a dual
pair for which all sets in $E$ satisfy the bipolar identity.

###### Proof.

We will apply lemma 3.8. Given a convex, balanced $V\subseteq E$. Suppose
$e\notin\overline{V}$. Because of regularity, there exists a
$\lambda\in\Lambda$ such that $\mathbf{m}(e)\cap(V+\lambda)=\emptyset$, i.e.
$e\notin\overline{V+\lambda}$. Since $\overline{V+\lambda}$ is closed, convex,
balanced and absorbing we can apply corollary 3.3, i.e. there exists a linear
functional $f\in E^{\prime}$ on $E$ such that $|f(e)|>1$ and
$f\in(\overline{V+\lambda})^{\circ}\subseteq\lambda^{\circ}\cap
V^{\circ}\subseteq E^{\prime}\cap V^{\circ}$. Hence, $e\notin(E^{\prime}\cap
V^{\circ})^{\circ}$. The opposite inclusion follows from lemma 3.6. ∎

By the transferring principle, all internal sets in the duality
$({}^{\star}{E},{}^{\star}{E^{\prime}})$ satisfy the bipolar identity. For
external sets, lemma’s 3.5 and 3.6 apply so that their bipolars are ⋆closed,
⋆convex, ⋆balanced sets; but they can be larger than the hulls. In many cases,
the bipolar identity will apply to external sets of interest too, with one
notable exception.

######  Lemma 3.10.

Given a dual pair $(E,F)$,
$\mathfrak{m}\in{}^{\filleddiamond}\mathfrak{M}({F})$, resp.
$\mathfrak{g}\in{}^{\filleddiamond}\mathfrak{G}({F})$ with
${}^{\filleddiamond}$base $(U_{j})_{j\in{}^{\star}{J}}$. Then
$\mathfrak{m}^{\circ}\in{}^{\filleddiamond}\mathfrak{G}({F})$, resp.
$\mathfrak{g}^{\circ}\in{}^{\filleddiamond}\mathfrak{M}({F})$ with base
$(U_{\phi}^{\circ})_{j\in{}^{\star}{J}}$.

###### Proof.

The polar mapping is order-reversing and $\dagger$-extendable. ∎

######  Lemma 3.11.

Given a dual pair $(E,F)$. Then
$({}^{\sigma}{E})^{\circ\circ}=\mathrm{qs}({}^{\star}{E})$.

###### Proof.

Take any $S\in\mathcal{P}_{\mathrm{fin}}(E)$. Note that $\mathrm{span}(S)$ is
closed for scalar multiplication, so that $\mathrm{span}(S)^{\circ}$ and
$\mathrm{span}(S)^{\circ\circ}$ are annihilators. Since $F$ separates
$\mathrm{span}(S)$, we have
$\\{{\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{\mathrm{span}(S)}}:f\in
F\\}=\mathrm{span}(S)^{*}.$ From this observation, applying well-known
principles of linear algebra, it is clear that
$\mathrm{span}(S)^{\circ\circ}=\mathrm{span}(S)$. Let $\\{e_{1},\dots
e_{n}\\}\subseteq S$ be a linearly independent basis of this space. Take $e\in
S^{\circ\circ}\subseteq\mathrm{span}(S)$, i.e. there exists
$(\alpha_{j})_{j\leq n}$ such that $e=\sum_{j=1}^{n}\alpha_{j}e_{j}$. Suppose
that $|\alpha_{k}|>1$ for some $k\leq n$. Applying again separation, there
must exist an $f\in F$ such that $\langle e_{j},f\rangle=0$ ($j\neq k$) and
$\langle e_{k},f\rangle=1$. Then $f\in S^{\circ}$ but $|\langle
e,f\rangle|=|\alpha_{k}|>1$, a contradiction. We conclude that
$S^{\circ\circ}=\mathrm{cobal}(S)$. Then the result follows from the fact that
the bipolar map is order-preserving and $\dagger$-extendable. ∎

######  Lemma 3.12.

Given a dual pair $(E,F)$ with $E$ a TVS. ${}^{\filleddiamond}$Monads and
${}^{\filleddiamond}$galaxies that have a base of sets that satisfy the
bipolar identity, satisfy the bipolar identity. In particular, for the dual
pair $(E,E^{\prime})$ ${}^{\filleddiamond}$monads and
${}^{\filleddiamond}$galaxies satisfy the bipolar identity.

###### Proof.

The bipolar mapping is order preserving and $\dagger$-extendable. If the
mapping coincides with $\overline{\mathrm{cobal}}$ on the sets in the base,
their $\dagger$-extensions must also coincide on the monad or galaxy. ∎

###  3.2 Dual topologies

In this section we work with a duality $(E,F)$ without any explicit topology
on $E$. If $\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$ is closed for
$\mathbb{K}$-scalar multiplication, then $\mathfrak{g}^{\circ}$ is closed for
$\mathbb{K}_{\mathfrak{f}}$-scalar multiplication and addition by lemma 3.4.
That means we set a topology (also denoted as $\mathfrak{g}$) on $E$ that
makes addition continuous, defined by
$\mathbf{m}_{\mathfrak{g}}(0):=\mathfrak{g}^{\circ}$, i.e. the polar topology
of $\mathfrak{g}$ or topology of uniform convergence on
$\mathfrak{g}$.444Recall that continuous addition is sufficient to use the
$\mathbf{m}$-equivalence notation from definition 2.6. We will in fact apply
this topology to $F^{*}$ in its entirety
($\mathbf{m}_{F^{*}}(0):=\mathfrak{g}^{\circ}$ where we take the polar for the
duality $(F^{*},F)$), even if it is only compatible with the vector structure
on $E$. This is the coarsest topology that makes every convergent
$\mathfrak{m}\in\mathfrak{M}({}^{\star}{E})$ uniformly convergent on
$\mathfrak{g}$, which is defined as (given $e\in E$):

$\mathfrak{m}\xrightarrow{\mathfrak{g}}e\iff\mathfrak{m}\subseteq
e+\mathfrak{g}^{\circ}\iff\forall V\in\mathrm{idl~{}}\mathfrak{g}:\exists
W\in\mathrm{fil~{}}\mathfrak{m}:W-e\subseteq V^{\circ}$

where we applied Cauchy principle on the second equivalence. Because of lemma
3.5 the topology is locally convex. For any base of $\mathfrak{g}$, its poles
are a base of closed, balanced, convex neighbourhoods of 0, i.e. of seminorms
in the polar topology. We will require additional conditions for this topology
to be compatible with the vector structure and Hausdorff.

######  Definition 3.13.

The weak topology or topology of pointwise convergence $\sigma(E,F)$ is
defined by

$\mathbf{m}_{E,\sigma}(0):=({}^{\sigma}{F})^{\circ}.$

######  Lemma 3.14.

$F=(E_{\sigma})^{\prime}$.

###### Proof.

We denote $\bullet$ as the polar for the duality $(E,E^{*})$. Since lemma 3.11
is independent of the dual space, we find:

${}^{\sigma}{(E_{\sigma})^{\prime}}={}^{\sigma}{E^{*}}\cap\mathbf{m}_{E,\sigma}(0)^{\bullet}={}^{\sigma}{E^{*}}\cap({}^{\sigma}{F})^{\circ\bullet}={}^{\sigma}{E^{*}}\cap\mathrm{qs}({}^{\star}{F})={}^{\sigma}{F}.$

∎

######  Definition 3.15.

A topology $\tau$ on $E$ is called compatible with the duality if
$(E_{\tau})^{\prime}=F$.

In particular, the weak topology is compatible with the duality. Hence, any
polar is weakly closed and the bipolar of any monad or galaxy is equal to the
weakly closed convex balanced hull. For topologies compatible with the
duality, the closure of any convex balanced set $V\subseteq E$ is equal to the
weak closure:
$\overline{V}^{\tau}=\overline{\mathrm{cobal}{(V)}}^{\tau}=V^{\circ\circ}=\overline{V}^{\sigma}$.

######  Proposition 3.16.

A topology $\tau$ on $E$ is polar for $(E,F)$ iff $\tau$ is locally convex and
$\mathbf{m}_{\tau}(0)$ is weakly ⋆closed.

###### Proof.

If $\tau$ is the polar topology of
$\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$, then
$\mathbf{m}_{\tau}(0)=\mathfrak{g}^{\circ}$, hence weakly ⋆closed. If
$\mathbf{m}_{\tau}(0)$ is convex and weakly closed, then it must be the polar
topology of $\mathbf{m}_{\tau}(0)^{\circ}$. ∎

######  Corrolary 3.17.

For any polar topology we have $\mathbf{m}(0)^{\circ\circ}=\mathbf{m}(0)$.

Hence, any polar topology $\tau$ is the topology of uniform convergence on
$\mathbf{m}_{\tau}(0)^{\circ}$. The polar topology of $\mathfrak{g}$ is also
the polar topology of $\overline{\mathrm{cobal}}(\mathfrak{g})$, i.e. uniform
convergence on the members of $\mathrm{idl~{}}\mathfrak{g}$ implies uniform
convergence on their closed convex balanced hulls. Since
$\mathbf{m}_{E,\sigma}(0)=\mathbf{m}_{E,\sigma}(0)^{\circ\circ}=\mathrm{qs}({}^{\star}{E})^{\circ}$,
the weak topology is the topology of uniform convergence of convex balanced
hulls of finite sets.

For any polar topology, $\approx_{\tau}$-saturated sets must be weakly closed,
which includes $\mathrm{pns}({}^{\star}{E})$ and
$\mathrm{Fin}({}^{\star}{E})$. But then their galaxies must also be weakly
closed, so that
$\mathrm{Bdd}({}^{\star}{E})^{\circ\circ}=\mathrm{Bdd}({}^{\star}{E})$ and
$\mathrm{pcomp}({}^{\star}{E})^{\circ\circ}=\mathrm{pcomp}({}^{\star}{E})$.

######  Proposition 3.18.

Given $\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$ such that
$\overline{\mathrm{cobal}}^{\sigma}(\mathfrak{g})$ is covering. The polar
topology of $\mathfrak{g}$ is Hausdorff. In particular the weak topology is
Hausdorff.

###### Proof.

Assume w.l.o.g. $\mathfrak{g}=\mathrm{cobal}(\mathfrak{g})$, then
$\overline{\mathfrak{g}}^{\sigma}$ is covering. Take any
$e\in\ker\mathbf{m}(0)$, then $\langle e,f\rangle\approx 0$ for each
$f\in\mathfrak{g}$; in fact this counts for any
$f\in\overline{\mathfrak{g}}^{\sigma}\supseteq{}^{\sigma}{F}$ since $e$ is
continuous for the weak topology on $F$. But if $f\in{}^{\sigma}{F}$, $\langle
e,f\rangle$ is standard and thus exactly $0$. Hence $e=0$. ∎

######  Proposition 3.19.

Given $\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$ closed for
$\mathbb{K}$-scalar multiplication. Then we have

$\begin{split}(\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g})^{\circ}=\mathrm{Fin}_{\mathfrak{g}}({}^{\star}{E})\textnormal{~{}
and
~{}}\mathrm{Fin}_{\mathfrak{g}}({}^{\star}{E})^{\circ}&=\mathbf{m}_{\mathbb{K}}(0)\cdot\overline{\mathrm{cobal}}(\mathfrak{g});\\\
\textnormal{in particular
}(\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\sigma}{E})^{\circ}=\mathrm{Fin}_{\sigma}({}^{\star}{E})\textnormal{~{}
and
~{}}\mathrm{Fin}_{\sigma}({}^{\star}{E})^{\circ}&=\mathbf{m}_{\mathbb{K}}(0)\cdot\mathrm{qs}({}^{\star}{E}).\end{split}$

Hence, $\mathrm{Fin}_{\mathfrak{g}}({}^{\star}{E})$ and
$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}$ satisfy the bipolar identity.

###### Proof.

Given $e\in{}^{\star}{E}$ we find, because of proposition 2.16:

$\begin{split}e\in(\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g})^{\circ}&\iff\forall\omega\in{}^{\star}{\mathbb{K}}_{\infty}:e\in\omega\cdot\mathfrak{g}^{\circ}(=\omega\cdot\mathbf{m}_{E,\mathfrak{g}}(0))\\\
&\iff\mathbf{m}_{\mathbb{K}}(0)\cdot
e\subseteq\mathbf{m}_{E,\mathfrak{g}}(0)\\\ &\iff
e\in\mathrm{Fin}_{\mathfrak{g}}({}^{\star}{E}).\end{split}$

Since
$\mathbf{m}_{\mathbb{K}}(0)\cdot\overline{\mathrm{cobal}}(\mathfrak{g})\subseteq(\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g})^{\circ\circ}$,
by proving the bipolar identity for
$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}$ in case
$\mathfrak{g}=\overline{\mathrm{cobal}}(\mathfrak{g})$ we get the result. Let
$\mathcal{U}$ be a base of $\mathfrak{g}$ existing out of closed, convex,
balanced sets. Given $\mathfrak{g}^{\prime}\in\mathfrak{G}({}^{\star}{F})$, we
have
$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}\subseteq\mathfrak{g}^{\prime}$
iff for any $U\in\mathcal{U}$, there exists an $\alpha\in\mathbb{K}$ and
$V\in\mathrm{idl~{}}\mathfrak{g}^{\prime}$ such that $U\subseteq\alpha V$. In
other words, there exist $(\alpha_{V})_{V\in\mathcal{U}}$ in $\mathbb{R}_{+}$
such that
$\cup_{V\in\mathcal{U}}{}^{\star}{(\alpha_{V}V)}\subseteq\mathfrak{g}^{\prime}$.
Now, since $\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}$ is a union of monads
and therefore chromatic, by proposition 1.52 it is equal to the intersection
of all galaxies that contain it. Hence,

$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}=\bigcap_{(\alpha_{V})_{V\in\mathcal{U}}}\bigcup_{V\in\mathcal{U}}{}^{\star}{(\alpha_{V}V)}.$

Then using lemma 1.31 we find that
$\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}$ satisfies the bipolar identity.
∎

######  Corrolary 3.20.

Given $\mathfrak{m}\subseteq{}^{\filleddiamond}\mathfrak{M}({}^{\star}{E})$
and $\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$ closed for
$\mathbb{K}$-scalar multiplication. Then $\mathfrak{m}$ is sbounded for the
polar topology of $\mathfrak{g}$ iff $\mathfrak{g}$ is sabsorbed by
$\mathfrak{m}^{\circ}$. In particular, $\mathfrak{m}$ is weakly sbounded iff
$\mathfrak{m}^{\circ}$ is sabsorbing.

###### Proof.

$\mathfrak{m}\subseteq\mathrm{Fin}_{\mathfrak{g}}({}^{\star}{E})\iff\mathfrak{m}^{\circ}\supseteq\mathbf{m}_{\mathbb{K}}(0)\cdot\mathfrak{g}$.
∎

######  Corrolary 3.21.

Given $\mathfrak{g}\in\mathfrak{G}({}^{\star}{F})$ closed for
$\mathbb{K}$-scalar multiplication. Then $\mathfrak{g}^{\circ}$ is linear iff
$\mathfrak{g}$ is weakly sbounded. In particular the weak topology is
compatible with the vector structure.

###### Proof.

$\mathfrak{g}^{\circ}\supseteq\mathbf{m}_{\mathbb{K}}(0)\cdot{}^{\sigma}{E}\iff\mathfrak{g}\subseteq\mathrm{Fin}_{\sigma}({}^{\star}{E})$.
∎

Hence, the finest polar topology compatible with the vector structure is
determined by the weakly bounded sets:

######  Definition 3.22.

The strong topology $\textnormal{b}(E,F)$ or topology of uniform convergence
on weakly bounded sets is defined by

$\mathbf{m}_{E,\textnormal{b}}(0):=\mathrm{Bdd}_{\sigma}({}^{\star}{E})^{\circ}.$

######  Proposition 3.23.

$\mathbf{m}_{E}(0)^{\circ}$ is the galaxy of equicontinuous subsets of $F$. In
other words, a set $H\subseteq F$ is equicontinuous iff
${}^{\star}{H}\subseteq\mathbf{m}_{E}(0)^{\circ}$ or equivalently
${}^{\star}{H}^{\circ}\supseteq\mathbf{m}_{E}(0)$.

###### Proof.

This follows from proposition 2.40. ∎

Hence, closed, convex, balanced hulls of equicontinuous sets are
equicontinuous. Each polar topology on $E$ is the topology of uniform
convergence on equicontinuous subsets of $F$. In fact, each Hausdorff local
convex topology is a dual topology for the duality $(E,E^{\prime})$, i.e.
uniform convergence on equicontinuous subsets of $E^{\prime}$.

######  Proposition 3.24.

$\mathrm{pns}_{\tau}({}^{\star}{E})$ satisfies the bipolar identity.

###### Proof.

As in the proof of proposition 2.31, we can write
$\mathrm{pns}({}^{\star}{E})$ as an intersection of ⋆closed, ⋆convex,
⋆balanced galaxies, so that lemma 1.31 implies the result. ∎

######  Proposition 3.25.

$\mathrm{pns}({}^{\star}{E})^{\circ}=\mathbf{m}(0)^{\circ}\cap{}^{\sigma}{E}^{\circ}$.

###### Proof.

Since $\mathbf{m}(0)\cup{}^{\sigma}{E}\subseteq\mathrm{pns}({}^{\star}{E})$ we
have
$\mathrm{pns}({}^{\star}{E})^{\circ}\subseteq\mathbf{m}(0)^{\circ}\cap{}^{\sigma}{E}^{\circ}$,
leaving only the opposite implication to prove. Take
$f\in\mathbf{m}(0)^{\circ}\cap{}^{\sigma}{E}^{\circ}$ and
$e\in\mathrm{pns}({}^{\star}{E})$. Since $f$ is scontinuous there exists a
$\lambda\in{}^{\sigma}{\Lambda}$ such that $\langle e^{\prime},f\rangle\leq 1$
for all $e^{\prime}\in\lambda$. But there exists a
$e_{\lambda}\in{}^{\sigma}{E}$ such that $e-e_{\lambda}\in\lambda$ and
therefore

$|\langle e,f\rangle|\leq|\langle e_{\lambda},f\rangle|+|\langle
e-e_{\lambda},f\rangle|\lessapprox 1.$

Since $\mathrm{pns}({}^{\star}{E})$ is closed for $\mathbb{K}$-scalar
multiplication $|\langle e,f\rangle|\approx 0$ and therefore
$f\in\mathrm{pns}({}^{\star}{E})^{\circ}$. ∎

######  Corrolary 3.26.

The weak topology is Henson-Moore, i.e.
$\mathrm{Fin}_{\sigma}({}^{\star}{E})=\mathrm{pns}_{\sigma}({}^{\star}{E}).$

###### Proof.

Take
$f\in\mathrm{pns}_{\sigma}({}^{\star}{E})^{\circ}=\mathbf{m}(0)^{\circ}\cap{}^{\sigma}{E}^{\circ}=\mathrm{qs}({}^{\star}{E})\cap\mathbf{m}_{F,\sigma}(0)$.
Then by the definition of $\mathrm{qs}({}^{\star}{E})$, there are finite
$e_{0},\dots,e_{n}\in{}^{\sigma}{E}$ such that
$f=\sum_{j=0}^{n}\alpha_{j}e_{j}$ with $\alpha_{j}\approx 0$ for all $j=0\dots
n$. Then for $e\in\mathrm{Fin}_{\sigma}({}^{\star}{E})$, $\langle
e,f\rangle=\sum_{j=1}^{n}\alpha_{j}\langle e_{j},f\rangle\approx 0$. So
$\mathrm{pns}_{\sigma}({}^{\star}{E})^{\circ}\subseteq\mathrm{Fin}_{\sigma}({}^{\star}{E})^{\circ}$
from which follows that
$\mathrm{Fin}_{\sigma}({}^{\star}{E})\subseteq\mathrm{pns}_{\sigma}({}^{\star}{E})$
since both sets are equal to their bipolar. ∎

######  Proposition 3.27.