# Generalizing $\beta$\- and $\lambda$-maps

Ana Belén Avilez  University of the Western Cape, Department of Mathematics
and Applied Mathematics,
7535 Bellville, South Africa
Email addresses<EMAIL_ADDRESS>
(Date: July 23, 2024)

###### Abstract.

We generalize the notions of $\beta$\- and $\lambda$-maps to general
selections of sublocales, obtaining different classes of localic maps. These
new classes of maps are used to characterize almost normality, extremal
disconnectedness, $F$-frames, $Oz$-frames, among others types of locales, in a
manner akin to the characterization of normal locales via $\beta$-maps. As a
byproduct we obtain a characterization of localic maps that preserve the
completely below relation (that is, the right adjoints of assertive frame
homomorphisms).

###### Key words and phrases:

Frame, locale, sublocale, $\beta$\- and $\lambda$-maps, Stone-Cech
compactification, Regular Lindel0̈f Correflection, variants of normality,
variants of extremal disconnectedness.

###### 2010 Mathematics Subject Classification:

06D22, 18F70, 54D35, 54C10, 54D15, 54G05.

## Introduction

Motivated by the idea of characterizing different types of locales using only
localic maps, we generalize the notions of $\beta$\- and $\lambda$-maps
obtaining a variety of classes of maps and offering several characterizations
of frames only on conditions on the localic maps.

In classical topology, a continuous map $k\colon X\to Y$ is an $N$-map (resp.
$WN$-map) if

$\mathsf{cl}_{\beta X}k^{-1}[F]=(k^{\beta})^{-1}[\mathsf{cl}_{\beta Y}F]$

for every closed subset (resp. zero subset) $F$ of $Y$, where $k^{\beta}$ is
the extension of $k$ that maps $\beta X$ to $\beta Y$. These maps were used in
[25] to give characterizations of normal and $\delta$-normally separated
spaces. In [10, 11] the corresponding notions in the point-free setting, $N$\-
and $WN$-homomorphisms, were firstly introduced. Later in [13, 15] the
authors, not only work with the Stone-Čech compactification, but similarly
define $\lambda$-maps (the analogous of $\beta$-maps or $N$-homomorphisms)
using the Lidelöfication of a frame.

In this paper, we continue the study that started in [1] where the author
generalizes the notion of $\beta$-maps to selections, in order to obtain a
general result that characterizes different types of frames. So far these maps
were only defined using selections of closed sublocales. Following the
reasoning of [19] we explore the dual results, when taking selections of open
sublocales. Moreover, the same study will be developed for generalized
$\lambda$-maps.

The main purpose of this paper, different from the point of view in [13, 15]
which was studying the liftings of frame homomorphisms under the Stone-Čech
compactification and Lindelöfication, we intend to provide new classes of
localic maps and explore the imporatance of these maps in its own right
(discarding, in a way, the relation they have with the Stone-Čech
compactification and Lindelöfication). This paper should be thought of as an
attempt to continue the study of different classes of localic maps.

Here is a brief outlay of the paper. In Section 1 we give some general
background and notation. In the following section we introduce a general
definition of a $\mathfrak{S}\gamma$-map where $\mathfrak{S}$ stands for a
selection of sublocales and $\gamma$ for a special kind of functor. In Section
3 we work with $\mathfrak{S}\beta$-maps, where $\beta$ denotes the Stone-Čech
compactification. We first recall and extend some of the known results, then
we take a closer look at $\mathfrak{S}\beta$-maps when the selection function
selects classes of open sublocales. This approach is new (as it was never done
classically or point-freely) and allows us to obtain dual results as the ones
already known and characterize the localic maps that preserve the completely
rather below relation. Doing a similar study as in the previous section, we
introduce and characterize $\mathfrak{S}\lambda$-maps in Section 4, where
$\lambda$ stands for the Lindelöfication of a frame. We then generalize the
definition of $\mathfrak{S}\lambda$-maps obtaining a whole range of classes of
localic maps and two characterizing theorems for special types of frames.
Finally, in Section 5 we look at special cases when embeddings of sublocales
are $\mathfrak{S}\beta$\- or $\mathfrak{S}\lambda$-maps.

##  1\. Notation and Terminology

###  1.1. The categories of frames and locales

Our notation and terminology for frames and locales is that of [23]. We recall
here some of the basic notions involved. A frame $L$ is a complete lattice in
which

$a\wedge\mathop{\textstyle\bigvee}S=\mathop{\textstyle\bigvee}\\{a\wedge b\mid
b\in S\\}\quad\mbox{ for any }a\in L\mbox{ and }S\subseteq L.$ (1)

A frame homomorphism preserves all joins (in particular, the bottom element
$0$ of the lattice) and all finite meets (in particular, the top element 1).

In any frame $L$, the mappings $(x\mapsto(a\wedge x))\colon L\to L$ preserve
suprema and hence they have right Galois adjoints $(y\mapsto(a\to y))\colon
L\to L$, satisfying

$a\wedge x\leq y\quad\text{iff}\quad x\leq a\to y$

and making $L$ a complete Heyting algebra. The pseudocomplement of $a\in L$ is
the element $a^{*}=a\to 0=\mathop{\textstyle\bigvee}\\{x\mid x\wedge a=0\\}$
and we write $L^{*}:=\left\\{a^{*}\mid a\in L\right\\}$. A regular element of
$L$ is an element of the form $a^{*}$ for some $a$ (equivalently, an element
$a$ such that $a^{**}=a$).

The rather below relation $\prec$ in $L$ is defined by $b\prec a$ iff
$b^{*}\vee a=1$. A frame is regular if $a=\mathop{\textstyle\bigvee}\\{b\in
L\mid b\prec a\\}$ for every $a\in L$, and it is normal if for any $a,b\in L$
such that $a\vee b=1$, there is a pair $u,v\in L$ such that $u\wedge v=0$ and
$a\vee u=b\vee v=1$.

The category of locales and localic maps is the dual category of frames and
frame homomorphisms. Thus a locale is a frame and localic maps can be
represented by the (uniquely defined) right adjoints $f=h_{*}\colon L\to M$ of
frame homomorphisms $h\colon M\to L$. They are precisely the meet preserving
maps $f\colon L\to M$ such that $f(h(a)\to b)=a\to f(b)$ and
$f(a)=1\Rightarrow a=1$.
Throughout the paper we will always use $f$ to refer to localic maps and $h$
will always denote the frame homomorphism left adjoint of $f$.

###  1.2. The coframes of sublocales of a locale

A sublocale of a locale $L$ is a subset $S\subseteq L$ closed under arbitrary
meets such that

$\forall x\in L\ \ \forall s\in S\ \ (x\to s\in S).$

They are precisely the subsets of $L$ for which the embedding $j\colon
S\hookrightarrow L$ is a localic map. The system $\mathsf{S}(L)$ of all
sublocales of $L$, partially ordered by inclusion, is a coframe, that is, its
dual lattice is a frame [23, Theorem III.3.2.1]. Infima and suprema are given
by

$\mathop{\textstyle\bigwedge}_{i\in J}S_{i}=\mathop{\textstyle\bigcap}_{i\in
J}S_{i},\quad\mathop{\textstyle\bigvee}_{i\in
J}S_{i}=\\{\mathop{\textstyle\bigwedge}M\mid
M\subseteq\mathop{\textstyle\bigcup}_{i\in J}S_{i}\\}.$

The least element is the void sublocale $\text{\sf O}=\\{1\\}$ and the
greatest element is the entire locale $L$. Since $\mathsf{S}(L)$ is a coframe,
there is a co-Heyting operator $S\smallsetminus T$ given by the formula
$\bigcap\left\\{A\mid S\subseteq T\vee A\right\\}$ and characterized by the
condition

$S\smallsetminus T\subseteq A\ \Leftrightarrow\ S\subseteq T\vee A.$

In particular, the co-pseudocomplement (usually called supplement) of $S$ is
the sublocale $S^{\\#}=L\smallsetminus S$ and we have that for any
$S,T\in\mathsf{S}(L)$,

$S\vee T=L\ \Leftrightarrow\ S^{\\#}\subseteq T\quad\text{ and }\quad S\cap
T=\text{\sf O}\ \Rightarrow\ S\subseteq T^{\\#}$ (2)

(we refer to [16] for more information on supplements in $\mathsf{S}(L)$).

For each $a\in L$, the sublocales

$\mathfrak{c}_{L}(a)={{\hbox to0.0pt{$\ $\hss}\hbox{$\uparrow$}}}a=\\{x\in
L\mid x\geq a\\}\quad\text{ and }\quad\mathfrak{o}_{L}(a)=\\{a\to b\mid b\in
L\\}$

are the _closed_ and _open_ sublocales of $L$, respectively (that we shall
denote simply by $\mathfrak{c}(a)$ and $\mathfrak{o}(a)$ when there is no
danger of confusion). For each $a\in L$, $\mathfrak{c}(a)$ and
$\mathfrak{o}(a)$ are complements of each other in $\mathsf{S}(L)$ and satisfy
the identities

$\mathop{\textstyle\bigcap}_{i\in
J}\mathfrak{c}(a_{i})={\mathfrak{c}}(\mathop{\textstyle\bigvee}_{i\in
J}a_{i}),\quad{\mathfrak{c}}(a)\vee{\mathfrak{c}}(b)={\mathfrak{c}}(a\wedge
b),$ (3) $\mathop{\textstyle\bigvee}_{i\in
J}{\mathfrak{o}}(a_{i})={\mathfrak{o}}(\mathop{\textstyle\bigvee}_{i\in
J}a_{i})\quad\mbox{ and
}\quad{\mathfrak{o}}(a)\cap{\mathfrak{o}}(b)={\mathfrak{o}}(a\wedge b).$ (4)

We denote by ${\mathfrak{o}}(L)$ and ${\mathfrak{c}}(L)$ the classes of all
open and closed sublocales of $L$.

Let $j_{S}^{*}$ be the left adjoint of the localic embedding $j_{S}\colon
S\hookrightarrow L$, given by
$j_{S}^{*}(a)=\mathop{\textstyle\bigwedge}\\{s\in S\mid s\geq a\\}$. The
closed (resp. open) sublocales ${\mathfrak{c}}_{S}(a)$ (resp.
${\mathfrak{o}}_{S}(a)$) of $S$ ($a\in S$) are precisely the intersections
${\mathfrak{c}}(a)\cap S$ (resp. ${\mathfrak{o}}(a)\cap S)$ and we have, for
any $a\in L$,

${\mathfrak{c}}(a)\cap
S={\mathfrak{c}}_{S}(j_{S}^{*}(a))\quad\mbox{and}\quad{\mathfrak{o}}(a)\cap
S={\mathfrak{o}}_{S}(j_{S}^{*}(a)).$ (5)

The closure $\overline{S}$ of a sublocale $S$ is the smallest closed sublocale
containing $S$, and the interior $S^{\circ}$ is the largest open sublocale
contained in $S$. There is a particularly simple formula for the closure:

$\overline{S}={\mathfrak{c}}(\mathop{\textstyle\bigwedge}S).$ (6)

Hence $\overline{{\mathfrak{o}}(a)}={\mathfrak{c}}(a^{*})$ and, consequently,
${\mathfrak{c}}(a)^{\circ}={\mathfrak{o}}(a^{*})$. More generally, for every
sublocale $S$, $\overline{S^{\\#}}=(S^{\circ})^{\\#}$ [16, Section 4]. A
sublocale is regular closed (resp. regular open) if it is of the form
${\mathfrak{c}}(a^{*})$ (resp. ${\mathfrak{o}}(a^{*})$) for some $a\in L$.
Note that, this allows us to look at the rather below relation in terms of
sublocales:

$b\prec
a\iff\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(a)\iff{\mathfrak{c}}(a)\subseteq{\mathfrak{c}}(b)^{\circ}.$
(7)

We say a sublocale is _clopen_ if it is both closed and open. A clopen
sublocale is of the form ${\mathfrak{c}}(a)$ (resp. ${\mathfrak{o}}(a)$) for
some _complemented_ $a\in L$ (that is, $a^{*}\vee a=0$). A sublocale $S$ of
$L$ is dense if $\overline{S}=L$, equivalently, by (6), if $0_{L}\in S$.

###  1.3. Image-Preimage adjunction of localic maps

Let $f\colon L\to M$ be a localic map. For any sublocale $S$ of $L$, its set
theoretic image $f[S]$ is a sublocale of $M$. On the other hand, the set
theoretic preimage $f^{-1}[T]$ of a sublocale $T$ of $M$ may not be a
sublocale of $L$. But since $f$ is meet preserving, $f^{-1}[T]$ is closed
under meets and thus there exists the largest sublocale of $L$ contained in
$f^{-1}[T]$, usually denoted as $f_{-1}[T]$ ([23, III.4]). This is the localic
preimage of $T$ that provides the image/preimage Galois adjunction

$\textstyle{\mathsf{S}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f[-]}$$\scriptstyle{\bot}$$\textstyle{\mathsf{S}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{-1}[-]}$

between coframes $\mathsf{S}(L)$ and $\mathsf{S}(M)$ of sublocales of $L$ and
sublocales of $M$, respectively. The right adjoint $f_{-1}[-]$ is a coframe
homomorphism that preserves complements while $f[-]$ is a colocalic map ([23,
III.9]). The adjunction between these two maps gives the following equivalence

$f[S]\subseteq T\iff S\subseteq f_{-1}[T]$ (8)

for every $S\in\mathsf{S}(L)$ and $T\in\mathsf{S}(M)$. Localic preimages of
closed (resp. open) sublocales are closed (resp. open). More specifically,
denoting by $h$ the frame homomorphism left adjoint to $f$, we have

$f_{-1}[{\mathfrak{c}}_{M}(a)]={\mathfrak{c}}_{L}(h(a))\quad\text{and}\quad
f_{-1}[{\mathfrak{o}}_{M}(a)]={\mathfrak{o}}_{L}(h(a))$ (9)

for any $a\in M$.

In the particular case of the embedding $j\colon S\hookrightarrow L$ of a
sublocale $S$ of $L$ the localic preimages are computed as the intersection;
that is, $j_{-1}[T]=S\cap T$ for every $T\in\mathsf{S}(L)$.

###  1.4. Zero and cozero sublocales

An $a\in L$ is a cozero element if
$a=\mathop{\textstyle\bigvee}_{n=1}^{\infty}a_{n}$ for some
$a_{n}{\mathop{\hskip 1.0pt\prec\hskip-2.0pt\prec\hskip 2.0pt}}a$
($n=1,2,\ldots$) where the completely below relation ${\mathop{\hskip
1.0pt\prec\hskip-2.0pt\prec\hskip 2.0pt}}$ is the interpolative modification
of the rather below relation: $b{\mathop{\hskip
1.0pt\prec\hskip-2.0pt\prec\hskip 2.0pt}}a$ if and only if there exists a
subset $\\{a_{q}\mid q\in[0,1]\cap\mathbb{Q}\\}\subseteq L$ with $a_{0}=b$ and
$a_{1}=a$ such that $a_{p}\prec a_{q}$ whenever $p<q$. Recall also that a
frame $L$ is said to be _completely regular_ if
$a=\mathop{\textstyle\bigvee}\\{b\in L\mid b{\mathop{\hskip
1.0pt\prec\hskip-2.0pt\prec\hskip 2.0pt}}a\\}$ for every $a\in L$.
Cozero elements form a $\sigma$-frame $\mathsf{Coz}~{}L\subseteq L$, that is,
a lattice in which all countable subsets have a join such that the
distributivity law (1) holds for any countable $S$. Furthermore,
$\mathsf{Coz}~{}L$ is a normal $\sigma$-frame, that is, $a\vee b=1$
($a,b\in\mathrm{Coz}\,L$) implies there exist $c$ and $d$ in
$\mathsf{Coz}~{}L$ such that $a\vee c=1=b\vee d$ and $c\wedge d=0$ [5].

The zero sublocales (resp. cozero sublocales) are the ${\mathfrak{c}}(a)$
(resp. ${\mathfrak{o}}(a)$) with $a\in\mathrm{Coz}\,L$. We denote by

$\mathsf{ZS}(L)\quad\mbox{ and }\quad\mathsf{CoZS}(L)$

the classes of zero and cozero sublocales, respectively. The former class is a
sub-$\sigma$-coframe of $\mathsf{S}(L)$ while the latter is a
sub-$\sigma$-frame. Since frame homorphisms preserve cozero elements, localic
preimages of cozero (resp. zero) sublocales are cozero (resp. zero)
sublocales.

Furthermore, a sublocale $S$ is both a zero and a cozero sublocale iff it is
clopen.

###  1.5. Completely separated sublocales

Two sublocales $S$ and $T$ of $L$ are _completely separated in L_ ([17,
4])111Complete separation is usually defined using real-valued functions; for
simplicity we opt to use one of its characterizations as a defintion (see for
instance [2, 2.7] for all the details). if they are contained in disjoint zero
sublocales of $L$; that is, if there exist $Z_{1},Z_{2}\in\mathsf{ZS}(L)$ such
that

$S\subseteq Z_{1},\quad T\subseteq Z_{2}\quad\text{and}\quad Z_{1}\cap
Z_{2}=\text{\sf O}.$

######  Remark 1.1.

  1. (1)

If $S\subseteq R$, $T\subseteq D$ and $R$ and $D$ are completely separated in
$L$, then so are $S$ and $T$.

  2. (2)

$S$ and $T$ are completely separated if and only if $\overline{S}$ and
$\overline{T}$ are completely separated.

  3. (3)

Disjoint zero sublocales are completely separated. In particular, disjoint
clopen sublocales are completely separated.

######  Proposition 1.2.

[20, 5.4.2] Let $a$ and $b$ be elements of a locale $L$, $a\prec\\!\\!\prec b$
if and only if ${\mathfrak{o}}(a)$ and ${\mathfrak{c}}(b)$ are completely
separated in $L$.

######  Remark 1.3.

If $S$ and $T$ are completely separated in $L$ there exist
$a,b\in\mathsf{Coz}~{}L$ such that

$S\subseteq{\mathfrak{c}}(a),\quad
T\subseteq{\mathfrak{c}}(b)\quad\text{and}\quad{\mathfrak{c}}(a)\cap{\mathfrak{c}}(b)=\text{\sf
O}.$

Then, since $\mathsf{Coz}~{}L$ is normal (which implies
$\prec=\prec\\!\\!\prec$) and $a\vee b=1$, there exist $u,v\in\mathsf{Coz}$
such that $u\wedge v=0$, $v\prec\\!\\!\prec a$ and $u\prec\\!\\!\prec b$.
Hence,

$S\subseteq{\mathfrak{c}}(a)\subseteq{\mathfrak{o}}(u),\quad
T\subseteq{\mathfrak{c}}(b)\subseteq{\mathfrak{o}}(u)\quad\text{and}\quad{\mathfrak{o}}(u)\cap{\mathfrak{o}}(v)=\text{\sf
O}.$

Moreover, by Proposition 1.2, ${\mathfrak{o}}(u)$ and ${\mathfrak{c}}(b)$ are
completely separated, and so are ${\mathfrak{o}}(v)$ and ${\mathfrak{c}}(a)$.
If we repeat this argument, then we can obtain another cozero sublocale
${\mathfrak{o}}(y)$ such that ${\mathfrak{o}}(u)$ and ${\mathfrak{o}}(y)$ are
completely separated and $S\subseteq{\mathfrak{o}}(u)$ and
$T\subseteq{\mathfrak{o}}(y)$.

###  1.6. Special classes of localic maps

A localic map $f\colon L\to M$ is said to be

  1. (1)

dense if $f[L]$ is a dense sublocale of $M$. Equivalently, if
$f(0_{L})=0_{M}$.

  2. (2)

closed if the image $f[S]$ is closed for each closed $S\in\mathsf{S}(L)$.
Equivalently, if $f(a\vee h(b))=f(a)\vee b$ for every $a\in L$ and $b\in M$.

  3. (3)

$\mathsf{z}$-closed if the image $f[S]$ is closed for each zero sublocale
$S\in\mathsf{S}(L)$. Equivalently, if $f(a\vee h(b))=f(a)\vee b$ for every
$a\in\mathsf{Coz}~{}L$ and $b\in M$.

  4. (4)

$\mathsf{reg}_{\mathfrak{c}}$-closed if the image $f[S]$ is closed for each
regular closed sublocale $S\in\mathsf{S}(L)$. Equivalently, if $f(a\vee
h(b))=f(a)\vee b$ for every $a\in L^{*}$ and $b\in M$.

  5. (5)

$\mathsf{z}$-preserving if the image $f[S]$ is a zero sublocale for each zero
sublocale $S\in\mathsf{S}(L)$. Equivalently, if $f(a\vee h(b))=f(a)\vee b$ for
every $a\in\mathsf{Coz}~{}L$ and $b\in M$ and $f(a)\in\mathsf{Coz}~{}M$ ([19,
Section 9]).

  6. (6)

$\mathsf{reg}_{\mathfrak{c}}$-preserving if the image $f[S]$ is a regular
closed for each regular closed sublocale $S\in\mathsf{S}(L)$. Equivalently, if
$f(a\vee h(b))=f(a)\vee b$ for every $a\in L^{*}$ and $b\in M$ and $f(a)\in
M^{*}$ ([19, Section 9]).

  7. (7)

_proper_ if it is closed and preserves directed joins.

Clearly, every closed map is $\mathsf{z}$-closed and
$\mathsf{reg}_{\mathfrak{c}}$-closed. Moreover, $\mathsf{z}$-preserving (resp.
$\mathsf{reg}_{\mathfrak{c}}$-preserving) maps are precisely the
$\mathsf{z}$-closed (resp. $\mathsf{reg}_{\mathfrak{c}}$-preserving) maps such
that $f(a)\in\mathsf{Coz}~{}M$ for every $a\in\mathsf{Coz}~{}L$ (resp.
$f(a)\in M^{*}$ for every $a\in L^{*}$).

######  Remark 1.4.

In general,
$\overline{f[S]}={\mathfrak{c}}_{M}(\mathop{\textstyle\bigwedge}f[S])={\mathfrak{c}}_{M}(f(\mathop{\textstyle\bigwedge}S))$,
so $\overline{f[{\mathfrak{c}}_{L}(a)]}={\mathfrak{c}}_{M}(f(a))$ for every
$a\in L$. Hence, if $f[{\mathfrak{c}}_{L}(a)]$ is closed for some $a\in L$, we
have that
$f[{\mathfrak{c}}_{L}(a)]=\overline{f[{\mathfrak{c}}_{L}(a)]}={\mathfrak{c}}_{M}(f(a))$.

A localic map $f\colon L\to M$ is said to be open if the image $f[S]$ is open
for each open $S\in\mathsf{S}(L)$. Other variants of openness have been
studied in [8] and [1], for instance nearly openness and
$\mathsf{coz}$-openness. Here we will use these notions and other variants. We
say $f$ is

  1. (1)

nearly open  if $h(a^{*})=h(a)^{*}$ for every $a\in M$. Equivalently,
$f_{-1}[\overline{{\mathfrak{o}}(a)}]=\overline{f_{-1}[{\mathfrak{o}}(a)]}$
for every $a\in L$.

  2. (2)

$\mathsf{coz}$-nearly open  if $h(a^{*})=h(a)^{*}$ for every
$a\in\mathsf{Coz}~{}M$. Equivalently,
$f_{-1}[\overline{{\mathfrak{o}}(a)}]=\overline{f_{-1}[{\mathfrak{o}}(a)]}$
for every $a\in\mathsf{Coz}~{}L$.

  3. (3)

$\mathsf{reg}_{\mathfrak{o}}$-nearly open  if $h(a^{*})=h(a)^{*}$ for every
$a\in M^{*}$. Equivalently,
$f_{-1}[\overline{{\mathfrak{o}}(a)}]=\overline{f_{-1}[{\mathfrak{o}}(a)]}$
for every $a\in L$ (this is condition $\mathbf{B}$ in [8]).

  4. (4)

$\mathsf{coz}$-open if $f[S]$ is open for each cozero sublocale
$S\in\mathsf{S}(L)$.

  5. (5)

$\mathsf{reg}_{\mathfrak{o}}$-open if $f[S]$ is open for each regular open
sublocale $S\in\mathsf{S}(L)$.

  6. (6)

$\mathsf{coz}$-preserving if the image $f[S]$ is a cozero sublocale for each
cozero sublocale $S\in\mathsf{S}(L)$.

  7. (7)

$f$ is $\mathsf{reg}_{\mathfrak{o}}$-preserving if the image $f[S]$ is regular
open for each regular open sublocale $S\in\mathsf{S}(L)$.

Every open map is nearly open, and every nearly open map is
$\mathsf{coz}$-nearly open and $\mathsf{reg}_{\mathfrak{o}}$-nearly open.
Furthermore, every open and every $\mathsf{coz}$-preserving (resp.
$\mathsf{reg}_{\mathfrak{o}}$-preserving) map is $\mathsf{coz}$-open (resp.
$\mathsf{reg}_{\mathfrak{o}}$-open). Note that $f$ is nearly open (resp.
$\mathsf{reg}_{\mathfrak{o}}$-nearly open) if and only if
$f[{\mathfrak{o}}_{L}(a)]\subseteq\overline{f[{\mathfrak{o}}_{L}(a)]}^{\circ}$
for every $a\in L$ (resp. $a\in L^{*}$).

##  2\. $\Gamma$-maps

Following the notation in [13], but adopting it to the localic setting,
suppose $\gamma\colon\mathsf{Loc}\to\mathsf{CRegLoc}$ is a functor such that
for each frame $L$ there is a localic map $\gamma_{L}\colon L\to\gamma L$ with
the property that every localic map $f\colon L\to M$ lifts; that is, there is
a localic map $f^{\gamma}\colon\gamma L\to\gamma M$ such that
$\gamma_{M}f=f^{\gamma}\gamma_{L}$. If we consider the respective frame
homorphisms (that is the left adjoints of the localic maps), we obtain the
following diagram

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\bot}$$\scriptstyle{\gamma_{L}}$$\scriptstyle{\dashv}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\gamma_{M}}$$\scriptstyle{\dashv}$$\textstyle{\gamma
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{L}^{*}}$$\scriptstyle{f^{\gamma}}$$\scriptstyle{\bot}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\gamma
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\gamma}}$$\scriptstyle{\gamma_{M}^{*}}$
(10)

where frame homomorphisms and localic maps commute respectively; that is

$h\cdot\gamma_{M}^{*}=\gamma_{L}^{*}\cdot
h^{\gamma}\quad\text{and}\quad\gamma_{M}\cdot f=f^{\gamma}\cdot\gamma_{L}.$
(11)

We say a localic map $f\colon L\to M$ is a _$\gamma$ -map_ if
$h^{\gamma}\cdot\gamma_{M}=\gamma_{L}\cdot h$. Note that if $a\in M$ is such
that $h^{\gamma}(\gamma_{M}(a))=\gamma_{L}(h(a))$, by taking the respective
closed sublocales, this is equivalent to

$\displaystyle
f_{-1}^{\gamma}\left[\overline{\gamma_{M}[{\mathfrak{c}}_{M}(a)]}^{\gamma
M}\right]$ $\displaystyle=f_{-1}^{\gamma}[{\mathfrak{c}}_{\gamma
M}(\gamma_{M}(a))]={\mathfrak{c}}_{\gamma
L}(h^{\gamma}(\gamma_{M}(a)))={\mathfrak{c}}_{\gamma L}(\gamma_{L}(h(a)))$
$\displaystyle=\overline{\gamma_{L}[{\mathfrak{c}}_{L}(h(a))]}^{\gamma
L}=\overline{\gamma_{L}\left[f_{-1}[{\mathfrak{c}}_{M}(a)])\right]}^{\gamma
L}.$

In general, we always have that
$h^{\gamma}(\gamma_{M}(a))\leq\gamma_{L}(h(a))$ for any $a\in M$. Indeed,
using the fact that $\gamma_{L}^{*}\dashv\gamma_{L}$ and (11) we get

$\displaystyle h^{\gamma}(\gamma_{M}(a))\leq\gamma_{L}(h(a))$
$\displaystyle\iff\gamma_{L}^{*}(h^{\gamma}(\gamma_{M}(a)))\leq\gamma_{L}(h(a))$
$\displaystyle\iff h(\gamma_{M}^{*}(\gamma_{M}(a)))\leq h(a)$

and $\gamma_{M}^{*}(\gamma_{M}(a))\leq a$ since
$\gamma_{M}^{*}\dashv\gamma_{M}$. Equivalently, it is always true that

$\overline{\gamma_{L}\left[f_{-1}[{\mathfrak{c}}_{M}(a)])\right]}^{\gamma
L}\subseteq
f_{-1}^{\gamma}\left[\overline{\gamma_{M}[{\mathfrak{c}}_{M}(a)]}^{\gamma
M}\right].$

Instead of working only with elements and closed sublocales, we would like to
extend the definition of $\gamma$-maps to other classes of sublocales. A
_selection function_ on sublocales is a function which assigns to each locale
$L$ a subset $\mathfrak{S}L$ of $\mathsf{S}(L)$. The standard selections we
will use in this paper are taking closed sublocales, open sublocales, zero
sublocales, cozero sublocales, regular closed sublocales and regular open
sublocales. We denote them respectively as follows:

$\mathfrak{S}_{\mathfrak{c}}\quad\mathfrak{S}_{\mathfrak{o}}\quad\mathfrak{S}_{\mathsf{ZS}}\quad\mathfrak{S}_{\mathsf{CoZS}}\quad\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}\quad\mathfrak{S}_{\mathsf{reg}{\mathfrak{o}}}.$

######  Definition 2.1.

Given a selection function on sublocales $\mathfrak{S}$, we say a localic map
$f\colon L\to M$ is a _$\mathfrak{S}\gamma$ -map_ if

$\overline{\gamma_{L}\left[f_{-1}[S]\right]}^{\gamma
L}=f_{-1}^{\gamma}\left[\overline{\gamma_{M}[S]}^{\gamma M}\right].$

holds for every $S\in\mathfrak{S}M$.

######  Remark 2.2.

Note that

$\displaystyle\overline{\gamma_{L}\left[f_{-1}[S]\right]}^{\gamma
L}={\mathfrak{c}}_{\gamma
L}\left(\mathop{\textstyle\bigwedge}\gamma_{L}[f_{-1}[S]]\right)={\mathfrak{c}}_{\gamma
L}\left(\gamma_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])\right)$

and

$\displaystyle f_{-1}^{\gamma}\left[\overline{\gamma_{M}[S]}^{\gamma
M}\right]=f_{-1}^{\gamma}\left[{\mathfrak{c}}_{\gamma
M}\left(\mathop{\textstyle\bigwedge}\gamma_{M}[S]\right)\right]=f_{-1}^{\gamma}\left[{\mathfrak{c}}_{\gamma
M}\left(\gamma_{M}(\mathop{\textstyle\bigwedge}S)\right)\right]={\mathfrak{c}}_{\gamma
L}\left(h^{\gamma}(\gamma_{M}(\mathop{\textstyle\bigwedge}S))\right).$

Thus, $f$ is a $\mathfrak{S}\gamma$-map if and only if

$h^{\gamma}(\gamma_{M}(\mathop{\textstyle\bigwedge}S))=\gamma_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])$

For any sublocale $S$ of $M$ the inclusion

$\overline{\gamma_{L}\left[f_{-1}[S]\right]}^{\gamma L}\subseteq
f_{-1}^{\gamma}\left[\overline{\gamma_{M}[S]}^{\gamma M}\right].$

always holds. Indeed, due to the image/preimage adjunction,
$f[f_{-1}[S]]\subseteq S$ thus

$h(\mathop{\textstyle\bigwedge}S)\leq
h(\mathop{\textstyle\bigwedge}f[f_{-1}[S]])=h(f(\mathop{\textstyle\bigwedge}f_{-1}[S]))\leq\mathop{\textstyle\bigwedge}f_{-1}[S].$

Hence,

$\displaystyle
h(\gamma_{M}^{*}(\gamma_{M}(\mathop{\textstyle\bigwedge}S)))\leq
h(\mathop{\textstyle\bigwedge}S)\leq\mathop{\textstyle\bigwedge}f_{-1}[S]$
$\displaystyle\iff\gamma_{L}^{*}(h^{\gamma}(\gamma_{M}(\mathop{\textstyle\bigwedge}S)))\leq\mathop{\textstyle\bigwedge}f_{-1}[S]$
$\displaystyle\iff
h^{\gamma}(\gamma_{M}(\mathop{\textstyle\bigwedge}S))\leq\gamma_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])$
$\displaystyle\iff\overline{\gamma_{L}\left[f_{-1}[S]\right]}^{\gamma
L}\subseteq f_{-1}^{\gamma}\left[\overline{\gamma_{M}[S]}^{\gamma M}\right].$

###  2.1. Two important $\gamma$-fications

The two $\gamma$-fications we will be working with are the Stone-Čech
compactification $\beta$ and the Lindelöfication $\lambda$. In general we do
not require the locales to be completely regular. Both constructions still
work with the exception that the maps $\beta_{L}$ and $\lambda_{L}$ may not be
embeddings. When complete regularity is required it will be specified
throughout the paper. We give a brief recap of the construction of these two
$\gamma$-fications.

Let us recall from [7] the Stone-Čech compatification of a frame $L$. An ideal
$I$ of $L$ is regular if for every $a\in I$ there is a $b\in I$ such that
$a\prec\\!\\!\prec b$. For any $a\in L$, the ideal
$\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}a=\left\\{x\in L\mid
x\prec\\!\\!\prec a\right\\}$ is regular. The lattice $\mathfrak{R}(L)$ of all
regular ideals of $L$ is a compact regular frame. Recall that a frame is
_compact_ if $1=\mathop{\textstyle\bigvee}S$ implies
$1=\mathop{\textstyle\bigvee}F$ for a finite $F\subseteq S$. The Stone-Čech
compactification of $L$ is the dense localic map

$\displaystyle\beta_{L}\colon L$ $\displaystyle\to\beta L=\mathfrak{R}(L)$
$\displaystyle a$
$\displaystyle\mapsto\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}a.$

The frame homorphism $\beta_{L}^{*}\colon\beta L\to L$ is given by taking
joins of ideals. If $L$ is completely regular $\beta_{L}$ is one-one localic
map. The functor $\beta\colon\mathsf{Loc}\to\mathsf{CRegKLoc}$ shows that
compact regular locales are reflective in the category of locales. For any
localic map $f\colon L\to M$ its Stone extension is the localic map
$f^{\beta}\colon\beta L\to\beta M$ given by the functor $\beta$. The frame
homomorphism $h^{\beta}\colon\beta M\to\beta L$, left adjoint of $f^{\beta}$,
is easily computed by the formula: $h^{\beta}(I)=\downarrow h[I]$ for any
regular ideal $I$ of $M$. The reflection gives the following diagram

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\bot}$$\scriptstyle{\beta_{L}}$$\scriptstyle{\dashv}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\beta_{M}}$$\scriptstyle{\dashv}$$\textstyle{\beta
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{L}^{*}}$$\scriptstyle{f^{\beta}}$$\scriptstyle{\bot}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\beta
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\beta}}$$\scriptstyle{\beta_{M}^{*}}$
(12)

where frame homomorphisms and localic maps commute respectively.

Madden and Vermeer [21] show that through the functor
$\lambda\colon\mathsf{Loc}\to\mathsf{CRLinLoc}$ the category of completely
regular Lindelöf locales is reflective in the category of locales. A locale is
_Lindelöf_ if $1=\mathop{\textstyle\bigvee}S$ implies
$1=\mathop{\textstyle\bigvee}T$ for some countable $T\subseteq S$. For any
locale $L$, $\lambda L$ is the completely regular Lindel0̈f frame of all
$\sigma$-ideals (that is, the ideals closed under countable joins) of
$\mathsf{Coz}~{}L$. For any $a\in L$ the set
$\downarrow_{\mathsf{coz}}a:=\left\\{x\in\mathsf{Coz}~{}L\mid x\leq
a\right\\}$ is a $\sigma$-ideal of $\mathsf{Coz}~{}L$. The Lindelöfication and
its respective frame homomorphism are given as follows:

$\begin{array}[]{rclcrcl}\lambda_{L}\colon L&\to&\lambda
L&&\lambda_{L}^{*}\colon\lambda L&\to&L\\\
a&\mapsto&\downarrow_{\mathsf{coz}}a&&I&\mapsto&\mathop{\textstyle\bigvee}I\end{array}$

The map $\lambda_{L}$ is always dense, and it is one-one if $L$ is completely
regular. For a localic map $f\colon L\to M$ the frame homorphism
$h^{\lambda}$, left adjoint of the localic map $f^{\lambda}\colon\lambda
L\to\lambda M$ given by $\lambda$, is described as follows:

$\displaystyle h^{\lambda}\colon\lambda M$ $\displaystyle\to\lambda L$
$\displaystyle I$ $\displaystyle\mapsto\downarrow_{\mathsf{coz}}h[I]$

where $\downarrow_{\mathsf{coz}}h[I]=\left\\{b\in\mathsf{Coz}~{}L\mid b\leq
h(x)\text{ for some }x\in I\right\\}$. The reflection gives the following
diagram

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\bot}$$\scriptstyle{\lambda_{L}}$$\scriptstyle{\dashv}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\lambda_{M}}$$\scriptstyle{\dashv}$$\textstyle{\lambda
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{L}^{*}}$$\scriptstyle{f^{\lambda}}$$\scriptstyle{\bot}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\lambda
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\lambda}}$$\scriptstyle{\lambda_{M}^{*}}$
(13)

where frame homomorphisms and localic maps respectively, commute.

##  3\. $\mathfrak{S}\beta$-maps

In this section we will study the $\mathfrak{S}\gamma$-maps when $\gamma$ is
the Stone-Čech compactification. The notion of $\mathfrak{S}\beta$-map
(referred only as $\mathfrak{S}$-maps in [1]) is a generalization of the
classical notions of $N$-, $WN$\- and $WZ$-maps ([25]).
$\mathfrak{S}_{\mathfrak{c}}\beta$-maps are the right adjoints of
$N$-homomorphisms of [10] (or $\beta$-maps in [13, 15]) and
$\mathfrak{S}_{\mathsf{ZS}}\beta$-maps are the right adjoints of
$WN$-homomorphisms in [10]. For these two selections, the equivalence
(i)$\iff$ (iii) of the Theorem below gives as a corollary [10, Lemma 5.2] and
[10, 6].

######  Theorem 3.1.

The following are equivalent for a localic map $f\colon L\to M$:

  1. (i)

$f$ is an $\mathfrak{S}\beta$-map.

  2. (ii)

If an open sublocale ${\mathfrak{o}}_{L}(a)$ is completely separated from
$f_{-1}[S]$ for some $S\in\mathfrak{S}M$, then there is an open sublocale
${\mathfrak{o}}_{M}(b)$ such that ${\mathfrak{o}}_{L}(a)\subseteq
f_{-1}[{\mathfrak{o}}_{M}(b)]$ and ${\mathfrak{o}}_{M}(b)$ is completely
separated from $S$ in $M$.

  3. (iii)

For every $a\in L$ such that
$a\prec\\!\\!\prec\mathop{\textstyle\bigwedge}f_{-1}[S]$ there exists $b\in M$
such that $a\leq h(b)$ and $b\prec\\!\\!\prec\mathop{\textstyle\bigwedge}S$.

###### Proof.

(i)$\iff$(iii): By Remark 2.2, $f$ is $\mathfrak{S}\beta$-map if and only if

$\beta_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])\leq
h^{\beta}(\beta_{M}(\mathop{\textstyle\bigwedge}S)).$

Using the definitions of $\beta_{L}$ and $h^{\beta}$, one sees that this
precisely means that

$\left\\{y\in L\mid
y\prec\\!\\!\prec\mathop{\textstyle\bigwedge}f_{-1}[S]\right\\}\subseteq\
\downarrow\left\\{h(x)\mid x\in M,\ x\prec\\!\\!\prec 0_{S}\right\\}.$

Equivalenty, if for every $y\in L$ such that
$y\prec\\!\\!\prec\mathop{\textstyle\bigwedge}f_{-1}[S]$ there is an $x\in M$
such that $y\leq h(x)$ and $x\prec\\!\\!\prec
0_{S}=\mathop{\textstyle\bigwedge}S$.
(ii)$\iff$ (iii): It follows from Proposition 1.2 and Remark 1.2. Indeed,

$\displaystyle a\prec\\!\\!\prec\mathop{\textstyle\bigwedge}f_{-1}[S]$
$\displaystyle\iff{\mathfrak{o}}_{L}(a)\text{ is completely separated from
}{\mathfrak{c}}_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])=\overline{f_{-1}[S]}$
$\displaystyle\iff{\mathfrak{o}}_{L}(a)\text{ is completely separated from
}f_{-1}[S]$

and

$\displaystyle b\prec\\!\\!\prec\mathop{\textstyle\bigwedge}S$
$\displaystyle\iff{\mathfrak{o}}_{M}(b)\text{ is completely separated from
}{\mathfrak{c}}_{M}(\mathop{\textstyle\bigwedge}S)=\overline{S}$
$\displaystyle\iff{\mathfrak{o}}_{M}(b)\text{ is completely separated from
}S.$

Furhtermore, from equations (4) and (9), it is clear that $a\leq h(b)$ if and
only if
${\mathfrak{o}}_{L}(a)\subseteq{\mathfrak{o}}_{L}(h(b))=f_{-1}[{\mathfrak{o}}_{M}(b)]$.
∎

######  Remark 3.2.

The fact that we are taking open sublocales in the characterization of
$\mathfrak{S}\beta$-maps (Theorem 3.1(ii)) is not relevant. In fact, one can
take closed, zero, cozero, regular open or regular closed sublocales. Indeed,
by definition and Remark 1.3, if $S$ and $T$ are completely separated there
are two completely separated zero (or cozero) sublocales that contain $S$ and
$T$. Furthermore, by Remarks 1.3 and 1.1 one can also obtain two completely
separated regular closed (or regular open) sublocales that contain $S$ and
$T$. Thus we get the following equivalent conditions to Theorem 3.1:

  * (iv)

$Z$ is completely separated form $f_{-1}[S]$ for some $S\in\mathfrak{S}M$ and
$Z\in\mathsf{ZS}(L)$, then there is $Z^{\prime}\in\mathsf{ZS}(M)$ such that
$Z\subseteq f_{-1}[Z^{\prime}]$ and $Z^{\prime}$ is completely separated from
$S$.

  * (v)

$C$ is completely separated form $f_{-1}[S]$ for some $S\in\mathfrak{S}M$ and
$C\in\mathsf{CoZS}(L)$, then there is $C^{\prime}\in\mathsf{CoZS}(M)$ such
that $C\subseteq f_{-1}[C^{\prime}]$ and $C^{\prime}$ is completely separated
from $S$.

  * (vi)

${\mathfrak{o}}_{L}(a^{*})$ is completely separated form $f_{-1}[S]$ for some
$S\in\mathfrak{S}M$, then there is a regular open sublocale
${\mathfrak{o}}_{M}(b^{*})$ such that ${\mathfrak{o}}_{L}(a^{*})\subseteq
f_{-1}[{\mathfrak{o}}_{M}(b^{*})]$ and ${\mathfrak{o}}_{M}(b^{*})$ is
completely separated from $S$.

  * (vii)

${\mathfrak{c}}_{L}(a^{*})$ is completely separated form $f_{-1}[S]$ for some
$S\in\mathfrak{S}M$, then there is a regular closed sublocale
${\mathfrak{c}}_{M}(b^{*})$ such that ${\mathfrak{c}}_{L}(a^{*})\subseteq
f_{-1}[{\mathfrak{c}}_{M}(b^{*})]$ and ${\mathfrak{c}}_{M}(b^{*})$ is
completely separated from $S$.

  * (viii)

${\mathfrak{c}}_{L}(a)$ is completely separated form $f_{-1}[S]$ for some
$S\in\mathfrak{S}M$, then there is a closed sublocale ${\mathfrak{c}}_{M}(b)$
such that ${\mathfrak{c}}_{L}(a)\subseteq f_{-1}[{\mathfrak{c}}_{M}(b)]$ and
${\mathfrak{c}}_{M}(b)$ is completely separated from $S$.

######  Remark 3.3.

  1. (1)

For any two selections $\mathfrak{S}_{1}$ and $\mathfrak{S}_{2}$, if
$\mathfrak{S}_{1}\subseteq\mathfrak{S}_{2}$ (that is, if
$\mathfrak{S}_{1}L\subseteq\mathfrak{S}_{2}L$ for every frame $L$) then any
$\mathfrak{S}_{2}\beta$-map is an $\mathfrak{S}_{1}\beta$-map.

  2. (2)

A localic map $f\colon L\to M$ is nearly open precisely when
$f_{-1}[\overline{{\mathfrak{o}}(a)}]=\overline{f_{-1}[{\mathfrak{o}}(a)]}$
for every $a\in L$, and two sublocales are completely separated if and only if
their closures are completely separated. Thus, if a localic map $f$ is nearly
open, then it is an $\mathfrak{S}_{\mathsf{reg}_{\mathfrak{c}}}\beta$-map if
and only if it is $\mathfrak{S}_{\mathfrak{o}}\beta$.

###  3.1. More characterizations of variants of normality

Let us provide characterizations of variants of normality using
$\mathfrak{S}\beta$-maps. A frame $L$ is _$\mathfrak{S}$ -normally separated_
when every $S\in\mathfrak{S}L$ is completely separated from every closed
sublocale disjoint from it. For instance,
$\mathfrak{S}_{\mathfrak{c}}$-normally separated locales are precisely the
normal locales, and $\mathfrak{S}_{\mathsf{ZS}}$-normally separated locales
correspond to $\delta$-normally separated locales (see for instance [10, 1]).
A further example is that of almost normal frames. Recall from [18, 24] that a
locale $L$ is _almost normal_ if for every $a,b^{*}\in L$ such that $a\wedge
b^{*}=0$ there are $u,v\in L$ such that $a\vee u=1=v\wedge b^{*}$ and $u\wedge
v=1$. Almost normal frames are precisely the
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$-locales:

######  Proposition 3.4.

The following are equivalent for a locale $L$:

  1. (i)

$L$ is almost normal.

  2. (ii)

The rather below relation $\prec$ interpolates (consequently,
$\prec=\prec\\!\\!\prec$).

  3. (iii)

If ${\mathfrak{c}}(a)\cap{\mathfrak{c}}(b^{*})=\text{\sf O}$ then
${\mathfrak{c}}(a)$ and ${\mathfrak{c}}(b^{*})$ are completely separated.

  4. (iv)

If ${\mathfrak{c}}(a)\subseteq{\mathfrak{c}}(b)^{\circ}$ for $a,b\in L$, then
there $x\in L$ such that

${\mathfrak{c}}(a)\subseteq{\mathfrak{o}}(x)\subseteq\overline{{\mathfrak{o}}(x)}\subseteq{\mathfrak{c}}(b)^{\circ}.$

  5. (v)

If $\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(a)$ for $a,b\in L$,
then there $x\in L$ such that

$\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(x)\subseteq\overline{{\mathfrak{o}}(x)}\subseteq{\mathfrak{o}}(a).$

###### Proof.

(i)$\iff$(ii) is proved in [24, VII.1.3.2].
(ii)$\iff$(iii): It suffices to note two facts. Firstly, recall that
${\mathfrak{c}}(a)\cap{\mathfrak{c}}(b^{*})=\text{\sf O}$ if and only if
$a\vee b^{*}=1$, that is $b\prec a$. Secondly, by (ii), $b\prec\\!\\!\prec a$
means ${\mathfrak{o}}(b)$ and ${\mathfrak{c}}(a)$ are completely separated
(recall Proposition 1.2). Equivalently, by Remark 1.1,
$\overline{{\mathfrak{o}}(b)}={\mathfrak{c}}(b^{*})$ and ${\mathfrak{c}}(a)$
are completely separated.
(iii)$\implies$(iv): Let ${\mathfrak{c}}(a)\subseteq{\mathfrak{c}}(b)^{\circ}$
for some $a,b\in L$, then ${\mathfrak{c}}(a)$ and ${\mathfrak{c}}(b^{*})$ are
disjoint; by assumption, they are completely separated. Thus, by Remark 1.3,
there are $x,y\in\mathsf{Coz}~{}L$ such that
${\mathfrak{c}}(a)\subseteq{\mathfrak{o}}(y)$,
${\mathfrak{c}}(b^{*})\subseteq{\mathfrak{o}}(x)$ and
${\mathfrak{o}}(x)\cap{\mathfrak{o}}(y)=\text{\sf O}$. Hence,

${\mathfrak{c}}_{L}(a)\subseteq{\mathfrak{o}}_{L}(y)\subseteq\overline{{\mathfrak{o}}(y)}\subseteq{\mathfrak{c}}_{L}(x)\subseteq{\mathfrak{c}}(b)^{\circ}.$

(iv)$\implies$(v): Let
$\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(a)$ for $a,b\in L$, then
${\mathfrak{c}}(a)\subseteq{\mathfrak{c}}(b)^{\circ}$. By assumption, there is
$y\in L$, such that
${\mathfrak{c}}(a)\subseteq{\mathfrak{o}}(y)\subseteq\overline{{\mathfrak{o}}(y)}\subseteq{\mathfrak{c}}(b)^{\circ}$.
Taking complements we obtain

$\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(y^{*})\subseteq{\mathfrak{c}}(y)\subseteq{\mathfrak{o}}(a)$

and
${\mathfrak{o}}(y^{*})\subseteq\overline{{\mathfrak{o}}(y^{*})}\subseteq{\mathfrak{c}}(y)$.
(v)$\iff$(ii): This is very easily seen from equation (7): for any $a,b\in L$

$b\prec a\iff\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(a)$

and $\prec$ interpolates if for any $b\prec a$ there is $x\in L$ such that

$b\prec x\prec
a\iff\overline{{\mathfrak{o}}(b)}\subseteq{\mathfrak{o}}(x)\subseteq\overline{{\mathfrak{o}}(x)}\subseteq{\mathfrak{o}}(a).$

∎

The following Theorem was proved in [1, Theorem 5.3] and it is a
generalization of the results in [25], for sake of completeness we write it
below:

######  Theorem 3.5.

Let $\mathfrak{S}$ be a selection on sublocales, and let $M$ be a locale such
that every $T\in\mathfrak{S}M$ is complemented. The following are equivalent:

  1. (i)

$M$ is $\mathfrak{S}$-normally separated.

  2. (ii)

Every $\mathsf{z}$-closed localic map $f\colon L\to M$ is an
$\mathfrak{S}\beta$-map.

  3. (iii)

Every closed localic map $f\colon L\to M$ is an $\mathfrak{S}\beta$-map.

  4. (iv)

Every proper localic map $f\colon L\to M$ is an $\mathfrak{S}\beta$-map.

One applies Theorem 3.5 to the selections $\mathfrak{S}_{\mathfrak{c}}$,
$\mathfrak{S}_{\mathsf{ZS}}$ and $\mathfrak{S}_{\mathsf{reg}_{\mathfrak{c}}}$
and obtains a characterization for normal, $\delta$-normally separated (see
Corollaries 5.4 and 5.5 of [1] or [10]) and almost normal frames.

Recall from [10] that a locale $L$ is _weakly $\delta$-normally_ separated if
whenever ${\mathfrak{c}}(a)\cap{\mathfrak{c}}(b)=\text{\sf O}$ with
$a\in\mathsf{Coz}~{}L$ and $b\in L^{*}$, then ${\mathfrak{c}}(a)$ and
${\mathfrak{c}}(b)$ are completely separated in $L$. A locale $L$ is _mildly
normal_ if for every regular elements $a$ and $b$ in $L$ such that $a\vee
b=1$, there exist $c,d\in L$ such that $c\wedge d=0$ and $a\vee c=b\vee d=1$
([22]); equivalently, if disjoint regular closed sublocales are completely
separated.

######  Proposition 3.6.

Let $M$ be a locale.

  1. (1)

$M$ is weakly $\delta$-normally separated if and only if every
$\mathsf{reg}_{\mathfrak{c}}$-preserving map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathsf{ZS}}$-map.

  2. (2)

$M$ is almost normal if and only if every
$\mathsf{reg}_{\mathfrak{c}}$-preserving map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{c}}\beta$-map.

  3. (3)

$M$ is mildly normal if and only if every
$\mathsf{reg}_{\mathfrak{c}}$-preserving map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$-map.

###### Proof.

(1): Let $M$ be a weakly $\delta$-normally separated frame and let $f\colon
L\to M$ be a $\mathsf{reg}_{\mathfrak{c}}$-preserving map. Let $a\in L$ and
$b\in\mathsf{Coz}M$ such that $\overline{{\mathfrak{o}}_{L}(a)}$ is completely
separated from $f_{-1}[{\mathfrak{c}}_{M}(b)]$. In particular we have

$\overline{{\mathfrak{o}}_{L}(a)}\subseteq
f_{-1}[{\mathfrak{c}}_{M}(b)]^{\\#}=f_{-1}[{\mathfrak{o}}_{M}(b)].$

Thus, by the image/preimage adjunction
$f[\overline{{\mathfrak{o}}_{L}(a)}]\subseteq{\mathfrak{o}}_{M}(b)$. That is,
$f[\overline{{\mathfrak{o}}_{L}(a)}]\cap{\mathfrak{c}}_{M}(b)=\text{\sf O}$.
Since $f$ is $\mathsf{reg}_{\mathfrak{c}}$-preserving,
$f[\overline{{\mathfrak{o}}_{L}(a)}]$ is regular closed, and $M$ is weakly
$\delta$-normally separated so $f[\overline{{\mathfrak{o}}_{L}(a)}]$ and
${\mathfrak{c}}_{M}(b)$ are completely separated in $M$, which proves that $f$
is a $\mathfrak{S}_{\mathsf{ZS}}$-map.
Conversely, assume that every $\mathsf{reg}_{\mathfrak{c}}$-preserving map
$f\colon L\to M$ is an $\mathfrak{S}_{\mathsf{ZS}}$-map. Let $a\in M$ and
$b\in\mathsf{Coz}~{}M$ such that
$\overline{{\mathfrak{o}}_{M}(a)}\cap{\mathfrak{c}}_{M}(b)=\text{\sf O}$. Let
us right $S:=\overline{{\mathfrak{o}}_{M}(a)}={\mathfrak{c}}_{M}(a^{*})$.
Consider the embedding $j\colon\overline{{\mathfrak{o}}_{M}(a)}\hookrightarrow
M$; it is $\mathsf{reg}_{\mathfrak{c}}$-closed. Indeed take a regular element
$x^{*_{S}}$ of $S$. Using properties of the Heyting operator ([23, Proposition
3.1.1]), we know

$x^{*_{S}}=x\rightarrow a^{*}=x\rightarrow(a\rightarrow 0)=(x\wedge
a)\rightarrow 0=(x\wedge a)^{*}.$

Thus $x^{*_{S}}$ is a regular element of $M$. Then, since closed sublocales of
closed sublocales are closed,
$j[{\mathfrak{c}}_{S}(x^{*_{S}})]={\mathfrak{c}}_{S}(x^{*_{S}})={\mathfrak{c}}_{M}(x^{*_{S}})$
is a regular closed sublocale of $M$. Thus, $j$ is
$\mathsf{reg}_{\mathfrak{c}}$-preserving and hence a
$\mathfrak{S}_{\mathsf{ZS}}$-map. Note that $\overline{{\mathfrak{o}}_{M}(a)}$
and $j{-1}[{\mathfrak{c}}_{M}(b)]=\text{\sf O}$ are trivially completely
separated in $S$. Then there exists a zero sublocale $Z\in\mathsf{ZS}(M)$ such
that $\overline{{\mathfrak{o}}_{M}(a)}\subseteq Z$ and $Z$ is completely
separated from ${\mathfrak{c}}_{M}(b)$. In particular,
$\overline{{\mathfrak{o}}_{M}(a)}$ and ${\mathfrak{c}}_{M}(b)$ are completely
separated in $M$.
For (2) and (3) one mimics the proof of (1) considering $b\in M$ or $b^{*}\in
M$, and substituting $\mathfrak{S}_{\mathsf{ZS}}$ for
$\mathfrak{S}_{\mathfrak{c}}$ and $\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$,
respectively. ∎

The following result is an improvement of Proposition 6.1 in [10] where the
authors impose two extra conditions, namely assertiveness and
$coz$-closedness222A word of caution to the reader: $coz$-closed maps of [10]
are not the same as $\mathsf{z}$-closed maps, as one would expect. More will
be commented in Remark 4.11.. As you can see this result is very similar to
the classical one in [25, Lemma 1.4]:

######  Proposition 3.7.

Let $f\colon L\to M$ be an onto $\mathfrak{S}_{\mathsf{ZS}}$-localic map.

  1. (1)

If $L$ is $\delta$-normally separated then so is $M$.

  2. (2)

If $L$ is weakly $\delta$-normally separated then so is $M$.

###### Proof.

(1): Take $Z\in\mathsf{ZS}(M)$ and $a\in M$ such that
$Z\cap{\mathfrak{c}}_{L}(a)=\text{\sf O}$. Then $f_{-1}[Z]\cap
f_{-1}[{\mathfrak{c}}_{M}(a)]=\text{\sf O}$. Since $L$ is $\delta$-normally
separated, there is $D\in\mathsf{ZS}(L)$ such that $f_{-1}[Z]\cap D=\text{\sf
O}$ and $f_{-1}[{\mathfrak{c}}_{L}(a)]\subseteq D$. Since $f$ is a
$\mathfrak{S}_{\mathsf{ZS}}$-map there is $Z^{\prime}\in\mathsf{ZS}(M)$ such
that $Z$ and $Z^{\prime}$ are completely separated in $M$ and $D\subseteq
f_{-1}[Z^{\prime}]$. we have

$f_{-1}[{\mathfrak{c}}_{M}(a)]\subseteq D\subseteq f_{-1}[Z^{\prime}]$

then $f[f_{-1}[{\mathfrak{c}}_{M}(a)]]\subseteq Z^{\prime}$. Thus, $Z$ and
$f[f_{-1}[{\mathfrak{c}}_{M}(a)]]$ are completely separated in $M$. In fact,
by Remark 1.1, $Z$ and $\overline{f[f_{-1}[{\mathfrak{c}}_{M}(a)]]}$ are
completely separated in $M$. Notice that

$\overline{f[f_{-1}[{\mathfrak{c}}_{M}(a)]]}=\overline{f[{\mathfrak{c}}_{L}(h(a))]]}={\mathfrak{c}}_{M}(f(h(a)))={\mathfrak{c}}_{M}(a)$

where the last equality holds because $f$ is onto. Hence, $Z$ and
${\mathfrak{c}}_{M}(a)$ are completely separated in $M$, as required.
The proof of (2) follows similarly as in (1), but one has to be careful with
regular elements. Let $Z\in\mathsf{ZS}(M)$ and $a\in M$ such that
$Z\cap{\mathfrak{c}}_{L}(a^{*})=\text{\sf O}$, then $f_{-1}[Z]\cap
f_{-1}[{\mathfrak{c}}_{M}(a^{*})]=\text{\sf O}$. Note that

${\mathfrak{c}}_{L}(h(a)^{*})=\overline{f_{-1}[{\mathfrak{o}}_{M}(a)]}\subseteq
f_{-1}[\overline{{\mathfrak{o}}_{M}(a)}]=f_{-1}[{\mathfrak{c}}_{M}(a^{*})].$

Hence, $f_{-1}[Z]\cap{\mathfrak{c}}_{L}(h(a)^{*})=\text{\sf O}$. Since $L$ is
weakly $\delta$-normally separated, there is $D\in\mathsf{ZS}(L)$ such that
$f_{-1}[Z]\cap D=\text{\sf O}$ and ${\mathfrak{c}}_{L}(h(a)^{*})\subseteq D$.
Since $f$ is a $\mathfrak{S}_{\mathsf{ZS}}\beta$-map there is
$Z^{\prime}\in\mathsf{ZS}(M)$ such that $Z$ and $Z^{\prime}$ are completely
separated in $M$ and $D\subseteq f_{-1}[Z^{\prime}]$. We have that

${\mathfrak{c}}_{L}(h(a)^{*})\subseteq D\subseteq f_{-1}[Z^{\prime}],$

so using the image/preimage adjunction and the fact that $Z^{\prime}$ is
closed, we get $\overline{f[{\mathfrak{c}}_{L}(h(a)^{*})]}\subseteq
Z^{\prime}$. Consequently, $Z$ and
$\overline{f[{\mathfrak{c}}_{L}(h(a)^{*})]}$ are completely separated in $M$.
Notice that
$\overline{f[{\mathfrak{c}}_{L}(h(a)^{*})]}={\mathfrak{c}}_{M}(f(h(a)^{*}))$
and

$f(h(a)^{*})=f(h(a)\to 0)=a\to f(0)=a\to 0=a^{*}$

where the second and third equality hold because $f$ is an onto localic map.
Thus, $Z$ and ${\mathfrak{c}}_{M}(a^{*})$ are completely separated in $M$, as
required. ∎

It is well-known that the image of a normal locale under a closed map is
normal ([24, VIII.5.2.2]). In fact, mimicking the proof of the previous
proposition, we also obtain the following result:

######  Proposition 3.8.

Let $f\colon L\to M$ be an onto $\mathfrak{S}_{\mathfrak{c}}\beta$-localic
map.

  1. (1)

If $L$ is normal then so is $M$.

  2. (2)

If $L$ is almost normal then so is $M$.

###  3.2. Characterizing variants of disconnectedness

We have generalized the notion of $\beta$-maps using selection of sublocales;
it is only natural to dualize the notions and results of the previous
subsection to selections of open sublocales.

We say a frame $L$ is _$\mathfrak{S}$ -disconnected_ if $S\in\mathfrak{S}L$ is
completely separated from every open sublocale disjoint from it.

######  Remark 3.9.

A locale $L$ is $\mathfrak{S}$-disconnected if and only if $\overline{S}$ is
clopen for every $S\in\mathfrak{S}L$. Indeed, in general we have
$S\cap\overline{S}^{\\#}=\text{\sf O}$ for any $S\in\mathfrak{S}L$; by
$\mathfrak{S}$-disconnectedness, $S$ and $\overline{S}^{\\#}$ are completely
separated. In particular, there exist $Z\in\mathsf{ZS}(L)$ and
$C\in\mathsf{CoZS}(L)$ such that $S\subseteq Z\subseteq
C\subseteq\overline{S}$. Hence,

$\overline{S}^{\circ}\subseteq\overline{S}\subseteq Z\subseteq
C\subseteq\overline{S}^{\circ}.$

Conversely, suppose $S\cap{\mathfrak{o}}(a)=\text{\sf O}$ for some
$S\in\mathfrak{S}L$ and $a\in L$, then
$\overline{S}\cap{\mathfrak{o}}(a)=\text{\sf O}$. Clearly
$S\subseteq\overline{S}$ and ${\mathfrak{o}}(a)\subseteq\overline{S}^{\\#}$.
By assumption, $\overline{S}$ and $\overline{S}^{\\#}$ are disjoint clopens,
and hence disjoint zero sublocales; thus, they completely separate $S$ and
${\mathfrak{o}}(a)$.

We obtain the dual result of Theorem 3.5:

######  Theorem 3.10.

Let $\mathfrak{S}$ be a selection on sublocales, and let $M$ be a locale such
that every $T\in\mathfrak{S}M$ is complemented. The following are equivalent:

  1. (i)

$M$ is $\mathfrak{S}$-disconnected.

  2. (ii)

Every $\mathsf{coz}$-open localic map $f\colon L\to M$ is an
$\mathfrak{S}\beta$-map.

  3. (iii)

Every open localic map $f\colon L\to M$ is an $\mathfrak{S}\beta$-map.

###### Proof.

(i)$\implies$(ii): Assume $M$ is $\mathfrak{S}$-disconnected. Let $f\colon
L\to M$ be a $\mathsf{coz}$-open localic map. Let $C\in\mathsf{CoZS}(L)$ and
$S\in\mathfrak{S}M$ such that $C$ and $f_{-1}[S]$ are completely separated in
$L$. In particular, $C\subseteq f_{-1}[S]^{\\#}=f_{-1}[S^{\\#}]$. By the
image/ preimage adjunction, we have that $f[C]\subseteq S^{\\#}$. Since every
sublocale in $\mathfrak{S}M$ is complemented, we get $f[C]\cap S=\text{\sf
O}$. By the $\mathsf{coz}$-openness of $f$ and the
$\mathfrak{S}$-disconnectedness of $M$, we know $f[C]$ and $S$ are completely
separated in $M$. In particular, by Remark 1.3, there is a cozero sublocale
$D\in\mathsf{CozS}(M)$ such that $D$ is completely separated from $S$ and
$f[C]\subseteq D$ (equivalently, $C\subseteq f_{-1}[D]$).
Clearly (ii)$\implies$ (iii), since very open localic map is
$\mathsf{coz}$-open.
(iii)$\implies$(i): Let $S\in\mathfrak{S}M$ and $a\in M$ such that
$S\cap{\mathfrak{o}}_{M}(a)=\text{\sf O}$. Take the open localic embedding
$j\colon{\mathfrak{o}}_{M}(a)\hookrightarrow L$; by assumption, it is an
$\mathfrak{S}\beta$-map. Trivially, ${\mathfrak{o}}_{M}(a)$ and $j_{-1}[S]$
are completely separated in ${\mathfrak{o}}_{M}(a)$ (since
$j_{-1}[S]=S\cap{\mathfrak{o}}_{M}(a)=\text{\sf O}$). Hence, there is $b\in M$
such that ${\mathfrak{o}}_{M}(b)$ and $S$ are completely separated in $M$, and
${\mathfrak{o}}_{M}(a)\subseteq
j_{-1}[{\mathfrak{o}}_{M}(b)]={\mathfrak{o}}_{M}(a)\cap{\mathfrak{o}}_{M}(b)$.
Thus, ${\mathfrak{o}}_{M}(a)\subseteq{\mathfrak{o}}_{M}(b)$, meaning
${\mathfrak{o}}_{M}(a)$ and $S$ are also completely separated, showing that
$M$ is $\mathfrak{S}$-disconnected. ∎

Recall that a frame $L$ is _extremally disconnected_ (resp. basically
disconnected) if $a^{*}\vee a^{**}=1$ for every $a\in L$ (resp.
$a\in\mathsf{Coz}~{}L$). Using the characterizations in [19] and [4,
Proposition 8.4.3] one can easily see that
$\mathfrak{S}_{\mathfrak{o}}$-disconnectedness is just extremal
disconnectedness, and a locale is $\mathfrak{S}_{\mathsf{CoZS}}$-disconnected
precisely if it is basically disconnected. Thus, applying Theorem 3.10 to
these selections we obtain the following two corollaries:

######  Corollary 3.11.

The following are equivalent for a locale $M$:

  1. (i)

$M$ is extremally disconnected.

  2. (ii)

Every $\mathsf{coz}$-open localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map.

  3. (iii)

Every open localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map.

######  Corollary 3.12.

The following are equivalent for a locale $M$:

  1. (i)

$M$ is basically disconnected.

  2. (ii)

Every $\mathsf{coz}$-open localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

  3. (iii)

Every open localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

A locale $L$ is said to be an _$F$ -frame_ if every pair of disjoint cozero
sublocales are completely separated in $L$ ([4, Proposition 8.4.10]).

######  Proposition 3.13.

Let $M$ be a locale.

  1. (1)

$M$ is an $F$-frame if and only if every $\mathsf{coz}$-preserving map is an
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

  2. (2)

$M$ is basically disconnected if and only if every $\mathsf{coz}$-preserving
map is an $\mathfrak{S}_{\mathfrak{o}}\beta$-map.

###### Proof.

Let $M$ be an $F$-frame (resp. basically disconnected frame) and $f\colon L\to
M$ be a $\mathsf{coz}$-preserving localic map. Let $C\in\mathsf{CoZS}(L)$ and
$D\in\mathsf{CoZS}(M)$ (resp. $D$ and open sublocale of $M$) such that $C$ and
$f_{-1}[D]$ are completely separated. In particular, $C\cap
f_{-1}[D]=\text{\sf O}$. Hence, $C\subseteq f_{-1}[D^{\\#}]$. By the
image/preimage adjunction $f[C]\subseteq D^{\\#}$, equivalently $f[C]\cap
D=\text{\sf O}$, and $f[C]$ is a cozero sublocale ($f$ is
$\mathsf{coz}$-preserving). Since $M$ is an $F$-frame (resp. basically
disconnected), $f[C]$ and $D$ are completely separated and $C\subseteq
f_{-1}[f[C]]$.
Conversely, suppose every $\mathsf{coz}$-preserving map is an
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map (resp.
$\mathfrak{S}_{\mathfrak{o}}\beta$-map). Let $C_{1},C_{2}\in\mathsf{CoZS}(L)$
(resp. $C_{1}\in\mathsf{CoZS}(L)$ and $C_{2}$ an open sublocale) such that
$C_{1}\cap C_{2}=\text{\sf O}$. Consider the localic embedding $j\colon
C_{1}\hookrightarrow L$. It is a $\mathsf{coz}$-preserving map. Indeed, every
$C\in\mathsf{CoZS}(C_{1})$, since $C_{1}$ is a cozero sublocale, is a cozero
sublocale of $L$ (recall [4, Proposition ]); that is,
$j[C]=C\in\mathsf{CoZS}(L)$. Trivially $C_{1}$ and $j_{-1}[C_{2}]$ are
completely separated in $C_{1}$, since $j_{-1}[C_{2}]=C_{1}\cap
C_{2}=\text{\sf O}$. By assumption, there is $D\in\mathsf{CoZS}(L)$ such that
$C_{1}\subseteq D$ and $D$ is completely separated from $C_{2}$. In
particular, $C_{1}$ and $C_{2}$ are completely separated, as required.

∎

A localic map $f\colon L\to M$ is _$\prec\\!\\!\prec$ -preserving_ if
$a\prec\\!\\!\prec b$ implies $f(a)\prec\\!\\!\prec f(b)$ for every $a,b\in
L$. The frame homomorphisms corresponding to these maps were introduced in
[10] and are called assertive. It seems natural (recall Theorem 3.1 (iii)) to
ask if $\mathfrak{S}\beta$-maps are related to $\prec\\!\\!\prec$-presserving
maps. In [15, Examples 3.13] the authors give examples showing that the
notions of $\mathfrak{S}_{\mathfrak{c}}\beta$-maps and
$\prec\\!\\!\prec$-preserving are not related. Nevertheless,
$\prec\\!\\!\prec$-preserving are precisely the
$\mathfrak{S}_{\mathfrak{o}}\beta$-maps.

######  Theorem 3.14.

Let $f\colon L\to M$ be a localic map, $f$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map if and only if it is
$\prec\\!\\!\prec$-preserving.

###### Proof.

Let $a,b\in L$ such that $a\prec\\!\\!\prec b$; in other words,
${\mathfrak{o}}_{L}(a)$ is completely separated from ${\mathfrak{c}}_{L}(b)$
(Proposition 1.2). We will show ${\mathfrak{o}}_{M}(f(a))$ and
${\mathfrak{c}}_{M}(f(a))$ are completely separated in $M$. Note that:

${\mathfrak{c}}_{L}(a)\subseteq f_{-1}[f[{\mathfrak{c}}_{L}(a)]]\subseteq
f_{-1}[\overline{f[{\mathfrak{c}}_{L}(a)]}]=f_{-1}[{\mathfrak{c}}_{M}(f(a))]$

By taking complements, we obtain
$f_{-1}[{\mathfrak{o}}_{M}(f(a))]\subseteq{\mathfrak{o}}_{L}(a)$. Thus,
$f_{-1}[{\mathfrak{o}}_{M}(f(a))]$ and ${\mathfrak{c}}_{L}(b)$ are completely
separated in $L$. Since $f$ is an $\mathfrak{S}_{\mathfrak{o}}\beta$-map,
there is $x\in M$ such that ${\mathfrak{c}}_{M}(x)$ is completely separated
from ${\mathfrak{o}}_{M}(f(a))$ in $M$, and ${\mathfrak{c}}_{L}(b)\subseteq
f_{-1}[{\mathfrak{c}}_{M}(x)]$. Furthermore,

${\mathfrak{c}}_{L}(b)\subseteq f_{-1}[{\mathfrak{c}}_{M}(x)]\iff
f[{\mathfrak{c}}_{L}(b)]\subseteq{\mathfrak{c}}_{M}(x)\iff{\mathfrak{c}}_{M}(f(b))=\overline{f[{\mathfrak{c}}_{L}(b)]}\subseteq{\mathfrak{c}}_{M}(x).$

Hence, ${\mathfrak{c}}_{M}(f(b))$ and ${\mathfrak{o}}_{M}(f(a))$ are
completely separated in $M$, as required.
Conversely, assume $f$ is $\prec\\!\\!\prec$-preserving and $b\in L$ and $a\in
M$ such that ${\mathfrak{c}}_{L}(b)$ and $f_{-1}[{\mathfrak{o}}_{M}(a)]$ are
completely separated. Equivalently, $h(a)\prec\\!\\!\prec b$, where $h$ is the
left adjoint (frame homomorphism) of $f$. By assumption,
$f(h(a))\prec\\!\\!\prec f(b)$. Since $a\leq f(h(a))$, we have
$a\prec\\!\\!\prec f(b)$ meaning ${\mathfrak{o}}_{L}(a)$ is completely
separated from ${\mathfrak{c}}_{M}(f(b))$. Furtheremore,

${\mathfrak{c}}_{L}(b)\subseteq{\mathfrak{c}}_{L}(h(f(b)))=f_{-1}[{\mathfrak{c}}_{M}(f(b))]$

again for the inclusion above we are using the fact that $f$ and $h$ are
adjoints. ∎

Using the definition of $\mathfrak{S}\beta$-map and Theorem 3.1, we put
together all the characterizing conditions of $\prec\\!\\!\prec$-preserving
maps. Recall the notation in diagram (12) and the fact that
$\mathop{\textstyle\bigwedge}{\mathfrak{o}}(a)=a^{*}$,

######  Corollary 3.15.

Let $f\colon L\to M$ be a localic map, the following are equivalent:

  1. (i)

$f$ is $\prec\\!\\!\prec$-preserving.

  2. (ii)

$\overline{\beta_{L}\left[f_{-1}[{\mathfrak{o}}_{M}(a)]\right]}^{\beta
L}=f_{-1}^{\beta}\left[\overline{\beta_{M}[{\mathfrak{o}}_{M}(a)]}^{\beta
M}\right]$ for every $a\in M$.

  3. (iii)

If an open sublocale ${\mathfrak{o}}_{L}(a)$ is completely separated from
$f_{-1}[{\mathfrak{o}}_{M}(x)]$ for some $x\in M$, then there is an open
sublocale ${\mathfrak{o}}_{M}(b)$ such that ${\mathfrak{o}}_{L}(a)\subseteq
f_{-1}[{\mathfrak{o}}_{M}(b)]$ and ${\mathfrak{o}}_{M}(b)$ is completely
separated from ${\mathfrak{o}}_{M}(x)$ in $M$.

  4. (iv)

$h^{\beta}(\beta_{M}(x^{*}))=\beta_{L}(h(x)^{*})$ for every $x\in M$.

  5. (v)

For every $a\in L$ such that $a\prec\\!\\!\prec h(x)^{*}$ there exists $b\in
M$ such that $a\leq h(b)$ and $b\prec\\!\\!\prec x^{*}$.

######  Remark 3.16.

Clearly, if a map is $\prec\\!\\!\prec$-preserving the image of complemented
elements is complemented. Thus, if $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map then $\overline{f[S]}$ is clopen for
every clopen sublocale $S$ of $L$.

######  Proposition 3.17.

Let $L$ be a an $\mathfrak{S}$-disconnected locale. If $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map then $\overline{f[S]}$ is clopen for
every $S\in\mathfrak{S}L$.

###### Proof.

Let $S\in\mathfrak{S}L$, by Remark 3.9, $\overline{S}$ is clopen. Hence, using
Theroem 3.14 (in particular, Remark 3.16), we get
$\overline{f[\overline{S}]}=\overline{f[S]}$ is clopen. ∎

From Proposition 3.17 one gets that if $L$ is extremally diconnected, then
every localic $\mathfrak{S}_{\mathfrak{o}}\beta$-map $f\colon L\to M$ is
nearly open.

We know that the image of an extremally disconnected frame under an open
localic map is extremally disconnected. In fact, we obtain the following
result:

######  Proposition 3.18.

Let $f\colon L\to M$ be a dense $\mathfrak{S}_{\mathfrak{o}}\beta$-localic
map.

  1. (1)

If $L$ extremally disconnected then so is $M$.

  2. (2)

If $L$ basically disconnected then so is $M$.

###### Proof.

Let $f\colon L\to M$ be an onto $\mathfrak{S}_{\mathfrak{o}}\beta$-localic
map, and let $L$ be an extremally disconnected (resp. basically disconnected)
locale. Let $a,b\in M$ (resp. $a\in M$ and $b\in\mathsf{Coz}~{}M$) such that
${\mathfrak{o}}_{M}(a)\cap{\mathfrak{o}}_{M}(b)=\text{\sf O}$, then
$f_{-1}[{\mathfrak{o}}_{M}(a)]\cap f_{-1}[{\mathfrak{o}}_{M}(b)]=\text{\sf
O}$. Since $L$ is extremally disconnected (resp. basically disconnected),
$f_{-1}[{\mathfrak{o}}_{M}(a)]$ and $f_{-1}[{\mathfrak{o}}_{M}(b)]$ are
completely separated in $L$. By Remark 1.1,
$\overline{f_{-1}[{\mathfrak{o}}_{M}(a)]}$ and $f_{-1}[{\mathfrak{o}}_{M}(b)]$
are completely separated. We know $f$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-localic map, so there is $x\in M$ such that
${\mathfrak{c}}_{M}(x)$ is completely separated from ${\mathfrak{o}}_{M}(b)$
and

$\overline{f_{-1}[{\mathfrak{o}}_{M}(a)]}\subseteq
f_{-1}[{\mathfrak{c}}_{M}(x)].$

Thus, by the image/preimage adjunction and the fact that
${\mathfrak{c}}_{M}(x)$ is closed, we have that

$\overline{f[\overline{f_{-1}[{\mathfrak{o}}_{M}(a)]}]}~{}\subseteq~{}{\mathfrak{c}}_{M}(x).$

Note that

$\overline{f[\overline{f_{-1}[{\mathfrak{o}}_{M}(a)]}]}=\overline{f[\overline{{\mathfrak{o}}_{L}(h(a))}]}=\overline{f[{\mathfrak{c}}_{L}(h(a)^{*})]}={\mathfrak{c}}_{M}(f(h(a)^{*}))={\mathfrak{c}}_{M}(a^{*})=\overline{{\mathfrak{o}}_{M}(a)}$

where the fourth equality holds because $f$ is dense which implies that
$f(0)=0$ and $f(h(a)^{*}))=f(h(a)\rightarrow 0)=a\rightarrow f(0)=a\rightarrow
0=a^{*}$. Hence, since
${\mathfrak{o}}_{M}(a)\subseteq\overline{{\mathfrak{o}}_{M}(a)}\subseteq{\mathfrak{c}}_{M}(x)$,
the sublocales ${\mathfrak{o}}_{M}(a)$ and ${\mathfrak{o}}_{M}(b)$ are
completely separated in $M$ showing that $M$ is extremally disconnected (resp.
basically disconnected). ∎

Since the localic preimage preserves cozero sublocales, the proof of
Proposition 3.19 follows similarly as the proof of Proposition 3.18. One takes
$a,b\in\mathsf{Coz}~{}L$ (resp. $a\in L$ and $b\in\mathsf{Coz}~{}L$) and uses
the fact that $L$ is an $F$-frame (resp. basically disconnected frame) and
that $f$ is an $\mathfrak{S}_{\mathsf{CoZS}}\beta$-localic map.

######  Proposition 3.19.

Let $f\colon L\to M$ be an dense $\mathfrak{S}_{\mathsf{CoZS}}\beta$-localic
map.

  1. (1)

If $L$ basically disconnected then so is $M$.

  2. (2)

If $L$ is an $F$-frame then so is $M$.

###  3.3. Special cases

In this section we take a look at when the notions of disconnectedness and
normality coincide. A locale is _Boolean_ if every element is complemented;
equivalently if every closed and every open sublocale is clopen. On the other
hand, a locale $L$ is a _$P$ -frame_ if every cozero element is complemented
([4]); again, this is equivalent to saying that every cozero or every zero
sublocale is clopen (in a $P$-frame the classes of cozero, zero and clopen
sublocales coincide).

######  Proposition 3.20.

$L$ is boolean if and only if it is $\mathfrak{S}_{\mathfrak{o}}$-normally
separated if and only if it is $\mathfrak{S}_{\mathfrak{c}}$-disconnected.

###### Proof.

It is obvious by the definitions that $\mathfrak{S}_{\mathfrak{o}}$-normally
separated and $\mathfrak{S}_{\mathfrak{c}}$-disconnected coincide. Now, by
Remark 3.9, $L$ is $\mathfrak{S}_{\mathfrak{c}}$-disconnected if and only if
$\overline{{\mathfrak{c}}(a)}={\mathfrak{c}}(a)$ is clopen for every $a\in L$.
∎

######  Proposition 3.21.

The following are equivalent for a locale $L$:

  1. (i)

$L$ is a $P$-frame.

  2. (ii)

$L$ is $\mathfrak{S}_{\mathsf{CoZS}}$-normally separated.

  3. (iii)

$L$ is $\mathfrak{S}_{\mathsf{ZS}}$-disconnected

###### Proof.

(i)$\implies$(ii): Let $C\in\mathsf{CoZS}(L)$ and $a\in L$ such that
$C\cap{\mathfrak{c}}(a)=\text{\sf O}$. Since $L$ is a $P$-frame $C$ is also a
zero sublocale; thus, $C$ and $C^{\\#}$ are disjoint zero sublocales and
${\mathfrak{c}}(a)\subseteq C^{\\#}$.
(ii)$\implies$(iii): Let ${\mathfrak{c}}(b)\in\mathsf{ZS}(L)$ and $a\in L$
such that ${\mathfrak{c}}(a)\cap{\mathfrak{o}}(b)=\text{\sf O}$. We have that,
${\mathfrak{o}}(a)\in\mathsf{CoZS}(L)$,
${\mathfrak{o}}(b)\subseteq{\mathfrak{o}}(a)$, and
${\mathfrak{c}}(a)\cap{\mathfrak{o}}(a)=\text{\sf O}$. By assumption,
${\mathfrak{c}}(b)$ and ${\mathfrak{o}}(b)$ are completeley separated; in
particular, so are ${\mathfrak{c}}(b)$ and ${\mathfrak{o}}(a)$, as required.
(iii)$\implies$(i): Let $a\in\mathsf{Coz}~{}L$, then
${\mathfrak{c}}(a)\cap{\mathfrak{o}}(a)=\text{\sf O}$. By assumption, they are
completely separated; in particular, we have
${\mathfrak{c}}(a)\cap\overline{{\mathfrak{o}}(a)}=\text{\sf O}$.
Equivalentely, $a\vee a^{*}=1$. Thus every cozero element of $L$ is
complemented. ∎

Then we obtain as corollaries of Theorems 3.1 and 3.10:

######  Corollary 3.22.

The following are equivalent for a locale $M$:

  1. (i)

$M$ is a $P$-frame.

  2. (ii)

Every closed localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

  3. (iii)

Every $\mathsf{z}$-closed localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

  4. (iv)

Every proper localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

  5. (v)

Every open localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{ZS}}\beta$-map.

  6. (vi)

Every $\mathsf{coz}$-open localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{ZS}}\beta$-map.

######  Corollary 3.23.

The following are equivalent for a locale $M$:

  1. (i)

$M$ is a boolean.

  2. (ii)

Every closed localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{o}}\beta$-map.

  3. (iii)

Every $\mathsf{z}$-closed localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{o}}\beta$-map.

  4. (iv)

Every proper localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{o}}\beta$-map.

  5. (v)

Every open localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{c}}\beta$-map.

  6. (vi)

Every $\mathsf{coz}$-open localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{c}}\beta$-map.

######  Remark 3.24.

Furthermore, since every closed and every open sublocale in a Boolean frame is
clopen, a map $f\colon L\to M$ with $M$ boolean is an
$\mathfrak{S}_{\mathfrak{c}}\beta$-map if and only if it is a
$\mathfrak{S}_{\mathfrak{o}}\beta$-map. Similarly, since zero sublocales and
cozero sublocales coincide in a $P$-frame $M$, then $f\colon L\to M$ is an
$\mathfrak{S}_{\mathsf{ZS}}\beta$-map if an only if it is an
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map.

The following proposition is an improvement of [11, Proposition 4.4], in the
sense that we are adding another equivalent condition.

######  Proposition 3.25.

The following are equivalent for a frame $M$:

  1. (i)

$M$ is boolean (resp. $P$-frame)

  2. (ii)

Every localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map (resp.
$\mathfrak{S}_{\mathsf{CoZS}}\beta$-map).

  3. (iii)

Every localic map $f\colon L\to M$ is an
$\mathfrak{S}_{\mathfrak{c}}\beta$-map. (resp.
$\mathfrak{S}_{\mathsf{ZS}}\beta$-map)

###### Proof.

Clearly, by Remark 3.24 (ii)$\iff$(iii) holds. By Corollary 3.23 and Corollary
3.22, (iii)$\implies$(i). Let us show (i)$\implies$(iii). Let $a\in L$ and
$b\in M$ (resp. $b\in\mathsf{Coz}M$) such that

${\mathfrak{o}}_{L}(a)\cap f_{-1}[{\mathfrak{c}}_{M}(b)]=\text{\sf O}.$

Then ${\mathfrak{o}}_{L}(a)\subseteq f_{-1}[{\mathfrak{o}}_{M}(b)]$, and since
$M$ is boolean (resp. $P$-frame), ${\mathfrak{o}}_{M}(b)$ and
${\mathfrak{c}}_{M}(b)$ are clopen and hence zero sublocales. Thus
${\mathfrak{o}}_{M}(b)$ and ${\mathfrak{c}}_{M}(b)$ are completely separated
in $M$. ∎

To close this section we offer a table showing the different frames that
$\mathfrak{S}$-normally separation and $\mathfrak{S}$-disconnectedness
describe:

Selection $\mathfrak{S}$ |  $\mathfrak{S}$-normally separated |  $\mathfrak{S}$-disconnected
---|---|---
$\mathfrak{S}_{\mathfrak{c}}$ | normal | Boolean
$\mathfrak{S}_{\mathsf{ZS}}$ |  $\delta$-normally separated |  $P$-frame
$\mathfrak{S}_{\mathfrak{o}}$ | Boolean | Extremally disconnected
$\mathfrak{S}_{\mathsf{CoZS}}$ |  $P$-frame | Basically disconnected
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$ | Almost normal | Extremally disconnected
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{o}}}$ | Extremally disconnected | Extremally disconnected

##  4\. $\mathfrak{S}\lambda$-maps

Now we will look at the $\mathfrak{S}\gamma$-maps when $\gamma$ is the
Lindelöfication $\lambda$. These maps are a generalization of the
$\lambda$-maps introduced in [13, 15]. Indeed, the
$\mathfrak{S}_{\mathfrak{c}}\lambda$-maps are precisely the right adjoints of
the $\lambda$-maps, and the characterization in [14, Example 1] comes as a
corollary of the equivalence (i)$\iff$(iv) in the theorem below.

######  Theorem 4.1.

Let $f\colon L\to M$ be a localic map, then the following are equivalent:

  1. (i)

$f$ is a $\mathfrak{S}\lambda$-map.

  2. (ii)

If $f_{-1}[S]\subseteq Z$ for $S\in\mathfrak{S}M$ and $Z\in\mathsf{ZS}(L)$
then there is $D\in\mathsf{ZS}(M)$ such that $S\subseteq D$ and
$f_{-1}[D]\subseteq Z$.

  3. (iii)

If $f_{-1}[S]\cap C=\text{\sf O}$ for $S\in\mathfrak{S}M$ and
$C\in\mathsf{CoZS}(L)$ then there is $D\in\mathsf{CoZS}(M)$ such that $S\cap
D=\text{\sf O}$ and $C\subseteq f_{-1}[D]$.

  4. (iv)

If $x\leq\mathop{\textstyle\bigwedge}f_{-1}[S]$ for $x\in\mathsf{Coz}~{}L$ and
$S\in\mathfrak{S}L$, there exists $y\in\mathsf{Coz}~{}M$ such that
$y\leq\mathop{\textstyle\bigwedge}S$ and $x\leq h(y)$.

Furthermore, if every sublocale in $\mathfrak{S}M$ is complemented the
following conditions are also equivalent:

  1. (v)

If $C\subseteq f_{-1}[S^{\\#}]$ for $S\in\mathfrak{S}M$ and
$C\in\mathsf{CoZS}(L)$ then there is $D\in\mathsf{CoZS}(M)$ such that
$D\subseteq S^{\\#}$ and $C\subseteq f_{-1}[D]$.

  1. (vi)

If $Z\vee f_{-1}[S^{\\#}]=L$ for $S\in\mathfrak{S}M$ and $Z\in\mathsf{ZS}(L)$
then there is $Z^{\prime}\in\mathsf{ZS}(M)$ such that $Z^{\prime}\vee
S^{\\#}=M$ and $f_{-1}[Z^{\prime}]\subseteq Z$.

###### Proof.

(i)$\iff$(ii): By Remark 2.2 $f$ is $\mathfrak{S}\lambda$-map if and only if

$\lambda_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])\leq
h^{\lambda}\lambda_{M}(\mathop{\textstyle\bigwedge}S).$

Since
$\lambda_{L}(\mathop{\textstyle\bigwedge}f_{-1}[S])=\left\\{b\in\mathsf{Coz}L\mid
b\leq\mathop{\textstyle\bigwedge}f^{-1}[S]\right\\}$ and

$\displaystyle
h^{\lambda}\lambda_{M}(\mathop{\textstyle\bigwedge}S)=\left\\{c\in\mathsf{Coz}L\mid
c\leq h(a)\text{ for some }a\in\mathsf{Coz}M\text{ with
}a\leq\mathop{\textstyle\bigwedge}S\right\\},$

then $f$ is $\mathfrak{S}\lambda$-map if and only if for every
$b\in\mathsf{Coz}L$ such that $b\leq\mathop{\textstyle\bigwedge}f_{-1}[S]$
there is $a\in\mathsf{Coz}M$ with $a\leq\mathop{\textstyle\bigwedge}S$ and
$b\leq h(a)$. Equivalently, using closed sublocales and (9), for every
$Z\in\mathsf{ZS}(L)$ such that $\overline{f_{-1}[S]}\subseteq Z$ there is
$D\in\mathsf{ZS}(M)$ such that $\overline{S}\subseteq D$ and
$f_{-1}[D]\subseteq Z$. Since zero sublocales are closed and a sublocale is
contained in a closed sublocale if and only if its closure is, we get (ii).
(ii)$\iff$(iii) is obvious.
(ii)$\iff$(iv) is immediate by taking closed sublocales and using equations
(3) and (9).
When every $S\in\mathfrak{S}M$ is complemented, using the fact that
$f_{-1}[-]$ preserves complements, one can readily see that
(ii)$\iff$(iii)$\iff$(v)$\iff$(vi). ∎

We say a locale $L$ is _ $\mathfrak{S}$-coz separating_ if whenever
${\mathfrak{o}}(a)\cap S=\text{\sf O}$ for $a\in L$ and $S\in\mathfrak{S}L$,
then there is $C\in\mathsf{CoZS}(L)$ such that ${\mathfrak{o}}(a)\subseteq C$
and $C\cap S=\text{\sf O}$. Equivalently, if $S\subseteq{\mathfrak{c}}(a)$ for
some $a\in L$ and $S\in\mathfrak{S}L$, then there is $Z\in\mathsf{ZS}(L)$ such
that $S\subseteq Z\subseteq{\mathfrak{c}}(a)$.

######  Remark 4.2.

Clearly, since completely separated sublocales are disjoint, for a selection
$\mathfrak{S}$ we have that:

  * •

$\mathfrak{S}\beta$-map $\implies\mathfrak{S}\lambda$-map;

  * •

$\mathfrak{S}$-disconnected $\implies\mathfrak{S}$-coz separating.

######  Example 4.3.

A localic map $f\colon L\to M$ is _$\mathsf{z}$ -heavy_ if
$f_{-1}[{\mathfrak{c}}_{M}(a)]=\text{\sf O}$ for some $a\in M$, then there is
$Z\in\mathsf{ZS}(M)$ such that ${\mathfrak{c}}_{M}(a)\subseteq Z$ and
$f_{-1}[Z]=\text{\sf O}$ (these are the right adjoints of $coz$-heavy frame
homomorphisms [12]). Every $\mathfrak{S}_{\mathfrak{c}}\lambda$-map is
$\mathsf{z}$-heavy. Indeed, if $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{c}}\lambda$-map and let $a\in M$ such that
$f_{-1}[{\mathfrak{c}}_{M}(a)]=\text{\sf O}$. By (ii) of 4.1, there is
$Z\in\mathsf{ZS}(M)$ such that ${\mathfrak{c}}_{M}(a)\subseteq Z$ and
$f_{-1}[Z]\subseteq\text{\sf O}$.

Following classical topology, we say a sublocale $S$ of $L$ is _normally
placed_ whenever the embedding $j\colon S\hookrightarrow L$ is
$\mathsf{z}$-heavy. That is, if ${\mathfrak{c}}_{L}(a)\cap S=\text{\sf O}$ for
some $a\in L$, then there is $Z\in\mathsf{ZS}(L)$ such that
${\mathfrak{c}}_{L}(a)\subseteq Z$ and $Z\cap S=\text{\sf O}$. Equivlently, if
$S\subseteq{\mathfrak{o}}_{L}(a)$ for some $a\in L$, then there is
$C\in\mathsf{CoZS}(L)$ such that $S\subseteq C\subseteq{\mathfrak{o}}_{L}(a)$.
Obviously, every cozero sublocale is normally placed. Note that every zero
sublocale is normally placed if and only if $L$ is $\delta$-normally
separated. Moreover, if every open sublocale is coz-heavy then $L$ is
perfectly normal. Recall that a locale $L$ is _perfectly normal_ if
$L=\mathsf{Coz}~{}L$; that is every open (resp. closed) sublocale is a cozero
(resp. zero) sublocale.

Let us now look at the analogous result of 3.10 for
$\mathfrak{S}\lambda$-maps:

######  Theorem 4.4.

Let $\mathfrak{S}$ be a selection of sublocales, and let $M$ be a locale such
that every $T\in\mathfrak{S}M$ is complemented. The following are equivalent:

  1. (i)

$M$ is $\mathfrak{S}$-coz separating.

  2. (ii)

Every $\mathsf{coz}$-open map is a $\mathfrak{S}\lambda$-map.

  3. (iii)

Every open map is a $\mathfrak{S}\lambda$-map.

###### Proof.

(i)$\implies$(ii): Let $f\colon L\to M$ be a $\mathsf{coz}$-open localic map.
We will use Theorem 4.1 to show $f$ is a $\mathfrak{S}\lambda$-map. Let
$S\in\mathfrak{S}$ and $C\in\mathsf{CoZS}(L)$ such that $C\cap
f_{-1}[S]=\text{\sf O}$, then $C\subseteq f_{-1}[S^{\\#}]$. By the
image/preimage adjunction we have $f[C]\subseteq S^{\\#}$; equivalently, since
$S$ is complemented, $f[C]\cap S=\text{\sf O}$. By assumption, there is
$D\in\mathsf{CoZS}(M)$ such that $D\cap S=\text{\sf O}$ and $f[C]\subseteq D$
(equiv. $C\subseteq f_{-1}[D]$), as required.
Since every open map is $\mathsf{coz}$-open then clearly (ii)$\implies$(iii).
(iii)$\implies$(i): Let $a\in M$ and $S\in\mathfrak{S}M$ such that
${\mathfrak{o}}(a)\cap S=\text{\sf O}$. Consider the open localic embedding
$j\colon{\mathfrak{o}}(a)\hookrightarrow M$. By assumption, it is an
$\mathfrak{S}\lambda$-map. Furthermore, the sublocale ${\mathfrak{o}}(a)$ is a
cozero sublocale of ${\mathfrak{o}}(a)$ and ${\mathfrak{o}}(a)\cap
j_{-1}[S]={\mathfrak{o}}(a)\cap S=\text{\sf O}$. Thus, there is
$D\in\mathsf{CoZS}(M)$ such that $D\cap S=\text{\sf O}$ and
${\mathfrak{o}}(a)\subseteq j_{-1}[D]={\mathfrak{o}}(a)\cap D$ (that is,
${\mathfrak{o}}(a)\subseteq D$). ∎

Let us look at examples of $\mathfrak{S}$-coz separating locales. Recall from
[6] that a locale $L$ is an _$Oz$ -frame_ if every regular element is a cozero
element333This is not how $Oz$-frames are traditionally defined, but we use
one of its characterizations to make it easier for the reader., and it is a
_weakly $Oz$-frame_ if every regular element $x\in L$ is equal to $c^{*}$ for
some $c\in\mathsf{Coz}~{}L$. By taking open sublocales in [6, Proposition 2.2]
and [6, Proposition 5.3] (c.f. [2, Section 4.3]) one can readily check that
indeed $Oz$\- and weakly $Oz$-frames coincide with
$\mathfrak{S}_{\mathfrak{o}}$\- and $\mathfrak{S}_{\mathsf{CoZS}}$-coz
separating frames respectively. Thus, we get the following two corollaries of
Theorem 4.4.

######  Corollary 4.5.

The following are equivalent for a locale $M$:

  1. (1)

$M$ is an $Oz$-frame.

  2. (2)

Every $\mathsf{coz}$-open map is a $\mathfrak{S}_{\mathfrak{o}}\lambda$-map.

  3. (3)

Every open map is a $\mathfrak{S}_{\mathfrak{o}}\lambda$-map.

######  Corollary 4.6.

The following are equivalent for a locale $M$:

  1. (1)

$M$ is a weakly $Oz$-frame.

  2. (2)

Every $\mathsf{coz}$-open map is a $\mathfrak{S}_{\mathsf{CoZS}}\lambda$-map.

  3. (3)

Every open map is a $\mathfrak{S}_{\mathsf{CoZS}}\lambda$-map.

###  4.1. Generalizing $\mathfrak{S}\lambda$-maps

In the characterization of $\mathfrak{S}\beta$-maps (Theorem 3.1(ii)), since
we are working with complete separation, one can choose to work with open,
closed, regular open, regular closed, zero or cozero sublocales (recall Remark
3.2). This is not the case for $\mathfrak{S}\lambda$-maps. Thus, it is worth
introducing more general classes of localic maps.
Let $\mathfrak{S}$ and $\mathfrak{T}$ be two selections of sublocales. We say
that a localic map $f\colon L\to M$ is _$\mathfrak{S}\mathfrak{T}\lambda$
-map_ if whenever $f_{-1}[S]\cap T=\text{\sf O}$ for $S\in\mathfrak{S}M$ and
$T\in\mathfrak{T}L$, there exists $T^{\prime}\in\mathfrak{T}M$ such that
$S\cap T^{\prime}=\text{\sf O}$ and $T\subseteq f_{-1}[T^{\prime}]$.

######  Remark 4.7.

  1. (1)

Clearly, $\mathfrak{S}\mathfrak{T}_{\mathsf{CoZS}}\lambda$-maps are precisely
the $\mathfrak{S}\lambda$-maps.

  2. (2)

For
$\mathfrak{T}=\mathfrak{T}_{\mathfrak{o}},\mathfrak{T}_{\mathfrak{c}},\mathfrak{T}_{\mathsf{CoZS}},\mathfrak{T}_{\mathsf{ZS}},\mathfrak{T}_{\mathsf{reg}{\mathfrak{o}}},\mathfrak{T}_{\mathsf{reg}{\mathfrak{c}}}$
every $\mathfrak{S}\beta$-map is a $\mathfrak{S}\mathfrak{T}\lambda$-map.

  3. (3)

If every $S\in\mathfrak{S}M$ is complemented and $S^{\\#}\in\mathfrak{T}M$,
then any map $f\colon L\to M$ is a $\mathfrak{S}\mathfrak{T}\lambda$-map. In
particular, if $\mathfrak{S}$ selects complemented sublocales and
$(\mathfrak{S}L)^{\\#}\subseteq\mathfrak{T}L$ for every frame $L$ (where
$(\mathfrak{S}L)^{\\#}=\\{S^{\\#}\mid S\in\mathfrak{S}L\\}$), then every
localic map is an $\mathfrak{S}\mathfrak{T}\lambda$-map.

  4. (4)

For two selections $\mathfrak{S}_{1}$ and $\mathfrak{S_{2}}$, if
$\mathfrak{S}_{2}\subseteq\mathfrak{S}_{1}$ then every
$\mathfrak{S}_{1}\mathfrak{T}\lambda$-map is an
$\mathfrak{S}_{2}\mathfrak{T}\lambda$-map.

  5. (5)

Two sublocales are completely separated if they are contained in disjoint zero
sublocales; hence,
$\mathfrak{S}_{\mathsf{ZS}}\mathfrak{T}_{\mathsf{ZS}}\lambda$-maps coincide
with $\mathfrak{S}_{\mathsf{ZS}}\beta$-maps.

  6. (6)

If a localic map $f\colon L\to M$ is $\mathfrak{T}$-preserving
($f[T]\in\mathfrak{T}M$ for every $T\in\mathfrak{T}L$), then it is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

######  Proposition 4.8.

Let $f\colon L\to M$ be a localic map, then

  1. (1)

$f$ is a $\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map
if and only if $f$ is nearly open.

  2. (2)

$f$ is a $\mathfrak{S}_{\mathsf{CoZS}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map
if and only if $f$ is $\mathsf{coz}$-nearly open.

  3. (3)

$f$ is a
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{o}}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map
if and only if $f$ is $\mathsf{reg}_{\mathfrak{o}}$-nearly open.

###### Proof.

Let us first note that a localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map if and
only if

$h(a)\wedge b=0\implies\exists\ d\in L\quad a\wedge d=0\text{ and }b\leq h(d)$
(14)

for every $a\in M$ and $b\in L$. Similarly, $f$ is a
$\mathfrak{S}_{\mathsf{CoZS}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map (resp.
$\mathfrak{S}_{\mathsf{reg}_{\mathfrak{o}}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map)
if and only if (14) holds for every $a\in\mathsf{Coz}~{}L$ (resp. $a\in
L^{*}$).
(1): Let $f\colon L\to M$ be an
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathfrak{o}}$ and let $a\in M$. We
always have that $h(a)\wedge h(a)^{*}=0$. By assumption, there is $d\in L$
such that $a\wedge d=0$ and $h(a)^{*}\leq h(d)$. Putting these two equations
together we get that $h(a)^{*}\leq h(d)\leq h(a^{*})$. Conversely, assume $f$
is nearly open, and let $a\in M$ and $b\in L$ such that $h(a)\wedge b=0$.
Hence, $b\leq h(a)^{*}=h(a^{*})$ and $a\wedge a^{*}=0$. Thus $f$ is an
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathfrak{o}}\lambda$-map.
Mimicking the argument above for $a\in\mathsf{Coz}~{}M$ and $a\in L^{*}$ one
gets the proof of (2) and (3) respectively. ∎

######  Remark 4.9.

Note that
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}$-are
precisely the maps where

$h(a)\wedge b^{*}=0\implies\exists d\in L\quad a\wedge d^{*}=0\text{ and
}b^{*}\leq h(d^{*})$

for every $a\in M$ and $b\in L$. Hence, if one take $b=h(a)$ one gets that
$h(a)^{*}\leq h(d^{*})\leq h(a^{*})$, which proves that
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}$-maps
are nearly open. Similarly, one can see that
$\mathfrak{S}_{\mathsf{CoZS}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}$-maps
are $\mathsf{coz}$-nearly open, and that
$\mathfrak{S}_{\mathsf{reg}_{\mathfrak{o}}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}$-maps
are $\mathsf{reg}_{\mathfrak{o}}$-nearly open.
Furthermore, assume $f$ is nearly open and $h(a)\wedge b^{*}=0$ for $a\in M$
and $b\in L$, then we get:

$\displaystyle h(a)\wedge b^{*}=0$ $\displaystyle\iff h(a)\leq b^{**}\iff
a\leq f(b^{**})\implies a\leq f(b^{**})^{**}$ $\displaystyle\iff a\wedge
f(b^{**})^{*}=0.$

Moreover, $b^{*}\leq h(f(b^{**})^{*})$. Indeed, we know that $b^{*}\leq
h(f(b^{**}))^{*}$ (because $h(f(b^{**}))\leq b^{**}$) and since $f$ is nearly
open $h(f(b^{**}))^{*}=h(f(b^{**})^{*})$. Thus, we have shown $f$ is
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}\lambda$-map.

The following Lemma will allow us to compare some of the localic maps that we
have introduced with closed and related localic maps. It is an easy and
probably known result among point-free topologists, but no explicit register
of it was found in the literature. In [10, 3] the authors prove (1), but
require regularity; in fact, only subfitness in the codomain is needed. Recall
from [23] that a local is _subfit_ if

$a\nleq b\implies\exists c\quad a\vee c=1\neq b\vee c$

for every $a,b\in L$.

######  Lemma 4.10.

Let $M$ be subfit locale and $f\colon L\to M$ a localic map then

  1. (1)

$f$ closed if and only if for every $a\in L$ and $b\in L$

$f(a\vee h(b))=1\implies f(a)\vee b=1.$

  2. (2)

$f$ is $\mathsf{z}$-closed if and only if for every $a\in\mathsf{Coz}~{}L$ and
$b\in L$

$f(a\vee h(b))=1\implies f(a)\vee b=1.$

  3. (3)

$f$ is $\mathsf{reg}_{\mathfrak{c}}$-closed if and only if for every $a\in
L^{*}$ and $b\in L$

$f(a\vee h(b))=1\implies f(a)\vee b=1.$

###### Proof.

(1): Clearly every closed map satisfies the equation in (1). Conversely, note
it only suffices to show that $f(a\vee h(b))\leq f(a)\vee b)$ since the other
inequality always holds. Suppose it is not true, that is $f(a\vee h(a))\nleq
f(a)\vee b$. Since $M$ is subfit there is $c\in M$ such that $f(a\vee
h(b))\vee c=1\neq c\vee f(a)\vee b$. Hence,

$1=h(f(a\vee h(b)))\vee h(c)\leq a\vee h(b)\vee h(c)=a\vee h(b\vee c)$

so $1=f(a\vee h(b\vee c))$. By assumption, we get $b\vee c\vee f(a)=1$
contradicting the fact that $1\neq c\vee f(a)\vee b$. Consequently, $f$ is
closed.
Clearly, every $\mathsf{z}$-closed (resp.
$\mathsf{reg}_{\mathfrak{c}}$-closed) map satisfies the equation in (2) (resp.
(3)). The converse follows exactly as the proof of (1), but one takes
$a\in\mathsf{Coz}~{}L$ (resp. $a\in L^{*}$). ∎

Note that $\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathfrak{c}}$-maps are
precisely those maps where

$a\vee h(b)=1\implies\exists d\in M\quad h(x)\leq a,\ x\vee b=1.$

Since $h(f(a))\leq a$, this is equivalent to

$a\vee h(b)=1\implies f(a)\vee b=1.$

From the adjunction between $f$ and $h$ we have that $1\leq f(a\vee h(b))\iff
1\leq a\vee h(b)$. Thus, a map is a
$\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathfrak{c}}$-map if and only if

$f(a\vee h(b))=1\implies f(a)\vee b=1$

for every $a\in L$ and $b\in M$. Consequently, by Proposition 4.10 (1), under
subfitness $\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathfrak{c}}$-maps and
closed localic maps coincide.

Using equations (3) and (9), one can easily see that a localic map is an
$\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathsf{ZS}}$-map (resp.
$\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{c}}}$-map)
if and only if

$a\vee h(b)=1\implies\exists d\in\mathsf{Coz}~{}M\quad b\vee d\ \text{ and }\
h(d)\leq a$ $(resp.\ a\vee h(b)=1\implies\exists d\in M^{*}\quad b\vee d\
\text{ and }\ h(d)\leq a)$

for every $a\in\mathsf{Coz}~{}L$ (resp. $a\in L^{*}$) and $b\in M$. Hence, by
Proposition 4.10, the following implications hold for a localic map $f\colon
L\to M$ when $M$ is subfit:

$\mathsf{z}\text{-preserving}\implies\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathsf{ZS}}\text{-map}\implies\mathsf{z}\text{-closed}$
$\mathsf{reg}_{\mathfrak{c}}\text{-preserving}\implies\mathfrak{S}_{\mathfrak{c}}\mathfrak{T}_{\mathsf{reg}_{\mathfrak{c}}}\text{-map}\implies\mathsf{reg}_{\mathfrak{c}}\text{-closed}.$

######  Remark 4.11.

A localic map $f\colon L\to M$ is a
$\mathfrak{S}_{\mathsf{ZS}}\mathfrak{T}_{\mathfrak{c}}\lambda$-map if and only
if

$h(a)\vee b=1\implies\exists d\in L\quad a\vee d=1\text{ and }h(d)\leq b$

for every $a\in\mathsf{Coz}~{}M$ and $b\in L$. Equivalently, since $f\vdash
h$,

$f(h(a)\vee b)=1\implies a\vee f(b)=1$

for every $a\in\mathsf{Coz}~{}M$ and $b\in L$. Assuming complete regularity
(see [15, Lemma 3.2]), these are precisely the right adjoints of the weakly
closed homomorphisms of [15] (or $coz$-closed in [10]). Nevertheless, if one
looks closely at the definition, these are not the same as the
$\mathsf{z}$-closed maps. The latter generalize the notion of $Z$-maps between
topological spaces [26], which are the continuous maps where images of zero
sets are closed.

Following the definition of $\mathfrak{S}$-coz separating, we say a locale is
_$\mathfrak{S}$ -$\mathfrak{T}$ separating_ if
$S\cap{\mathfrak{o}}(a)=\text{\sf O}$ for some $S\in\mathfrak{S}L$, then ther
is $T\in\mathfrak{T}L$ such that $S\cap T=\text{\sf O}$ and
${\mathfrak{o}}(a)\subseteq T$.

A localic map $f\colon L\to M$ is a _$\mathfrak{T}$ -open_ map if $f[T]$ is
open for every $T\in\mathfrak{T}L$. For the selections
$\mathfrak{T}_{\mathfrak{o}}$,$\mathfrak{T}_{\mathsf{reg}_{\mathfrak{o}}}$ and
$\mathfrak{T}_{\mathsf{CoZS}}$, these are the open,
$\mathsf{reg}_{\mathfrak{o}}$-open and $\mathsf{coz}$-open maps respectively.

######  Theorem 4.12.

Let $\mathfrak{S}$ and $\mathfrak{T}$ be two selections of sublocales such
that $\mathfrak{T}L\subseteq{\mathfrak{o}}(L)$ and $L\in\mathfrak{T}L$ for
every locale $L$. Let $M$ be a locale such that every $S\in\mathfrak{S}M$ is
complemented, then the following are equivalent:

  1. (i)

$M$ is $\mathfrak{S}$-$\mathfrak{T}$ separating.

  2. (ii)

Every $\mathfrak{T}$-open localic map $f\colon L\to M$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

  3. (iii)

Every open localic map $f\colon L\to M$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

###### Proof.

(i)$\implies$(ii): Assume $M$ is $\mathfrak{S}$-$\mathfrak{T}$-separating, and
let $f\colon L\to M$ be a $\mathfrak{T}$-open localic map. Let
$S\in\mathfrak{S}M$ and $T\in\mathfrak{T}L$ such that $T\cap
f_{-1}[S]=\text{\sf O}$. Then, since the localic preimage preserves
complementes, $T\subseteq f_{-1}[S]^{\\#}=f_{-1}[S^{\\#}]$. By the
image/preimage adjunction, this means $f[T]\subseteq S^{\\#}$. Since $S$ is
complemented this is equivalent to $f[T]\cap S=\text{\sf O}$. By assumption,
and since $f[T]$ is open, there is $T^{\prime}\in\mathfrak{T}M$ such that
$f[T]\subseteq T^{\prime}$ (equiv. $T\subseteq f_{-1}[T^{\prime}]$) and
$T^{\prime}\cap S=\text{\sf O}$. Thus, $f$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.
The implication (ii)$\implies$(iii) is immediate, since every open map is
$\mathfrak{T}$-open (we are assuming
$\mathfrak{T}L\subseteq{\mathfrak{o}}(L)$).
(iii)$\implies$(i): Let $a\in M$ and $S\in\mathfrak{S}M$ such that
${\mathfrak{o}}_{M}(a)\cap S=\text{\sf O}$. Consider the open localic
embedding $j\colon{\mathfrak{o}}_{M}(a)\hookrightarrow M$. By assumption, $j$
is a $\mathfrak{S}\mathfrak{T}\lambda$-map. Since ${\mathfrak{o}}_{M}(a)\cap
S=\text{\sf O}={\mathfrak{o}}_{M}(a)\cap j_{-1}[S]$ and
${\mathfrak{o}}_{M}(a)\in\mathfrak{T}{\mathfrak{o}}_{M}(a)$, there exists
$T\in\mathfrak{T}M$ such that ${\mathfrak{o}}_{M}(a)\subseteq
j_{-1}[T]=T\cap{\mathfrak{o}}_{M}(a)$ (equiv. ${\mathfrak{o}}_{M}(a)\subseteq
T$) and $T\cap S=\text{\sf O}$. Consequently, $M$ is
$\mathfrak{S}$-$\mathfrak{T}$ separating. ∎

Similarly, now using closed sublocales, we say that a locale is
_$\mathfrak{S}$ -$\mathfrak{T}$* separating_ if
$S\cap{\mathfrak{c}}(a)=\text{\sf O}$ for some $S\in\mathfrak{S}L$, then there
is $T\in\mathfrak{T}L$ such that $S\cap T=\text{\sf O}$ and
${\mathfrak{c}}(a)\subseteq T$. A localic map $f\colon L\to M$ is a
_$\mathfrak{T}$ -closed map_ if $f[T]$ is closed for every
$T\in\mathfrak{T}L$. For the selections
$\mathfrak{T}_{\mathfrak{c}}$,$\mathfrak{T}_{\mathsf{reg}_{\mathfrak{c}}}$ and
$\mathfrak{T}_{\mathsf{ZS}}$, these are the closed,
$\mathsf{reg}_{\mathfrak{c}}$-closed and $\mathsf{z}$-closed maps
respectively.

######  Theorem 4.13.

Let $\mathfrak{S}$ and $\mathfrak{T}$ be two selections of sublocales such
that $\mathfrak{T}L\subseteq{\mathfrak{c}}(L)$ and $L\in\mathfrak{T}L$ for
every locale $L$. Let $M$ be a locale such that every $S\in\mathfrak{S}M$ is
complemented, then the following are equivalent:

  1. (i)

$M$ is $\mathfrak{S}$-$\mathfrak{T}$* separating.

  2. (ii)

Every $\mathfrak{T}$-closed localic map $f\colon L\to M$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

  3. (iii)

Every closed localic map $f\colon L\to M$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

  4. (iv)

Every proper localic map $f\colon L$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.

###### Proof.

(i)$\implies$(ii): Assume $M$ is $\mathfrak{S}$-$\mathfrak{T}$* separating,
and let $f\colon L\to M$ be a $\mathfrak{T}$-closed localic map. Let
$S\in\mathfrak{S}M$ and $T\in\mathfrak{T}L$ such that $T\cap
f_{-1}[S]=\text{\sf O}$. Then, since the localic preimage preserves
complementes, $T\subseteq f_{-1}[S]^{\\#}=f_{-1}[S^{\\#}]$. By the
image/preimage adjunction, this means $f[T]\subseteq S^{\\#}$. Since $S$ is
complemented this is equivalent to $f[T]\cap S=\text{\sf O}$. By assumption,
and since $f[T]$ is closed, there is $T^{\prime}\in\mathfrak{T}M$ such that
$f[T]\subseteq T^{\prime}$(equiv. $T\subseteq f_{-1}[T^{\prime}]$) and
$T^{\prime}\cap S=\text{\sf O}$. Thus, $f$ is an
$\mathfrak{S}\mathfrak{T}\lambda$-map.
The implications (ii)$\implies$(iii) and (iii)$\implies$(iv) are immediate,
since every closed map is $\mathfrak{T}$-closed (we are assuming
$\mathfrak{T}L\subseteq{\mathfrak{c}}(L)$), and every proper map is closed.
(iv)$\implies$(i): Let $a\in M$ and $S\in\mathfrak{S}M$ such that
${\mathfrak{c}}_{M}(a)\cap S=\text{\sf O}$. Consider the proper localic
embedding $j\colon{\mathfrak{c}}_{M}(a)\hookrightarrow M$. By assumption, $j$
is a $\mathfrak{S}\mathfrak{T}\lambda$-map. Since ${\mathfrak{c}}_{M}(a)\cap
S=\text{\sf O}={\mathfrak{c}}_{M}(a)\cap j_{-1}[S]$ and
${\mathfrak{c}}_{M}(a)\in\mathfrak{T}{\mathfrak{c}}_{M}(a)$, there exists
$T\in\mathfrak{T}M$ such that ${\mathfrak{c}}_{M}(a)\subseteq
j_{-1}[T]=T\cap{\mathfrak{c}}_{M}(a)$ (equiv. ${\mathfrak{c}}_{M}(a)\subseteq
T$) and $T\cap S=\text{\sf O}$. Consequently, $M$ is
$\mathfrak{S}$-$\mathfrak{T}$* separating. ∎

The following two tables show all the $\mathfrak{S}\mathfrak{T}$-separating
and $\mathfrak{S}\mathfrak{T}$*-separating that describe our usual selections.
Thus, Theorem 4.12 and Theorem 4.13 offer characterizations for some of the
respective frames below.

Table 1. $\mathfrak{S}\mathfrak{T}$-separating

$\mathfrak{S}\diagdown\mathfrak{T}$ | $\mathfrak{T}_{\mathfrak{o}}$ | $\mathfrak{T}_{\mathfrak{c}}$ | $\mathfrak{T}_{\mathsf{CoZS}}$ | $\mathfrak{T}_{\mathsf{ZS}}$ | $\mathfrak{T}_{\mathsf{reg}{\mathfrak{o}}}$ | $\mathfrak{T}_{\mathsf{reg}{\mathfrak{c}}}$
---|---|---|---|---|---|---
$\mathfrak{S}_{\mathfrak{o}}$ | – | – | Oz | Oz | – | –
$\mathfrak{S}_{\mathfrak{c}}$ | – | Boolean | Perfectly Normal | Boolean | Boolean | Boolean
$\mathfrak{S}_{\mathsf{CoZS}}$ | – | – | Weak Oz | – | – | –
$\mathfrak{S}_{\mathsf{ZS}}$ | – |  $P$-frame | – |  $P$-frame | Almost $P$ |  $P$-frame
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{o}}}$ | – | – | Oz | Oz | – | –
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$ | – | ED | Oz | ED | – | ED

Note: – denotes every frame

Table 2. $\mathfrak{S}\mathfrak{T}$*-separating

$\mathfrak{S}\diagdown\mathfrak{T}$ | $\mathfrak{T}_{\mathfrak{o}}$ | $\mathfrak{T}_{\mathfrak{c}}$ | $\mathfrak{T}_{\mathsf{CoZS}}$ | $\mathfrak{T}_{\mathsf{ZS}}$ | $\mathfrak{T}_{\mathsf{reg}{\mathfrak{o}}}$ | $\mathfrak{T}_{\mathsf{reg}{\mathfrak{c}}}$
---|---|---|---|---|---|---
$\mathfrak{S}_{\mathfrak{o}}$ | Boolean | – | Boolean | Perfectly Normal | Boolean | Boolean
$\mathfrak{S}_{\mathfrak{c}}$ | – | – | Normal | Normal |  |  Disjoint closed sublocales are contianed in disjoint regular closed sublocales.
$\mathfrak{S}_{\mathsf{CoZS}}$ |  $P$-frame | – |  $P$-frame | – |  $P$-frame | Almost$P$
$\mathfrak{S}_{\mathsf{ZS}}$ | – | – | – |  $\delta$-NS |  |
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{o}}}$ | ED | – | ED | Oz | ED | –
$\mathfrak{S}_{\mathsf{reg}{\mathfrak{c}}}$ | – | – |  |  | – |

Note: – denotes every frame, and no obvious characterizing class of locales
was found when the cell is left blank.

##  5\. Embeddings

This section offers some examples and characterizations as to when is a
localic embedding an $\mathfrak{S}\beta$\- or
$\mathfrak{S}\mathfrak{T}\lambda$-map.

In [12], the authors introduce the notions of $C_{1}$\- and strong
$C_{1}$-quotients. Here we present them in terms of sublocales. We say a
sublocale is _$C_{1}$ -embedded_ if for every $Z\in\mathsf{ZS}(S)$ and
$F\in\mathsf{ZS}(L)$ such that $Z\cap(F\cap S)=\text{\sf O}$, then $Z$ and $F$
are completely separated in $L$ (equiv. the left adjoint of $j\colon
S\hookrightarrow L$ is a $C_{1}$-quotient).
A sublocale $S$ is _strongly $C_{1}$-embedded_ if for every
$Z\in\mathsf{ZS}(S)$ and $a\in L$ such that ${\mathfrak{c}}_{L}(a)\cap
S\in\mathsf{ZS}(S)$ and $Z\cap({\mathfrak{c}}_{L}(a)\cap S)=\text{\sf O}$,
then $Z$ and ${\mathfrak{c}}_{L}(a)$ are completely separated in $L$ (equiv.
the left adjoint of $j\colon S\hookrightarrow L$ is a strong
$C_{1}$-quotient).

It is already shown in [12, Corollary 3.3] that $j\colon S\hookrightarrow L$
is an $\mathfrak{S}_{\mathsf{ZS}}\beta$-map if and only if $S$ is
$C_{1}$-embedded. For strong $C_{1}$-embeddings we have the following result:

######  Proposition 5.1.

Let $S$ be a sublocale of $L$. The localic embedding $j\colon S\hookrightarrow
L$ is $\mathfrak{S}_{\mathfrak{c}}\beta$-map if and only if $S$ is strongly
$C_{1}$.

###### Proof.

Let $j$ be an $\mathfrak{S}_{\mathfrak{c}}\beta$-map and let $a\in L$ and
$Z\in\mathsf{ZS}(S)$ such that ${\mathfrak{c}}_{L}(a)\cap S\in\mathsf{ZS}(S)$
and $Z\cap({\mathfrak{c}}_{L}(a)\cap S)=\text{\sf O}$. In particular, $Z$ and
$j_{-1}[{\mathfrak{c}}_{L}(a)]={\mathfrak{c}}_{L}(a)\cap S$ are completely
separated in $S$ (they are disjoint zero sublocales of $S$). By assumption
there is $D\in\mathsf{ZS}(L)$ completely separated from
${\mathfrak{c}}_{L}(a)$ and $Z\subseteq j_{-1}[D]=D\cap S$. In particular, $Z$
and ${\mathfrak{c}}_{L}(a)$ are completely separated in $L$.
Conversely, assume $S$ is strongly $C_{1}$-embedded in $L$. Let
$Z\in\mathsf{ZS}(S)$ and $a\in L$ such that $Z$ and ${\mathfrak{c}}_{L}(a)\cap
S$ are completely separated in $S$. Then there is $D\in\mathsf{ZS}(S)$ such
that $Z\cap D=\text{\sf O}$ and ${\mathfrak{c}}_{L}(a)\cap S\subseteq D$.
Since $D$ is a closed sublocale of $S$, there is a closed sublocale
$D^{\prime}$ of $L$ such that $D=D^{\prime}\cap S$. Then

$({\mathfrak{c}}_{L}(a)\vee D^{\prime})\cap S=({\mathfrak{c}}_{L}(a)\cap
S)\vee(D^{\prime}\cap S)=D.$

Using the fact tha $S$ is strongly $C_{1}$-embedded we can conclude
${\mathfrak{c}}_{L}(a)\vee D^{\prime}$ and $Z$ are completely separated in
$L$. This means there is $Z^{\prime}\in\mathsf{ZS}(L)$ such that $Z\subseteq
Z^{\prime}$ (in particular, $Z\subseteq Z^{\prime}\cap S$), and $Z^{\prime}$
is completely separated from ${\mathfrak{c}}_{L}(a)\vee D^{\prime}$; thus, it
is also completely separated from ${\mathfrak{c}}_{L}(a)$. ∎

Recall that a sublocale $S$ of $L$ is _$C^{*}$ -embedded_ if every bounded
continuous real-valued function on the locale $S$ can be extended to $L$ (see
for instance [4, 2, 19]). For our purposes it is more useful to use one of its
well-known characterizations: $S$ is $C^{*}$-embedded iff whenever two
sublocales of $S$ are completely separated in $S$ then they are completely
separated in $L$ [3, Theorem 6.1].

######  Proposition 5.2.

Let $S$ be a sublocale of $L$ and $j\colon S\hookrightarrow L$ its localic
embedding, $j$ is an $\mathfrak{S}_{\mathfrak{o}}\beta$-map if and only if $S$
is $C^{*}$-embedded and $\overline{S}$ is clopen.

###### Proof.

Suppose $j$ is an $\mathfrak{S}_{\mathfrak{o}}\beta$-map. Immediately, by
Remark 3.16, we obtain that $\overline{S}=\overline{j[S]}$ is clopen. To show
$S$ is $C^{*}$-embedded consider
${\mathfrak{c}}_{S}(a),{\mathfrak{c}}_{S}(b)\in\mathsf{ZS}(S)$ such that
${\mathfrak{c}}_{S}(a)\cap{\mathfrak{c}}_{S}(b)=\text{\sf O}$, we will prove
they are completely separated in $L$. From Remark 1.3, there is
${\mathfrak{o}}_{S}(x)$ such that
${\mathfrak{c}}_{S}(a)\subseteq{\mathfrak{o}}_{S}(x)$ and
${\mathfrak{o}}_{S}(x)$ is completely separated from ${\mathfrak{c}}_{S}(b)$
in $S$; that is $x\prec\\!\\!\prec_{S}b$444To avoid confusion, in this proof
we use the subscript on the completely below relation to denote the
corresponding locale.. Since $j$ is $\prec\\!\\!\prec$-preserving (Theorem
3.14), we get $x\prec\\!\\!\prec_{L}b$. Meaning ${\mathfrak{o}}_{L}(x)$ and
${\mathfrak{c}}_{L}(b)$ are completely separated in $L$. Clearly,
${\mathfrak{c}}_{S}(b)\subseteq{\mathfrak{c}}_{L}(b)$ and
${\mathfrak{c}}_{S}(a)\subseteq{\mathfrak{o}}_{S}(x)\subseteq{\mathfrak{o}}_{L}(x)$.
Thus, ${\mathfrak{c}}_{S}(a)$ and ${\mathfrak{c}}_{S}(b)$ are completely
separated in $L$.
Conversely, suppose $S$ is $C^{*}$-embedded and $\overline{S}$ is clopen. Let
$a\in L$ and $D\in\mathsf{ZS}(S)$ such that $D$ is completely separated from
${\mathfrak{o}}_{L}(a)\cap S=j_{-1}[{\mathfrak{o}}_{L}(a)]$ in $S$. Since $S$
is $C^{*}$-embedded, $D$ and ${\mathfrak{o}}_{L}(a)\cap S$ are completely
separated in $L$. Thus, there are $Z_{1},Z_{2}\in\mathsf{ZS}(L)$ such that

$D\subseteq Z_{1},\quad{\mathfrak{o}}_{L}(a)\cap S\subseteq Z_{2}\quad\text{
and}\quad Z_{1}\cap Z_{2}=\text{\sf O}.$

From Remark 1.3, there is a $C\in\mathsf{CoZS}(L)$ such that $Z_{1}\subseteq
C$ and $C$ is completely separated from $Z_{2}$ in $L$. In particular, we have
that $C\cap{\mathfrak{o}}_{L}(a)\cap S\subseteq C\cap Z_{2}=\text{\sf O}$.
Hence, $S\subseteq(C\cap{\mathfrak{o}}_{L}(a))^{\\#}$. Since
$(C\cap{\mathfrak{o}}_{L}(a))^{\\#}$ is a closed sublocale, we observe that
$\overline{S}\subseteq(C\cap{\mathfrak{o}}_{L}(a))^{\\#}$; consequently,
$C\cap{\mathfrak{o}}_{L}(a)\subseteq\overline{S}^{\\#}$. Thus,
$\overline{S}\cap
C\cap{\mathfrak{o}}_{L}(a)\subseteq\overline{S}\cap\overline{S}^{\\#}=\text{\sf
O}$, which implies that

${\mathfrak{o}}_{L}(a)\subseteq(\overline{S}\cap
C)^{\\#}=\overline{S}^{\\#}\vee C^{\\#}.$

Note that $\overline{S}^{\\#}\vee C^{\\#}$ and $Z_{1}\cap\overline{S}$ are
zero sublocales (since $\overline{S}$ is clopen); furthermore, they are
disjoint:

$\displaystyle(\overline{S}^{\\#}\vee C^{\\#})\cap(Z_{1}\cap\overline{S})$
$\displaystyle=(\overline{S}^{\\#}\cap Z_{1}\cap\overline{S})\vee(C^{\\#}\cap
Z_{1}\cap\overline{S})$
$\displaystyle\subseteq(\overline{S}^{\\#}\cap\overline{S})\vee(C^{\\#}\cap
Z_{1})=\text{\sf O}.$

We have then shown that ${\mathfrak{o}}_{L}(a)$ is completely separated from
$Z_{1}\cap\overline{S}$, and, since $D\subseteq S$ and $D\subseteq Z_{1}$, we
have that $D\subseteq S\cap Z_{1}\cap\overline{S}=j_{-1}[\overline{S}\cap
Z_{1}]$, as required. ∎

A sublocale $S$ is _$z$ -embedded_ if for every $Z\in\mathsf{ZS}(S)$ there is
$Z^{\prime}\in\mathsf{ZS}(L)$ such that $Z=Z^{\prime}\cap S$ ([3, 20]).
Equivalently, if for every $C\in\mathsf{CoZS}(S)$ there is
$C^{\prime}\in\mathsf{CoZS}(L)$ such that $C=C^{\prime}\cap S$.

######  Proposition 5.3.

Let $S$ be a sublocale of $L$ and $j\colon S\hookrightarrow L$ its localic
embedding, then the following are equivalent:

  1. (i)

$j$ is a $\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{CoZS}}\lambda$-map.

  2. (ii)

$S$ is $z$-embedded and whenever $S\subseteq{\mathfrak{c}}(a)$ for some $a\in
L$ there is $D\in\mathsf{CoZS}(L)$ such that $S\subseteq
D\subseteq{\mathfrak{c}}_{L}(a)$.

###### Proof.

(i)$\implies$(ii): We will show $S$ is $z$-embedded using the characterization
in [1, Corollary 2.5.3]. That is, given disjoint
$C_{1},C_{2}\in\mathsf{CoZS}(S)$, we will prove there are disjoint
$D_{1},D_{2}\in\mathsf{CoZS}(L)$ such that $C_{i}\subseteq D_{i}\cap S$ for
$i=1,2$. Since $C_{2}$ is a cozero (in particular an open) sublocale of $S$,
there is $a\in L$ such that $C_{2}={\mathfrak{o}}_{L}(a)\cap
S=j_{-1}[{\mathfrak{o}}_{L}(a)]$. Hence, $C_{1}\cap
j_{-1}[{\mathfrak{o}}_{L}(a)]=\text{\sf O}$. By assumption there is
$D_{1}\in\mathsf{CoZS}(L)$ such that $C_{1}\subseteq j_{-1}[D_{1}]=D_{1}\cap
S$ and $D_{1}\cap{\mathfrak{o}}(a)=\text{\sf O}$. In particular, we have that

$j_{-1}[D_{1}]\cap j_{-1}[{\mathfrak{o}}_{L}(a)]=j_{-1}[D_{1}]\cap
C_{2}=\text{\sf O}.$

Using again the fact that $j$ is a
$\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{CoZS}}\lambda$-map, there is
$D_{2}\in\mathsf{CoZS}(L)$ such that $C_{2}\subseteq j_{-1}[D_{2}]=D_{2}\cap
S$ and $D_{1}\cap D_{2}=\text{\sf O}$. Moreover, if
$S\subseteq{\mathfrak{c}}_{L}(a)$ for some $a\in L$, then
$S\cap{\mathfrak{o}}_{L}(a)=S\cap j_{-1}[S]=\text{\sf O}$. By assumption,
since $S$ is a cozero sublocale of $S$, there is $C\in\mathsf{CoZS}(L)$ such
that $S\subseteq C$ and $C\cap{\mathfrak{o}}_{L}(a)=\text{\sf O}$. Thus,
$S\subseteq C\subseteq{\mathfrak{c}}_{L}(a)$.
(ii)$\implies$(i): Let $a\in S$ and $C\in\mathsf{CoZS}(L)$ such that
${\mathfrak{o}}_{S}(a)\cap j_{-1}[C]={\mathfrak{o}}_{L}(a)\cap S\cap
C=\text{\sf O}$. Since $S$ is $z$-embedded there is
$C^{\prime}\in\mathsf{CoZS}(L)$ such that $C^{\prime}\cap S=C$. Thus,
${\mathfrak{o}}_{L}(a)\cap S\cap C^{\prime}={\mathfrak{o}}_{L}(a)\cap S\cap
C=\text{\sf O}$; equivalently, $S$ is contained in the closed sublocale
$C^{\prime\\#}\vee{\mathfrak{c}}_{L}(a)$. By assumption there is
$D\in\mathsf{CoZS}(L)$ such that $S\subseteq D\subseteq
C^{\prime\\#}\vee{\mathfrak{c}}_{L}(a)$. Hence,

$D\cap C^{\prime}\cap{\mathfrak{o}}_{L}(a)=\text{\sf O}\quad\text{and}\quad
C=S\cap C^{\prime}=D\cap C^{\prime}\cap S=j_{-1}[D\cap C^{\prime}]$

and $D\cap C^{\prime}$ is a cozero sublocale of $L$. ∎

Let us consider a very special sublocale. Given a locale $L$ the smallest
dense sublocale (or _Booleanization_) of $L$ is

$\mathfrak{B}L=\left\\{x\in L\mid x=x^{**}\right\\}=\left\\{x^{*}\mid x\in
L\right\\}.$

Let $j\colon\mathfrak{B}L\hookrightarrow L$ be the localic embedding from the
booleanization of $L$ to $L$. In [11, Proposition 4.4] it is proved that $j$
is a $\mathfrak{S}_{\mathsf{ZS}}\beta$-map if and only if $L$ is a $P$-frame.
Furthermore, Proposition 5.2 together with [12, Corollary 3.18] show that $j$
is a $\mathfrak{S}_{\mathfrak{c}}\beta$-map if and only if $L$ is a boolean.

######  Proposition 5.4.

Let $L$ be a locale and $j\colon\mathfrak{B}L\hookrightarrow L$.

  1. (1)

$j$ is a $\mathfrak{S}_{\mathfrak{o}}\beta$-map if and only if $L$ is
extremally disconnected.

  2. (2)

$j$ is a $\mathfrak{S}_{\mathsf{CoZS}}\beta$-map if and only if $L$ is
basically disconnected.

###### Proof.

Let us denote by $h\colon L\to\mathfrak{B}L$ the left adjoint of the localic
embedding $j$ given by $h(a)=a^{**}$.
(1): Assume $j\colon\mathfrak{B}L\hookrightarrow L$ is an
$\mathfrak{S}_{\mathfrak{o}}\beta$-map, let us show $L$ is extremally
disconnected. Let $b\in L$, then
${\mathfrak{c}}_{L}(b^{**})\cap{\mathfrak{o}}_{L}(b)=\text{\sf O}$ (because
$b\leq b^{**}$). Taking preimages we get

$j_{-1}[{\mathfrak{c}}_{L}(b^{**})]\cap
j_{-1}[{\mathfrak{o}}_{L}(b)]=\text{\sf O}$

and, since $\mathfrak{B}L$ is boolean, both sublocales above are clopen.
Because they are disjoint, they are also completely separated. By assumption,
there is $a\in L$ such that

$j_{-1}[{\mathfrak{c}}_{L}(b^{**})]\subseteq j_{-1}[{\mathfrak{c}}_{L}(a)]$
(15)

and ${\mathfrak{c}}_{L}(a)$ is completely separated from
${\mathfrak{o}}_{L}(b)$. In fact, by Remark 1.1, ${\mathfrak{c}}_{L}(a)$ and
$\overline{{\mathfrak{o}}_{L}(b)}={\mathfrak{c}}(b^{*})$ are completely
separated. In particular, they are disjoint closed sublocales, so $a\vee
b^{*}=1$. Furthermore, from (15), we have that

${\mathfrak{c}}_{\mathfrak{B}L}(b^{**})={\mathfrak{c}}_{\mathfrak{B}L}(h(b^{**}))\subseteq{\mathfrak{c}}_{\mathfrak{B}L}(h(a))={\mathfrak{c}}_{\mathfrak{B}L}(a^{**});$

thus, $a^{**}\leq b^{**}$. Since $a\vee b^{*}=1$, we get $1=a\vee b^{*}\leq
a^{**}\vee b^{*}\leq b^{**}\vee b^{*}$. Hence, $b^{*}$ is complemented.
Conversely, suppose $L$ is extremally disconnected, and let
$a^{*}\in\mathfrak{B}L$ and $b\in L$ such that
${\mathfrak{c}}_{\mathfrak{B}L}(a^{*})$ and $j_{-1}[{\mathfrak{o}}_{L}(b)]$
are completely separated. In particular,
${\mathfrak{c}}_{\mathfrak{B}L}(a^{*})\subseteq
j_{-1}[{\mathfrak{c}}_{L}(b)]$. Note that

$j_{-1}[{\mathfrak{c}}_{L}(b^{**})]={\mathfrak{c}}_{\mathfrak{B}L}(h(b^{**}))={\mathfrak{c}}_{\mathfrak{B}L}(b^{**})={\mathfrak{c}}_{\mathfrak{B}L}(h(b))=j_{-1}[{\mathfrak{c}}_{L}(b)],$

then ${\mathfrak{c}}_{\mathfrak{B}L}(a^{*})\subseteq
j_{-1}[{\mathfrak{c}}_{L}(b^{**})]$. Moreover, ${\mathfrak{c}}_{L}(b^{**})$
and ${\mathfrak{o}}_{L}(b^{**})$ are disjoint clopen (since $L$ is extremally
disconnected) and hence disjoint zero sublcoales; thus, they are completely
separated in $L$. Further, since
${\mathfrak{o}}_{L}(b)\subseteq{\mathfrak{o}}_{L}(b^{**})$,
${\mathfrak{c}}_{L}(b^{**})$ and ${\mathfrak{o}}_{L}(b)$ are completely
separated in $L$, as required.
(2): The proof follows analogously like in (1) but we take $b\in\mathsf{Coz}L$
and since the $L$ is basically disconnected the argument follows since
$b^{**}$ is complemented. ∎

Putting together Proposition 5.2 and Proposition 5.4 we get:

######  Corollary 5.5.

$L$ is extremally disconnected if and only if $\mathfrak{B}L$ is
$C^{*}$-embedded.

A further example: since $\mathfrak{B}L$ is dense, by Proposition 5.3, $j$ is
a $\mathfrak{S}_{\mathfrak{o}}\mathfrak{T}_{\mathsf{CoZS}}\lambda$-map if and
only if $L$ is coole (that is, a frame whose booleanization is $z$-embedded
[9]).

## Acknowledgements

The author gratefully acknowledges the financial support from the UWC Research
Chair in Mathematics and Applied Mathematics – Topology for Tomorrow. The
author also acknowledges the support of a scholarship from the Oppenheimer
Memorial Trust (OMT) to support the full-time Postdoctoral studies (OMT Ref.
2023-1907).

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