# Dimensional types and $P$-spaces

Wojciech Bielas  Institute of Mathematics, University of Silesia in Katowice,
ul. Bankowa 14, 40-007 Katowice<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>,  Andrzej Kucharski  and
Szymon Plewik

###### Abstract.

We investigate the category of discrete topological spaces, with emphasis on
inverse systems of height $\omega_{1}$. Their inverse limits belong to the
class of $P$-spaces, which allows us to explore dimensional types of these
spaces.

###### Key words and phrases:

$P$-space, $G_{\delta}$-modification, dimensional type, topological rank,
labeled tree

###### 2020 Mathematics Subject Classification:

Primary: 54G10, ; Secondary: 54A10, 03E35

##  1\. Introduction

The purpose of this paper is to discuss the connection between inverse limits
of height $\omega_{1}$, which extend the category of discrete topological
spaces. Results are concentrated on dimensional types of some $P$-spaces. The
name “$P$-space” was used by L. Gillman and M. Henriksen [6]. If a space $X$
is completely regular and every countable intersection of open sets of $X$ is
open, then $X$ is called $P$-space. A. K. Misra [14] proposed investigation of
$T_{1}$-spaces which satisfy this last condition and called them $P$_-spaces_
, too. If $X$ is a $T_{1}$-space, then $X$ endowed with the topology generated
by all $G_{\delta}$-sets is called the $G_{\delta}$_-modification_ of $X$, and
it is denoted by $(X)_{\delta}$. Thus, any $T_{1}$-space is a $P$-space if and
only if it is its own $G_{\delta}$-modification. Following M. Fréchet [5], K.
Kuratowski [12] or W. Sierpiński [17], etc., we are convinced that pairs of
topological spaces which embed into each other are interesting in themselves.
This relation appeared under various names: topological rank, see [12, p.
112]; dimensional type, see [17, p. 130], etc. If $X$ is homeomorphic to a
subspace of $Y$, i.e. $Y$ contains a homeomorphic copy of $X$, in short
$X\subset_{h}Y$, then the dimensional type of $X$ is less or equal to the
dimensional type of $Y$. If $X\subset_{h}Y$ and $Y\subset_{h}X$, then we write
$X=_{h}Y$. If $X\subset_{h}Y$ and $Y$ does not contain a homeomorphic copy of
$X$, then $X$ has smaller dimensional type than $Y$. In the paper [2], the
relation “$\subset_{h}$” was called “topological inclusion relation” and was
used to examine topological arrow relation of the form $X\to(Y)^{1}_{2}$. One
of the results of this paper, which improves some results from [3], is
$(2^{\omega_{1}})_{\delta}\to(\mathbf{\Sigma})^{1}_{2}$, see Section 7.

We use the standard notation and terminology of [12], [7], or [4], as well as
of papers [11] and [3] with some minor changes. A dense in itself space is
called a crowded space. A partition is a cover consisting of pairwise disjoint
open sets, hence elements of a partition are clopen sets, here a clopen set
means a closed and open set. We refer the readers to the book [4, pp. 98–104]
for details about limits of inverse systems. We use inverse systems
$\\{X_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$, where each $X_{\alpha}$ is
a discrete space.

The paper is organised as follows. In Section 2 we briefly sketch some facts
about $P$-spaces. For the description of inverse systems, see [4], but notions
related to trees are taken from [7]. In Sections 3 and 4, we discuss the
$G_{\delta}$-modifications of Cantor cubes and its subspaces, where we show
that the Baire number varies among them. Theorem 13 says that, under the
Continuum Hypothesis, any two dense subspaces of $(2^{\omega_{1}})_{\delta}$
of cardinality $\mathfrak{c}$ are homeomorphic. In Sections 5–7, we
investigate $P$-spaces of cardinality $\omega_{1}$, $P$-spaces of weight
$\omega_{1}$ and $P$-spaces of cardinality and weight $\omega_{1}$. Any
$P$-space of weight $\omega_{1}$ can be embedded into
$(2^{\omega_{1}})_{\delta}$, hence the space $(2^{\omega_{1}})_{\delta}$ has
the greatest dimensional type in the class of all $P$-spaces of weight
$\omega_{1}$, whenever the Continuum Hypothesis is assumed. In Section 8, we
introduce the notion of a $\lambda$-thin labeling and prove that any two
$P$-spaces which have a $\lambda$-thin labeling are homeomorphic. Also we
prove that if a $P$-space $X$ has an $\omega$-thin labeling (or an
$\omega_{1}$-thin labeling), then $X$ has the smallest dimensional type in the
class of all crowded $P$-spaces of weight $\omega_{1}$. Finally, we give a few
remarks about rigid Lindelöf $P$-spaces.

##  2\. $P$-spaces, inverse limits and trees

Before proceeding, let us note the following observation.

######  Proposition 1.

Any regular $P$-space is $0$-dimensional.

###### Proof.

Fix a regular $P$-space $X$ and $x\in V\subseteq X$, where $V$ is an open set.
By regularity of $X$, there exists a sequence $(U_{n})$ of open sets such that
$x\in\operatorname{cl}U_{n}\subseteq U_{n-1}.$ The set

$V^{*}=\bigcap_{n}\operatorname{cl}U_{n}=\bigcap_{n}U_{n}\subseteq V$

is clopen and $x\in V^{*}\subseteq V$. Therefore, $X$ has a base consisting of
clopen sets. ∎

The family of all clopen sets of a regular $P$-space is a $\sigma$-algebra.
For these reasons, from now on we assume that a $P$-space is completely
regular.

######  Proposition 2.

If $X$ is a $P$-space and a clopen subset $U\subseteq X$ has a limit point,
then $U$ contains uncountably many pairwise disjoint clopen subsets.

###### Proof.

Let $U\subseteq X$ be a clopen set with a limit point $y\in U$. By Proposition
1, there exists an uncountable base $\mathcal{B}$ at $y$ consisting of clopen
subsets. Choose a strictly decreasing sequence

$\\{V_{\alpha}\subseteq U\colon\alpha<\omega_{1}\\}\subseteq\mathcal{B}.$

The family $\\{V_{\alpha}\setminus V_{\alpha+1}\colon\alpha<\omega_{1}\\}$ is
as desired. ∎

######  Proposition 3.

In a $P$-space, any countable family consisting of open covers has a common
refinement.

###### Proof.

It suffices to consider a family $\\{\mathcal{P}_{n}\colon n<\omega\\}$ of
covers, each one consists of clopen sets. For a point $x$, choose
$V_{n,x}\in\mathcal{P}_{n}$ such that $x\in V_{n,x}$. The intersection

$V_{x}=\bigcap\\{V_{n,x}\colon n<\omega\\}$

is a clopen set, so the family of all $V_{x}$ is a desired refinement. ∎

Let $\\{X_{\alpha}\colon\alpha<\omega_{1}\\}$ be a family of discrete spaces.
The $G_{\delta}$-modification $(\prod_{\alpha<\omega_{1}}X_{\alpha})_{\delta}$
of a product $\prod_{\alpha<\omega_{1}}X_{\alpha}$ with the Tychonoff topology
has a base

$\\{[f]\colon f\in\prod_{\beta<\alpha}X_{\beta}\mbox{ and
}\alpha<\omega_{1}\\},\mbox{ where
}[f]=\\{g\in\prod_{\alpha<\omega_{1}}X_{\alpha}\colon f\subseteq g\\}.$

Assume that there are given bonding maps $\pi^{\alpha}_{\beta}\colon
X_{\alpha}\to X_{\beta}$ such that

$\gamma<\beta<\alpha<\omega_{1}\mbox{ implies
}\pi^{\beta}_{\gamma}\circ\pi^{\alpha}_{\beta}=\pi^{\alpha}_{\gamma}.$

We have an inverse system
$\mathbb{P}=\\{X_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$ and the inverse
limit $\varprojlim\mathbb{P}$. By [4, Proposition 2.5.5.], the inverse limit
$\varprojlim\mathbb{P}$ is a $P$-space. Given an inverse system
$\mathbb{P}=\\{X_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$, we would like
to enrich it by injections $\iota^{\alpha}_{\beta}\colon X_{\beta}\to
X_{\alpha}$, for each $\beta<\alpha$, such that

  * (a)

$\iota^{\alpha}_{\beta}\circ\iota^{\beta}_{\gamma}=\iota^{\alpha}_{\gamma}$,
where $\gamma<\beta<\alpha<\omega_{1}$;

  * (b)

$\pi^{\alpha}_{\beta}\circ\iota^{\alpha}_{\beta}=\operatorname{id}_{X_{\beta}}$,
where $\beta<\alpha<\omega_{1}$,

see Diagram 1.

$\textstyle{X_{\gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\alpha}_{\gamma}}$$\scriptstyle{\iota^{\beta}_{\gamma}}$$\textstyle{X_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\beta}_{\gamma}}$$\scriptstyle{\iota^{\alpha}_{\beta}}$$\textstyle{X_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\pi^{\alpha}_{\beta}}$$\scriptstyle{\pi^{\alpha}_{\gamma}}$
Diagram 1.

Given commutative diagrams, as Diagram 1 with
$\gamma\leq\beta\leq\alpha<\omega_{1}$, we get an enriched inverse system
$\mathbb{P}=\\{X_{\alpha},\pi^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$.
We use the same symbol as for the system
$\\{X_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$, since both of them gives
the same inverse limit. But the injections allow us to define the following
$P$-spaces. Let

$\mathbf{\Sigma}_{\mathbb{P}}=\\{(p_{\alpha})\in\varprojlim\mathbb{P}\colon\exists_{\gamma<\omega_{1}}\forall_{\beta>\gamma}\;p_{\beta}=\iota^{\beta}_{\gamma}(p_{\gamma})\\}\subseteq\varprojlim\mathbb{P}\subseteq\prod_{\alpha<\omega_{1}}X_{\alpha}$

and let $\mathbf{\Sigma}^{\mathbb{P}}$ be a subspace of
$\mathbf{\Sigma}_{\mathbb{P}}$, which consists of threads
$(p_{\alpha})\in\mathbf{\Sigma}_{\mathbb{P}}$ such that if $\alpha$ is an
infinite limit ordinal, then there exists $\beta<\alpha$ such that
$p_{\alpha}=\iota^{\alpha}_{\beta}(p_{\beta})$. We say that a thread
$(p_{\alpha})\in\varprojlim\mathbb{P}$ has a jump at an ordinal $\beta$,
whenever $\iota^{\beta+1}_{\beta}(p_{\beta})\neq p_{\beta+1}$.

######  Lemma 4.

Any thread in $\mathbf{\Sigma}^{\mathbb{P}}$ has finitely many jumps.

###### Proof.

Suppose that a thread $(p_{\alpha})$ belongs to
$\mathbf{\Sigma}^{\mathbb{P}}$. If there exists a limit ordinal $\gamma$ which
is a supremum of infinitely many jumps for this thread, then
$\iota^{\gamma}_{\beta}(p_{\beta})\neq p_{\gamma}$ for any $\beta<\gamma$; a
contradiction with $(p_{\alpha})\in\mathbf{\Sigma}^{\mathbb{P}}$. ∎

Let
$\mathbb{Q}=\\{Q_{\alpha},\pi^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$
and
$\mathbb{R}=\\{R_{\alpha},r^{\alpha}_{\beta},e^{\alpha}_{\beta},\omega_{1}\\}$
be enriched inverse sequences.

$\textstyle{Q_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\alpha}_{\beta}}$$\scriptstyle{s_{\beta}}$$\textstyle{Q_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\alpha}_{\beta}}$$\scriptstyle{s_{\alpha}}$$\textstyle{R_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{\alpha}_{\beta}}$$\textstyle{R_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\alpha}_{\beta}}$
Diagram 2.

######  Lemma 5.

Let $s_{\alpha}\colon Q_{\alpha}\to R_{\alpha}$ be one-to-one functions such
that Diagram 2 is commutative, whenever $\beta<\alpha<\omega_{1}$, then

$\Sigma_{\mathbb{Q}}\subset_{h}\Sigma_{\mathbb{R}}\mbox{ and
}\Sigma^{\mathbb{Q}}\subset_{h}\Sigma^{\mathbb{R}},\mbox{ and
}\varprojlim\mathbb{Q}\subset_{h}\varprojlim\mathbb{R}.$

Moreover, if each $s_{\alpha}$ is a bijection, then $\Sigma_{\mathbb{Q}}$ is
homeomorphic to $\Sigma_{\mathbb{R}}$ and $\Sigma^{\mathbb{Q}}$ is
homeomorphic to $\Sigma^{\mathbb{R}}$, and $\varprojlim\mathbb{Q}$ is
homeomorphic to $\varprojlim\mathbb{R}$.

###### Proof.

If $(u_{\alpha})\in\varprojlim\mathbb{Q}$, then the formula
$(u_{\alpha})\mapsto(s_{\alpha}(u_{\alpha}))$ defines an embedding from
$\varprojlim\mathbb{Q}$ to $\varprojlim\mathbb{R}$. The restrictions of this
embedding give embeddings $\Sigma_{\mathbb{Q}}\to\Sigma_{\mathbb{R}}$ and
$\Sigma^{\mathbb{Q}}\to\Sigma^{\mathbb{R}}$. But if all $s_{\alpha}$ are
bijections, then these embeddings are homeomorphisms. ∎

Any inverse system
$\mathbb{P}=\\{X_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$ can be
interpreted as a tree of height $\omega_{1}$, we refer the readers for basic
notions about trees to the book [7]. Namely, assume that the sets $X_{\alpha}$
are pairwise disjoint. Let

$T=\bigcup\\{X_{\alpha}\colon\alpha<\omega_{1}\\}$

and we put $x\leq y$, whenever $x\in X_{\alpha}$ and $y\in X_{\beta}$, and
$x=\pi^{\alpha}_{\beta}(y)$. Let $[T]$ be the family of all branches of length
$\omega_{1}$. If $A\in[T]$, then $A=\\{p_{\alpha}\colon p_{\alpha}\in
X_{\alpha}\mbox{ and }\alpha<\omega_{1}\\}$ and
$(p_{\alpha})\in\varprojlim\mathbb{P}$. So, the mapping $A\mapsto(p_{\alpha})$
is a bijection between $[T]$ and $\varprojlim\mathbb{P}$, which is also a
homeomorphism, whenever $[T]$ is endowed with the topology generated by the
family

$\\{\\{b\in[T]\colon x\in b\\}\colon x\in T\\}.$

Some authors use a notion tree topology for the topology just defined on
$[T]$, compare [16, p. 14].

The interpretation of an inverse limit as a tree which yields a topological
space, consisting of branches of length $\omega_{1}$, leads us to the notion
of labeling. Namely, a surjection $E\colon T\to Y\subseteq[T]$ is called
labeling, if for every $x,y\in T$ we have $x\in E(x)$ and the following
implication holds:

$x\leq y\mbox{ and }y\in E(x)\Rightarrow E(x)=E(y).$

If $E\colon T\to Y\subseteq[T]$ is a labeling, then $(T,\leq,E)$ is a labelled
tree, which corresponds to the enriched inverse system
$\\{X_{\alpha},\pi^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$,
where $X_{\alpha}$ is the $\alpha$-th level of $T$, and
$\pi^{\alpha}_{\beta}\colon X_{\alpha}\to X_{\beta}$ is such that
$\pi^{\alpha}_{\beta}(x)\leq x$ for every $x\in X_{\alpha}$. The injections
$\iota^{\alpha}_{\beta}$ are defined as follows. If $\alpha<\beta$ and $x\in
X_{\alpha}$, then $\iota^{\beta}_{\alpha}(x)$ is the unique element of
$E(x)\cap X_{\beta}$. Finally $Y=\\{E(x)\colon x\in T\\}$.

##  3\. The $G_{\delta}$-modifications of the Cantor cubes

For an infinite cardinal $\kappa$, let $2^{\kappa}$ denote the Cantor cube
with the product topology. Thus $(2^{\kappa})_{\delta}$ is the
$G_{\delta}$-modification of the Cantor cube $2^{\kappa}$. Recall that if
$f\colon A\to\\{0,1\\}$ is a function and $A\subseteq\kappa$, then $[f]$
denotes the family of all extensions of $f$ with the domain $\kappa$ and
values in $\\{0,1\\}$. The next lemma is well-known, for example for readers
familiar with box products.

######  Lemma 6.

If $\kappa$ is an infinite cardinal, then the family

$\\{[f]\colon f\in 2^{A}\mbox{ and }A\in[\kappa]^{\omega}\\}$

is a base for $(2^{\kappa})_{\delta}$.∎

Recall that the Baire number of a crowded topological space $X$ is the
smallest cardinal $\kappa$ such that $X$ cannot be covered by a family of
cardinality less than $\kappa$ and consisting of nowhere dense subsets. The
Baire number of $(2^{\kappa})_{\delta}$ is always at least $\omega_{2}$.

######  Proposition 7.

If $\kappa$ is an uncountable cardinal, then any union of at most $\omega_{1}$
nowhere dense subsets of $(2^{\kappa})_{\delta}$ is a boundary set.

###### Proof.

Let $\mathcal{F}=\\{F_{\beta}\colon\beta<\omega_{1}\\}$ be a family of nowhere
dense subsets of $(2^{\kappa})_{\delta}$. Fix a non-empty open set
$U\subseteq(2^{\kappa})_{\delta}$ and then choose a function $f_{0}\colon
A_{0}\to\\{0,1\\}$ such that $[f_{0}]\subseteq U\setminus F_{0}$ and
$A_{0}\in[\kappa]^{\omega}$. Suppose that for $\beta<\alpha$ there are defined
functions $f_{\beta}\colon A_{\beta}\to\\{0,1\\}$ such that
$[f_{\gamma}]\supseteq[f_{\beta}]$ and $A_{\beta}\in[\kappa]^{\omega}$
whenever $\gamma<\beta<\alpha$. Choose $A_{\alpha}\in[\kappa]^{\omega}$ and
$f_{\alpha}\colon A_{\alpha}\to\\{0,1\\}$ such that

$[f_{\alpha}]\cap F_{\alpha}=\emptyset\mbox{ and
}\bigcup\\{f_{\beta}\colon\beta<\alpha\\}\subseteq f_{\alpha}.$

If $\bigcup\\{f_{\beta}\colon\beta<\omega_{1}\\}\subseteq f\in 2^{\kappa}$,
then $f\in U\setminus\bigcup\mathcal{F}$, i.e. the complement of
$\bigcup\mathcal{F}$ is a dense subset of $(2^{\kappa})_{\delta}$. ∎

##  4\. The $G_{\delta}$-modification of $2^{\omega_{1}}$

For each $\alpha<\omega_{1}$, assume that the set $2^{\alpha}$ is equipped
with the discrete topology and let $\pi^{\alpha}_{\beta}(f)=f|_{\beta}$,
whenever $\beta<\alpha$ and $f\in 2^{\alpha}$. We have

$\prod_{\alpha<\omega_{1}}2^{\alpha}\supseteq\varprojlim\\{2^{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}\stackrel{{\scriptstyle\varphi}}{{\to}}(2^{\omega_{1}})_{\delta},$

where $\varphi((p_{\alpha}))=\bigcup\\{p_{\alpha}\colon\alpha<\omega_{1}\\}$
for each
$(p_{\alpha})\in\varprojlim\\{2^{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$.
Since $\varphi(\pi^{-1}_{\alpha}(p_{\alpha}))=[p_{\alpha}]$, then the
bijection $\varphi$ is a homeomorphism.

If $\alpha<\omega_{1}$ and $f\in 2^{\alpha}$, then put

$f^{*}(\gamma)=\begin{cases}f(\gamma),&\mbox{when }\gamma<\alpha;\\\
0,&\mbox{when }\alpha\leq\gamma<\omega_{1}.\end{cases}$

Let $\mathbf{\Sigma}=\\{f^{*}\colon f\in 2^{\alpha}\mbox{ and
}\alpha<\omega_{1}\\}$. Thus, if any $X_{\alpha}=2^{\alpha}$ is endowed with
the discrete topology and $\iota^{\alpha}_{\beta}(f)=f^{*}|_{\alpha}$, then
$\mathbf{\Sigma}=\mathbf{\Sigma}_{\mathbb{P}}$, where
$\mathbb{P}=\\{2^{\alpha},\pi^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$.
Clearly, the mapping $[f]\mapsto f^{*}$ is a labeling.

######  Proposition 8.

The Baire number of the subspace
$\mathbf{\Sigma}\subseteq(2^{\omega_{1}})_{\delta}$ is $\omega_{1}$.

###### Proof.

Each set $A_{\alpha}=\\{f^{*}\colon f\in
2^{\alpha}\\}\subseteq\mathbf{\Sigma}$ is a discrete subset and
$\mathbf{\Sigma}=\bigcup\\{A_{\alpha}\colon\alpha<\omega_{1}\\}$, hence the
Baire number of $\mathbf{\Sigma}$ is at most $\omega_{1}$.

If $\\{F_{n}\colon n<\omega\\}$ is an increasing sequence of nowhere dense
subsets of $\mathbf{\Sigma}$, then inductively define a function
$f_{n}\colon\alpha_{n}\to\\{0,1\\}$ such that $(\alpha_{n})$ is an increasing
sequence of countable ordinals such that $[f_{n}]\subseteq[f_{n-1}]\setminus
F_{n}$. We get $f^{*}\in\mathbf{\Sigma}\setminus\bigcup\\{F_{n}\colon
n<\omega\\}$, whenever $f=\bigcup\\{f_{n}\colon n<\omega\\}$. ∎

A small modification of the proof above gives that any first category subset
of a dense subspace of $(2^{\omega_{1}})_{\delta}$ is nowhere dense.

######  Corollary 9.

$(2^{\omega_{1}})_{\delta}$ is not homeomorphic to a subspace of
$\mathbf{\Sigma}$.

###### Proof.

Any crowded subspace of $\mathbf{\Sigma}$, being a union of at most
$\omega_{1}$ many discrete subspaces, has the Baire number not greater than
$\omega_{1}$. But $(2^{\omega_{1}})_{\delta}$ has the Baire number at least
$\omega_{2}$. ∎

Recall that the family $\mathfrak{B}=\\{[f]\colon f\in 2^{\alpha}\mbox{ and
}\alpha<\omega_{1}\\}$ is a base for $(2^{\omega_{1}})_{\delta}$. Below, we
present a modified proof from [10], cf. [1, 3.1. Theorem].

######  Proposition 10.

If $\omega_{2}\leq\mathfrak{c}$, then the space $(2^{\omega_{1}})_{\delta}$ is
the union of an increasing sequence $\\{D_{\alpha}\colon\alpha<\omega_{2}\\}$
of nowhere dense subsets.

###### Proof.

If $V=[f]$ and $f\in 2^{\alpha}$, then let $\\{V_{\nu}\subseteq
V\colon\nu<\omega_{2}\\}$ be a family consisting of pairwise disjoint elements
of $\mathfrak{B}$. Put

$D_{\alpha}=(2^{\omega_{1}})_{\delta}\setminus\bigcup\\{V_{\nu}\colon\alpha<\nu<\omega_{2}\mbox{
and }V\in\mathfrak{B}\\}.$

The sequence $\\{D_{\alpha}\colon\alpha<\omega_{2}\\}$ is increasing and its
elements are nowhere dense sets. If $x\in(2^{\omega_{1}})_{\delta}$, then $x$
belongs to elements of the form $[x|_{\alpha}]\in\mathfrak{B}$ only. In other
words, $x$ belongs to $\omega_{1}$ many elements $V\in\mathfrak{B}$. Thus,
there exists $\alpha_{x}<\omega_{2}$ such that $x\in D_{\alpha_{x}}$, which
implies
$\bigcup\\{D_{\alpha}\colon\alpha<\omega_{2}\\}=(2^{\omega_{1}})_{\delta}$. ∎

In fact, we have shown that the Baire number of $(2^{\omega_{1}})_{\delta}$
equals $\omega_{2}$ is consistent with ZFC, for example, when
$|2^{\omega_{1}}|=\omega_{2}$.

The space $\mathbf{\Sigma}$ is a counterpart of a $P$-space of cardinality and
weight $\omega_{1}$, which appears in Lemma 2.2 and Corollary 2.3 in [3]. If
the Continuum Hypothesis fails, the space $\mathbf{\Sigma}$ being of
cardinality $\mathfrak{c}$, is not homeomorphic to a $P$-space of cardinality
$\omega_{1}$.

######  Theorem 11.

Any dense subset of $(2^{\omega_{1}})_{\delta}$ contains a homeomorphic copy
of the space $\mathbf{\Sigma}$.

###### Proof.

Let $Y\subseteq(2^{\omega_{1}})_{\delta}$ be a dense subset. Inductively,
define a sequence of functions $S_{\alpha}\colon 2^{\alpha}\to 2^{\alpha}$,
for $0<\alpha<\omega_{1}$, such that the following conditions are fulfilled.

  * (A).

If $f\in 2^{\alpha}$, then $S_{\alpha+1}$ restricted to the set
$\\{f^{\frown}0,f^{\frown}1\\}$ is a bijection onto the set
$\\{S_{\alpha}(f)^{\frown}0,S_{\alpha}(f)^{\frown}1\\}$, where
$f^{\frown}i=f\cup\\{(\alpha,i)\\}$.

  * (B).

If $\alpha$ is a limit ordinal and $f\in 2^{\alpha}$, then

$S_{\alpha}(f)=\bigcup\\{S_{\beta}(f|_{\beta})\colon\beta<\alpha\\},$

in particular $S_{\alpha}(f)\in 2^{\alpha}$.

  * (C).

If $g\in\mathbf{\Sigma}$, then

$\bigcup\\{S_{\alpha}(g|_{\alpha})\colon\alpha<\omega_{1}\\}\in Y.$

If $f\in 2^{1}$, then choose $y(f)\in Y\cap[f]$, and put
$S_{\alpha}(f^{*}|_{\alpha})=y(f)|_{\alpha}$ for each $\alpha<\omega_{1}$.
Thus, $y(f)=\bigcup\\{S_{\alpha}(f^{*}|_{\alpha})\colon\alpha<\omega_{1}\\}$
and $S_{1}(f)=f$.

Fix $\alpha<\omega_{1}$ and assume that bijections $S_{\beta}\colon
2^{\beta}\to 2^{\beta}$ are defined, whenever $\beta<\alpha$, such that
conditions (A)–(C) are fulfilled, in particular, for $g\in 2^{\beta}$ and
$\beta<\alpha$, the values $S_{\gamma}(g^{*}|_{\gamma})$ are defined such that

$\bigcup\\{S_{\gamma}(g^{*}|_{\gamma})\colon\gamma<\omega_{1}\\}\in Y.$

If $\alpha$ is a limit ordinal and $g\in 2^{\alpha}$, and $S_{\alpha}(g)$ has
not been defined, i.e. $\beta<\alpha$ implies $g^{*}\neq(g|_{\beta})^{*}$,
then choose

$y(g)\in Y\cap[\bigcup\\{S_{\beta}(g|_{\beta})\colon\beta<\alpha\\}]$

and put $S_{\gamma}(g^{*}|_{\gamma})=y(g)|_{\gamma}$ for
$\alpha\leq\gamma<\omega_{1}$. We get

$S_{\alpha}(g^{*}|_{\alpha})=S_{\alpha}(g)=\bigcup\\{S_{\beta}(g|_{\beta})\colon\beta<\alpha\\}\in
2^{\alpha}$

and $\bigcup\\{S_{\gamma}(g^{*}|_{\gamma})\colon\gamma<\omega_{1}\\}=y(g)\in
Y.$

If $\beta<\alpha$ and $g\in 2^{\beta}$, then all the values
$S_{\gamma}(g^{*}|_{\gamma})$ are defined by induction assumption. Since
$g^{*}=(g^{\frown}0)^{*}$, it remains to define $S_{\alpha}(g^{\frown}1)$,
where $\alpha=\beta+1$, and $S_{\gamma}((g^{\frown}1)^{*}|_{\gamma})$ for
$\alpha<\gamma<\omega_{1}$. Namely, let $i\in\\{0,1\\}$ be such that
$S_{\beta}(g)^{\frown}i\neq S_{\beta+1}(g^{\frown}0)\in 2^{\beta+1}$. Then put

$S_{\beta+1}(g^{\frown}1)=S_{\beta}(g)^{\frown}i,$

and choose $y(g)\in Y\cap[S_{\beta+1}(g^{\frown}1)]$, and put

$S_{\gamma}((g^{\frown}1)^{*}|_{\gamma})=y(g)|_{\gamma},$

whenever $\beta+1<\gamma<\omega_{1}$.

$\textstyle{2^{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{\beta}}$$\textstyle{2^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\alpha}_{\beta}}$$\scriptstyle{S_{\alpha}}$$\textstyle{2^{\beta}}$$\textstyle{2^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\alpha}_{\beta}}$
Diagram 3.

By the definition, Diagram 3, where $\pi^{\alpha}_{\beta}(f)=f|_{\beta}$, is
commutative.

Equipping each $2^{\alpha}$ with the discrete topology, we obtain that
$(2^{\omega_{1}})_{\delta}=\varprojlim\\{2^{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$
and we get an automorphism

$S\colon(2^{\omega_{1}})_{\delta}\to(2^{\omega_{1}})_{\delta},$

where $S(f)=\bigcup\\{S_{\alpha}(f|_{\alpha})\colon\alpha<\omega_{1}\\}$. By
condition (C), the image $S[\mathbf{\Sigma}]\subseteq Y$ is a homeomorphic
copy of $\mathbf{\Sigma}$. ∎

For any partition $(2^{\omega_{1}})_{\delta}=A\cup B$, we have
$\mathbf{\Sigma}\subset_{h}A$ or $\mathbf{\Sigma}\subset_{h}B$, in other
words, the topological arrow relation
$(2^{\omega_{1}})\to(\mathbf{\Sigma})^{1}_{2}$ is fulfilled. Namely, there
exists $f\in 2^{\alpha}$, where $\alpha<\omega_{1}$, such that $A\cap[f]$ or
$B\cap[f]$ is dense in $[f]$, since both sets $A$ and $B$ cannot be nowhere
dense. But the subspace $[f]\subseteq(2^{\omega_{1}})_{\delta}$ is
homeomorphic to $(2^{\omega_{1}})_{\delta}$.

######  Corollary 12.

The family

$\\{Y\subseteq(2^{\omega_{1}})_{\delta}\colon Y\mbox{ is a dense subset}\\}$

contains the least element with respect to the relation $\subset_{h}$.∎

But under the Continuum Hypothesis, we conclude the following.

######  Theorem 13.

If the Continuum Hypothesis is assumed, then any two dense subsets of
$(2^{\mathfrak{c}})_{\delta}$ of cardinality $\mathfrak{c}$ are homeomorphic.

###### Proof.

Let
$X=\\{x_{\alpha}\colon\alpha<\omega_{1}=\mathfrak{c}\\}\subseteq(2^{\mathfrak{c}})_{\delta}$
be a dense subset and let
$\mathfrak{M}_{X}=\\{\mathcal{P}_{\alpha}\colon\alpha<\mathfrak{c}\\}$, where
$\mathcal{P}_{\alpha}=\\{[f]\cap X\colon f\in 2^{\alpha}\\}$. If
$V\in\mathcal{P}_{\alpha}$, since $X\subseteq(2^{\mathfrak{c}})_{\delta}$ is
dense, then there exists $\beta=\inf\\{\nu\colon x_{\nu}\in V\\}$. Put
$E(V)=x_{\beta}$, and then put $E(U)=x_{\beta}$, whenever $E(V)\in U\subseteq
V$ and $U\in\bigcup\mathfrak{M}_{X}$. The function
$E\colon\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\mathfrak{c}\\}\to X$ is a
labeling. Indeed, fix $\alpha<\omega_{1}$, then the set
$\\{x_{\gamma}\colon\gamma<\alpha\\}$ is closed, being countable. But
$\bigcup\mathfrak{M}_{X}$ is a base for $X$, hence there exists
$\beta<\omega_{1}$ and $V\in\mathcal{P}_{\beta}$ such that $x_{\alpha}\in V$
and $V\cap\\{x_{\gamma}\colon\gamma<\alpha\\}=\emptyset$. Then
$E(V)=x_{\alpha}$, thus $E$ is a surjection. We have a labeling
$E\colon\bigcup\mathfrak{M}_{X}\to X$ and bijections $S_{\alpha}$ defined
analogously as in the proof of 11 together with Lemma 5 give needed
homeomorphism from $\mathbf{\Sigma}$ to $X$, where $[f]\mapsto f^{*}$ is a
labeling with the image $\mathbf{\Sigma}$. ∎

In order to avoid a modification of our proof of Theorem 11, we end this
section by stating without proof that any two dense subsets of
$(2^{\omega_{1}})_{\delta}$ which have labelings are homeomorphic.

##  5\. $P$-spaces of cardinality $\omega_{1}$

The below lemma is a counterpart of [15, Theorem 3].

######  Lemma 14.

If $\mathcal{B}$ is a base consisting of clopen sets of a $P$-space $X$ of
cardinality $\omega_{1}$, then any open cover of $X$ has a refinement, which
is a partition and consists of elements of $\mathcal{B}$.

###### Proof.

Let $X=\\{x_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-space. Fix an open
cover $\mathcal{P}$ of $X$. Since clopen subsets of $X$ constitute a
$\sigma$-algebra, define inductively a desired partition
$\\{V_{\alpha}\colon\alpha<\omega_{1}\\}$ as follows. If $x_{\alpha}\in
U\in\mathcal{P}$ and
$x_{\alpha}\notin\bigcup\\{V_{\beta}\colon\beta<\alpha\\}$, then choose
$V_{\alpha}\in\mathcal{B}$, satisfying $x_{\alpha}\in V_{\alpha}\subseteq U$
and $V_{\alpha}\cap\bigcup\\{V_{\beta}\colon\beta<\alpha\\}=\emptyset$,
otherwise put $V_{\alpha}=V_{0}$. ∎

Since Proposition 3 and Lemma 14, we clearly have the following.

######  Corollary 15.

If $X$ is a $P$-space of cardinality $\omega_{1}$, then for any countable
family $\mathcal{P}$ consisting of open covers there exists a partition of
$X$, which refines any cover from $\mathcal{P}$.∎

Recall that a cover is point-countable, whenever each point belongs to at most
countable many elements of this cover.

######  Proposition 16.

Any base for a $P$-space of cardinality $\omega_{1}$ contains a point-
countable cover.

###### Proof.

Let $X=\\{x_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-space. Fix a base
$\mathcal{B}$ of open subsets. Choose $U_{0}\in\mathcal{B}$ such that
$x_{0}\in U_{0}$, then choose a clopen set $V_{0}$ such that $x_{0}\in
V_{0}\subseteq U_{0}$. Suppose that sets $U_{\beta}\in\mathcal{B}$ and clopen
sets $V_{\beta}$ are defined for $\beta<\alpha$ in such a way that

  * •

$\\{x_{\beta}\colon\beta<\alpha\\}\subseteq\bigcup\\{V_{\beta}\colon\beta<\alpha\\}$;

  * •

If $\gamma<\beta<\alpha$, then $V_{\gamma}\cap U_{\beta}=\emptyset$.

If $\delta<\omega_{1}$ is the minimal ordinal such that
$x_{\delta}\notin\bigcup\\{V_{\beta}\colon\beta<\alpha\\}$, then choose
$U_{\alpha}\in\mathcal{B}$ such that

$x_{\delta}\in U_{\alpha}\mbox{ and
}U_{\alpha}\cap\bigcup\\{V_{\beta}\colon\beta<\alpha\\}=\emptyset$

and fix a clopen set $V_{\alpha}$ such that $x_{\delta}\in V_{\alpha}\subseteq
U_{\alpha}$. By the definition, the family
$\\{V_{\alpha}\colon\alpha<\omega_{1}\\}$ is a partition on $X$. But the
family $\\{U_{\alpha}\colon\alpha<\omega_{1}\\}\subseteq\mathcal{B}$ is a
point-countable open cover. Indeed, if $x\in V_{\alpha}$, then $x\notin
U_{\gamma}$ for each $\gamma>\alpha$, hence the set $\\{\beta\colon x\in
U_{\beta}\\}$ is countable. ∎

As far as we are aware, covering properties of $P$-spaces have not been deeply
investigated. If a $P$-space is of cardinality or weight $\omega_{1}$, then it
is paracompact. We do not know when a paracompact $P$-space is totally
paracompact, for the definition of totally paracompactness, see [13].

##  6\. $P$-spaces of weight $\omega_{1}$

If $X$ is a $P$-space of weight $\omega_{1}$, then any open cover of $X$ has a
refinement which is a partition. Indeed, $X$ has a base of cardinality
$\omega_{1}$ which consists of clopen sets. If ${\mathcal{U}}$ is an open
cover of $X$, then let $\\{V_{\alpha}\colon\alpha<\omega_{1}\\}$ be a
refinement of ${\mathcal{U}}$ which consists of clopen sets. For each
$\alpha<\omega_{1}$, put

$U_{\alpha}=V_{\alpha}\setminus\bigcup\\{V_{\beta}\colon\beta<\alpha\\}.$

The sets $U_{\alpha}$ constitute the desired partition.

Under the Continuum Hypothesis, by Lemma 6, the space
$(2^{\omega_{1}})_{\delta}$ is of weight $\omega_{1}=\mathfrak{c}$. But
without the Continuum Hypothesis, any open cover of
$(2^{\omega_{1}})_{\delta}$ has a refinement which is a partition. Indeed, the
family

$\mathcal{B}=\\{[f]\colon f\in 2^{\alpha}\mbox{ and }\alpha<\omega_{1}\\}$

is a base for $(2^{\omega_{1}})_{\delta}$. If ${\mathcal{U}}$ is an open cover
of $(2^{\omega_{1}})_{\delta}$, then let $\mathcal{V}\subseteq\mathcal{B}$ be
a refinement of ${\mathcal{U}}$. The family of all maximal elements of
$\mathcal{V}$, with respect to the inclusion, is the desired partition. In
particular, we see that $(2^{\omega_{1}})_{\delta}$ is a paracompact space.

If $X$ is a $P$-space, then a family
$\mathfrak{M}_{X}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ is called
$P$-matrix, whenever

  1. (1)

Each $\mathcal{P}_{\alpha}$ is a partition of $X$.

  2. (2)

If $\beta<\alpha$, then $\mathcal{P}_{\alpha}$ refines $\mathcal{P}_{\beta}$.

  3. (3)

The union $\bigcup\mathfrak{M}_{X}$ is a base for $X$.

  4. (4)

If $\alpha$ is an infinite limit ordinal and $U\in\mathcal{P}_{\alpha}$, then

$U=\bigcap\\{V\in\mathcal{P}_{\beta}\colon\beta<\alpha\mbox{ and }U\subseteq
V\\}.$

If a $P$-space $X$ has a $P$-matrix
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$, then any open cover of
$X$ has a refinement which is a partition, since a slightly modified argument
used for $(2^{\omega_{1}})_{\delta}$ works. Indeed, if ${\mathcal{U}}$ is an
open cover of $X$, then let
$\mathcal{V}\subseteq\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$
be a refinement of ${\mathcal{U}}$. The family of all maximal elements of
$\mathcal{V}$, with respect to the inclusion, is the desired partition.

######  Lemma 17.

Any $P$-space of weight $\omega_{1}$ has a $P$-matrix.

###### Proof.

Let $X$ be a $P$-space with a base $\\{U_{\alpha+1}\colon\alpha<\omega_{1}\\}$
consisting of clopen sets. Let $\mathcal{P}_{0}=\\{X\\}$. Assume that
partitions $\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$ are already defined.
Let

$\mathcal{P}_{\alpha}^{*}=\\{\bigcap L\colon L\mbox{ is a maximal chain in
}\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\\}.$

If $\alpha$ is an infinite limit ordinal, then put
$\mathcal{P}_{\alpha}=\mathcal{P}_{\alpha}^{*}$. If $\alpha$ is not a limit
ordinal, then let $\mathcal{P}_{\alpha}$ be a partition which refines
$\\{U_{\alpha},X\setminus U_{\alpha}\\}$ and the partition
$\mathcal{P}_{\alpha}^{*}$. The family
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ is the desired $P$-matrix.
∎

Let $(\gamma_{\alpha})$ be the increasing enumeration of all countable
infinite limit ordinals. Put $Q_{\alpha}=2^{\gamma_{\alpha}}$. Since
$(2^{\omega_{1}})_{\delta}=\varprojlim\\{2^{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$
and the family of countable limit ordinals is cofinal in $\omega_{1}$, we get

$(2^{\omega_{1}})_{\delta}=\varprojlim\\{Q_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\},$

where ${\pi}^{\alpha}_{\beta}\colon Q_{\alpha}\to Q_{\beta}$ and
$\pi^{\alpha}_{\beta}(f)=f|_{\gamma_{\beta}}$: the symbol
$\pi^{\alpha}_{\beta}$ has been used in two different meanings, but this does
not lead to confusion. Note that, each $Q_{\alpha}$ is of cardinality
$\mathfrak{c}$.

######  Theorem 18.

Any $P$-space of weight $\omega_{1}$ can be embedded into
$(2^{\omega_{1}})_{\delta}$.

###### Proof.

Let $X$ be a $P$-space of weight $\omega_{1}$ and let
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-matrix for $X$.
Thus, we have an inverse system
$\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$, where each
$r^{\alpha}_{\beta}$ is the restriction of the inclusion and each
$\mathcal{P}_{\alpha}$ is equipped with the discrete topology. Each $x\in X$
determines the thread in
$\varprojlim\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$, since
$\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ is a base for $X$.
Hence $X$ can be embedded into
$\varprojlim\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$, by
Proposition 2.5.5 [4].

To show that
$\varprojlim\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$ can be
embedded into
$(2^{\omega_{1}})_{\delta}=\varprojlim\\{Q_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$,
where $Q_{\alpha}=2^{\gamma_{\alpha}}$, we shall define a sequence of
injections $S_{\alpha}\colon\mathcal{P}_{\alpha}\to Q_{\alpha}$ such that
Diagram 4 is commutative.

$\textstyle{\mathcal{P}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{\beta}}$$\textstyle{\mathcal{P}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\alpha}_{\beta}}$$\scriptstyle{S_{\alpha}}$$\textstyle{Q_{\beta}}$$\textstyle{Q_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\alpha}_{\beta}}$
Diagram 4.

Let $S_{0}\colon\mathcal{P}_{0}\to Q_{0}$ be an arbitrary injection. Assume
that for each $\beta<\alpha$ an injection $S_{\beta}$ is defined such that
appropriate diagrams are commutative, i.e. $S_{\gamma}\circ
r^{\beta}_{\gamma}=\pi^{\beta}_{\gamma}\circ S_{\beta}$ for
$\gamma<\beta<\alpha$. If $\alpha$ is a limit ordinal and
$U\in\mathcal{P}_{\alpha}$, then

$S_{\alpha}(U)=\bigcup\\{S_{\beta}(V)\colon V\in\mathcal{P}_{\beta}\mbox{ and
}V\supseteq U\mbox{ and }\beta<\alpha\\}.$

Let $\alpha=\beta+1$. For each $V\in\mathcal{P}_{\beta}$, let
$\mathcal{P}_{V}=\\{U\in\mathcal{P}_{\alpha}\colon U\subseteq V\\}$ and let
$Q_{V}=\\{f\in Q_{\alpha}\colon S_{\beta}(V)\subseteq f\\}$. Choose arbitrary
injections $S_{V}\colon\mathcal{P}_{V}\to Q_{V}$ for each
$V\in\mathcal{P}_{\beta}$. Then put $S_{\alpha}=\bigcup\\{S_{V}\colon
V\in\mathcal{P}_{\beta}\\}$.

Since each $S_{\alpha}$ is an injection, hence by Lemma 2.5.9, [4], we obtain
the desired embedding. ∎

Under the Continuum Hypothesis, the space $(2^{\omega_{1}})_{\delta}$ has the
greatest dimensional type in the class of all $P$-spaces of weight
$\omega_{1}$. But if the Continuum Hypothesis fails, then our argumentation
does not work, since $(2^{\omega_{1}})_{\delta}$ is of weight $\mathfrak{c}$.

##  7\. $P$-spaces of cardinality and weight $\omega_{1}$

Let $X$ be a $P$-space with a $P$-matrix
$\mathfrak{M}_{X}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$. If $X$
has weight $\omega_{1}$, then such a $P$-matrix exists by Lemma 17. But, if
$Z\subseteq X$ is a dense subset of cardinality and weight $\omega_{1}$, then
there exists a labeling $E\colon\bigcup\mathfrak{M}_{X}\to Z$.

######  Lemma 19.

If $X$ is a $P$-space of cardinality and weight $\omega_{1}$, then there
exists a labeling $E\colon\bigcup\mathfrak{M}_{X}\to X$ for any $P$-matrix
$\mathfrak{M}_{X}$.

###### Proof.

Let $X=\\{x_{\alpha}\colon\alpha<\omega_{1}\\}$ and let
$\mathfrak{M}_{X}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a
$P$-matrix. If $V\in\mathcal{P}_{\alpha}$, then let $\beta=\inf\\{\nu\colon
x_{\nu}\in V\\}$ and then put $E(V)=x_{\beta}$. The function
$E\colon\bigcup\mathfrak{M}_{X}\to X$ is a labeling, which can be checked as
in the proof of Theorem 13. ∎

The below theorem, under the Continuum Hypothesis, follows from [3, Lemma
2.2].

######  Theorem 20.

Let $Y$ be a $P$-space with a $P$-matrix

$\mathfrak{M}_{Y}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$

such that each $\mathcal{P}_{\alpha}$ is of cardinality at most
$\mathfrak{c}$. If there exists a labeling $E\colon\mathfrak{M}_{Y}\to Y,$
then $Y$ can be embedded into $\mathbf{\Sigma}$.

###### Proof.

Fix a $P$-space $Y$ with a $P$-matrix

$\mathfrak{M}_{Y}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\},$

such that each $\mathcal{P}_{\alpha}$ is of cardinality at most
$\mathfrak{c}$. Let $E\colon\bigcup\mathfrak{M}_{Y}\to Y$ be a labeling.
Analogously as in the proof of Theorem 18 we shall define Diagram 5, which is
a version of Diagram 2, where $\iota^{\alpha}_{\beta}\colon Q_{\beta}\to
Q_{\alpha}$ and $\pi^{\alpha}_{\beta}\colon Q_{\alpha}\to Q_{\beta}$ are
defined just before Theorem 18.

$\textstyle{\mathcal{P}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\alpha}_{\beta}}$$\scriptstyle{S_{\beta}}$$\textstyle{\mathcal{P}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\alpha}_{\beta}}$$\scriptstyle{S_{\alpha}}$$\textstyle{Q_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\alpha}_{\beta}}$$\textstyle{Q_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\alpha}_{\beta}}$
Diagram 5.

But injections
$\eta^{\alpha}_{\beta}\colon\mathcal{P}_{\beta}\to\mathcal{P}_{\alpha}$ are
determined by the labeling $E\colon\bigcup\mathfrak{M}_{Y}\to Y$, i.e.
$\eta^{\alpha}_{\beta}(U)\in\mathcal{P}_{\alpha}$ and
$r^{\alpha}_{\beta}(V)\in\mathcal{P}_{\beta}$ are unique elements such that
$E(U)\in\eta^{\alpha}_{\beta}(U)$ and $r^{\alpha}_{\beta}(V)\supseteq V$, for
$U\in\mathcal{P}_{\beta}$ and $V\in\mathcal{P}_{\alpha}$. Thus, we have
defined two enriched inverse systems
$\mathbb{Q}=\\{Q_{\alpha},\pi^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$
and
$\mathbb{P}=\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\eta^{\alpha}_{\beta},\omega_{1}\\}$,
so it remains to define injections $S_{\alpha}$, which will be done by a
modification of the proof of Theorem 11. Namely, let
$S_{0}\colon\mathcal{P}_{0}\to Q_{0}$ be an injection. Let
$\alpha<\omega_{1}$. Assume that we have defined a sequence of injections
$S_{\gamma}\colon\mathcal{P}_{\gamma}\to Q_{\gamma}$ for $\gamma<\alpha$, such
that the diagrams obtained from Diagram 5 by replacing $\alpha$ with $\gamma$
are commutative, where $\beta<\gamma<\alpha$.

If $\alpha=\gamma+1$, then fix $V\in\mathcal{P}_{\gamma}$. Choose an injection

$S^{V}_{\alpha}\colon\\{U\in\mathcal{P}_{\alpha}\colon U\subseteq
V\\}\to\\{W\in Q_{\alpha}\colon W\subseteq S_{\gamma}(V)\\}$

such that
$S^{V}_{\alpha}(\eta^{\alpha}_{\gamma}(V))=\iota^{\alpha}_{\gamma}(S_{\gamma}(V))$.
Put $S_{\alpha}(U)=S^{V}_{\alpha}(U)$, where $V$ is a unique element of
$\mathcal{P}_{\gamma}$ containing $U$.

If $\alpha$ is a limit ordinal and $V\in\mathcal{P}_{\alpha}$, then
$V=\bigcap\\{U\in\mathcal{P}_{\beta}\colon\beta<\alpha\mbox{ and }V\subseteq
U\\},$ so put

$S_{\alpha}(V)=\bigcup\\{S_{\beta}(U)\colon\beta<\alpha\mbox{ and }V\subseteq
U\in\mathcal{P}_{\beta}\\}.$

By Lemma 2.5.9 [4], the function
$S\colon\varprojlim\mathbb{P}\to\varprojlim\mathbb{Q}$, given by the formula
$S((x_{\alpha}))=(S_{\alpha}(x_{\alpha}))$, is an embedding such that

$S|_{\mathbf{\Sigma}_{\mathbb{P}}}\colon\mathbf{\Sigma}_{\mathbb{P}}\to\mathbf{\Sigma}_{\mathbb{Q}}=\mathbf{\Sigma}.$

The proof is completed, since $Y$ has to be homeomorphic to
$\mathbf{\Sigma}_{\mathbb{P}}$. Indeed, if $f_{\alpha}\colon
Y\to\mathcal{P}_{\alpha}$ are functions such that $x\in f_{\alpha}(x)$, then
the function $f\colon Y\to\mathbf{\Sigma}_{\mathbb{P}}$, given by the formula
$x\mapsto f(x)=(f_{\alpha}(x))$, is a homeomorphism. ∎

######  Corollary 21.

Any $P$-space of cardinality and weight $\omega_{1}$ can be embedded into the
space $\mathbf{\Sigma}$.

###### Proof.

By Lemmas 17 and 19, any $P$-space of cardinality and weight $\omega_{1}$ has
a $P$-matrix and a labeling as it is required in Theorem 20. ∎

If $X$ and $Y$ are topological spaces, then $X\to(Y)^{1}_{2}$ means that $Y$
can be embedded into one of $A$ or $B$ for any subspaces $A$ and $B$ such that
$X=A\cup B$. If $Z\subseteq(2^{\omega_{1}})_{\delta}$ is a dense subset, then
any $P$-space of cardinality and weight $\omega_{1}$ can be embedded into $Z$,
see A. Dow [3]. Thus, Theorem 11 and Corollary 21 provide another proof of
Dow’s result. Also, Theorem 11 implies
$(2^{\omega_{1}})_{\delta}\to(\mathbf{\Sigma})^{1}_{2}$, which gives an
example concerning Question 6.2 stated in [2].

##  8\. On Lindelöf and nowhere Lindelöf $P$-spaces

Recall that a space is Lindelöf if its every open cover has a countable
subcover. We say that a topological space is nowhere Lindelöf, whenever it
does not contain a non-empty open subset with the Lindelöf property. Assume
that $\lambda$ is an infinite cardinal number and a $P$-matrix
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ satisfies conditions
(1)–(4). We shall add another condition.

(5-$\lambda$). Each $\mathcal{P}_{\alpha}$ is of cardinality $\lambda$ and if
$\beta<\alpha$, then any $V\in\mathcal{P}_{\beta}$ contains $\lambda$ many
elements of $\mathcal{P}_{\alpha}$.

We are particularly interested in
$\lambda\in\\{\omega,\omega_{1},\mathfrak{c}\\}$. If $X$ is a $P$-space with a
$P$-matrix $\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ which satisfies
condition (5-$\lambda$), then we have an inverse system
$\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$ defined analogously
as $\\{Q_{\alpha},\pi^{\alpha}_{\beta},\omega_{1}\\}$ just before Theorem 18.
If each $\mathcal{P}_{\alpha}$ is equipped with the discrete topology and
$r^{\alpha}_{\beta}\colon\mathcal{P}_{\alpha}\to\mathcal{P}_{\beta}$, where
$r^{\alpha}_{\beta}(U)\in\mathcal{P}_{\beta}$ is a unique element containing
$U$, then we get the inverse limit

$\varprojlim\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\},$

which contains a homeomorphic copy of $X$ as a dense subset. Thus,
$\varprojlim\\{\mathcal{P}_{\alpha},r^{\alpha}_{\beta},\omega_{1}\\}$ is a
crowded $P$-space of weight and density equal to

$\max\\{\lambda,\omega_{1}\\}=|\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}|.$

######  Proposition 22.

If a Lindelöf (nowhere Lindelöf) $P$-space $X$ is of cardinality and weight
$\omega_{1}$, then $X$ has a $P$-matrix
$\\{\mathcal{P}_{\beta}\colon\beta<\omega_{1}\\}$ such that if $\beta<\alpha$,
then any $V\in\mathcal{P}_{\beta}$ contains countably (respectively
$\omega_{1}$) many elements of $\mathcal{P}_{\alpha}$.

###### Proof.

If $X$ is a Lindelöf space, then a $P$-matrix constructed as in the proof of
Lemma 17 is suitable. But, if $X$ is a nowhere Lindelöf space, it suffices to
modify the construction of a $P$-matrix
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ from the proof of Lemma
17, defining a new $P$-matrix
$\\{\mathcal{P}^{*}_{\alpha}\colon\alpha<\omega_{1}\\}$. Namely, each
partition $\mathcal{P}^{*}_{\alpha+1}$ is such that any
$V\in\mathcal{P}^{*}_{\alpha}$ contains $\omega_{1}$ many elements of
$\mathcal{P}^{*}_{\alpha+1}$. But, if $\alpha$ is a limit ordinal, then
$\mathcal{P}^{*}_{\alpha}$ is defined analogously as $\mathcal{P}_{\alpha}$. ∎

Let $X$ be a $P$-space with a $P$-matrix
$\mathfrak{M}_{X}=\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ which
satisfies condition (5-$\lambda$). Suppose that there exists a labeling
$E\colon\bigcup\mathfrak{M}_{X}\to X$ which satisfy the following condition.

$(*)$.  If $\alpha<\omega_{1}$ and $\mathcal{L}$ is a chain contained in
$\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$ and
$\bigcap\mathcal{L}\neq\emptyset$, then there exists $V\in\mathcal{L}$ such
that $E(V)\in\bigcap\mathcal{L}$.

In this case we say that $X$ has a $\lambda$-thin labeling. By the definition,
any $P$-space with a $\lambda$-thin labeling is a crowded space since every
base set contains infinitely many pairwise disjoint subsets. Also, if $X$ has
a $\lambda$-thin labeling $E\colon\mathfrak{M}_{X}\to X$, then

$E[\mathcal{P}_{\alpha}]=E[\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}],$

for each limit ordinal $\alpha$. Applying Lemma 4, one can check that an
arbitrary $P$-space with a $\lambda$-thin labeling has to be of first
category.

######  Lemma 23.

If a crowded $P$-space $X$ is of weight $\omega_{1}$, then there exists
$Z\subseteq X$ such that $Z$ has an $\omega$-thin labeling.

###### Proof.

Applying Lemma 17, choose a dense subset $Z\subseteq X$ of cardinality
$\omega_{1}$ with a $P$-matrix
$\mathfrak{M}_{Z}=\\{\mathcal{Q}_{\alpha}\colon\alpha<\omega_{1}\\}$. Let
$E\colon\bigcup\mathfrak{M}_{Z}\to Z$ be a labeling, which exists by Lemma 19.
Without loss of generality, because $Z$ is crowded, assume that if
$\beta<\alpha$ and $V\in\mathcal{Q}_{\beta}$, then $V$ contains infinitely
many elements of $\mathcal{Q}_{\alpha}$. Choose a family
$\mathcal{P}_{0}\subseteq\mathcal{Q}_{0}$ such that
$|\mathcal{P}_{0}|=\omega$. For each $V\in\mathcal{P}_{0}$, choose a point
$E(V)\in V$. Suppose families $\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$
and points

$\\{E(V)\colon V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\\}$

are defined. If $V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$, then
put

$\mathcal{L}_{V}=\\{W\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\colon
E(V)\in W\\}.$

Then choose a family $\mathcal{R}_{V}\subseteq\mathcal{Q}_{\alpha+1}$
consisting of $\omega$ many pairwise disjoint clopen subsets of
$\bigcap\mathcal{L}_{V}\subseteq V$ such that $E(V)\in\bigcup\mathcal{R}_{V}$.
For each $W\in\mathcal{R}_{V}$ such that $E(V)\notin W$, choose a point
$E(W)\in W$. Let

$\mathcal{P}_{\alpha+1}=\bigcup\\{\mathcal{R}_{V}\colon\mathcal{R}_{V}\subseteq\mathcal{Q}_{\alpha+1}\mbox{
and }V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\\}.$

Let $Z$ be the set of all points $E(V)$, which are defined above. Any base set
of $Z$ contains infinitely many pairwise disjoint subsets, hence $Z$ is
crowded. Putting $\mathcal{P}^{*}_{\alpha}=\\{V\cap Z\colon
V\in\mathcal{P}_{\alpha}\\}$, define the function
$E^{*}\colon\bigcup\\{\mathcal{P}^{*}_{\alpha}\colon\alpha<\omega_{1}\\}\to Z$
by the formula $V\cap Z\mapsto E(V)$. The map $E^{*}$ is a labeling. ∎

######  Theorem 24.

If a $P$-space $Y$ has an $\omega$-thin labeling, then $Y$ is a Lindelöf
space.

###### Proof.

Let $\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-matrix for
$Y$ and

$E\colon\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}\to Y$

be an $\omega$-thin labeling. Fix an open cover $\mathcal{U}$ of $Y$. We can
assume that $\mathcal{U}$ is a partition of $Y$, since $Y$ is of cardinality
and weight $\omega_{1}$, see Lemma 14. Let $\alpha_{0}<\omega_{1}$ be an
ordinal number such that if $V\in\mathcal{P}_{0}$, then there exist
$\beta\leq\alpha_{0}$, $W_{V}\in\mathcal{P}_{\beta}$ and $U_{V}$ such that

$E(V)\in W_{V}\subseteq U_{V}\in\mathcal{U}.$

Assume that an ordinal $\alpha_{n}$ is defined such that if
$V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta\leq\alpha_{n-1}\\}$, then there
exist $\gamma\leq\alpha_{n}$, $W_{V}\in\mathcal{P}_{\gamma}$ and $U_{V}$ such
that

$E(W_{V})=E(V)\in W_{V}\subseteq U_{V}\in\mathcal{U}.$

Let $\alpha_{n+1}>\alpha_{n}$ be a countable ordinal such that if
$V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta\leq\alpha_{n}\\}$, then there
$W_{V}\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta\leq\alpha_{n+1}\\}$ and
$U_{V}$, fulfilling

$E(V)\in W_{V}\subseteq U_{V}\in\mathcal{U}.$

Since $E$ is an $\omega$-thin labeling, using conditions (5-$\omega$) and
($*$), we check that if $\alpha=\sup\\{\alpha_{n}\colon n>0\\}$, then the
partition $\mathcal{P}_{\alpha}$ refines $\mathcal{U}$. ∎

Translating the above proof to the language of category theory, we get that
$P$-matrix for $Y$ is a Fraïssé $\omega_{1}$-sequence in the category of all
open covers of $Y$, with refining pairs of covers as morphisms. We recommend
the paper [9] for details about Fraïssé sequences.

If $\mathbb{P}=\\{\mathcal{P}_{\alpha},\pi^{\beta}_{\alpha},\omega_{1}\\}$ is
an inverse system, where $\mathcal{P}_{\alpha}$ are countable discrete spaces,
then $\varprojlim\mathbb{P}$ is not necessary a Lindelöf space. This has been
observed in [8], compare [11, Lemma 2]. Let us present a sketch of proof. Let
$\mathbb{T}_{A}$ be an Aronszajn tree. Let
$\\{\ell_{\alpha}\colon\alpha<\omega_{1}\\}$ be a sequence of branches of
$\mathbb{T}_{A}$ with different height. Then, each $\ell_{\alpha}$ is extended
by a copy of a tree determined by a $P$-space which has an $\omega$-thin
labeling. The family of all just extended branches gives a tree of height
$\omega_{1}$ with all levels countable, i.e. we get the desired inverse limit
which is not Lindelöf.

######  Proposition 25.

If a crowded $P$-space $X$ is of weight $\omega_{1}$, then there exists
$Y\subseteq X$ such that $Y$ is nowhere Lindelöf.

###### Proof.

Let $X$ be a crowded $P$-space with a $P$-matrix
$\\{\mathcal{Q}_{\alpha}\colon\alpha<\omega_{1}\\}$. If $V\in\mathcal{Q}_{0}$,
then choose a family $\mathcal{P}_{V}$ consisting of $\omega_{1}$ pairwise
disjoint open sets such that $\bigcup\mathcal{P}_{V}\subseteq V$. Let
$\mathcal{P}_{0}=\bigcup\\{\mathcal{P}_{V}\colon V\in\mathcal{Q}_{0}\\}$ and
let $Y_{0}\subseteq X$ be such that $Y_{0}\cap V$ is a singleton for each
$V\in\mathcal{P}_{0}$. Assume that families
$\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$ are defined. If
$V\in\mathcal{Q}_{\alpha}$ and there exists $W_{\beta}\in\mathcal{P}_{\beta}$,
for each $\beta<\alpha$, such that

$V\cap\bigcap\\{W_{\beta}\colon\beta<\alpha\\}\neq\emptyset,$

then choose a family $\mathcal{P}_{V}$ consisting of $\omega_{1}$ pairwise
disjoint open sets such that $\bigcup\mathcal{P}_{V}\subseteq
V\cap\bigcap\\{W_{\beta}\colon\beta<\alpha\\}$, otherwise
$\mathcal{P}_{V}=\emptyset$.

Let

$\mathcal{P}_{\alpha}=\bigcup\\{\mathcal{P}_{V}\colon
V\in\mathcal{Q}_{\alpha}\\}$

and let $Y_{\alpha}\subseteq X$ be such that
$\bigcup\\{Y_{\beta}\colon\beta<\alpha\\}\subseteq Y_{\alpha}$ and
$Y_{\alpha}\cap V$ is a singleton for each $V\in\mathcal{P}_{\alpha}$. Thus
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ is a $P$-matrix for the
subset $Y=\bigcup\\{Y_{\alpha}\colon\alpha<\omega_{1}\\}\subseteq X$. Put
$E(V)=Y_{\alpha}\cap V$, whenever $V\in\mathcal{P}_{\alpha}$. Thus,
$E\colon\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}\to Y$ is a
desired labeling. ∎

######  Corollary 26.

If a crowded $P$-space $X$ is of weight $\omega_{1}$, then there exists
$Z\subseteq X$ such that $Z$ has an $\omega_{1}$-thin labeling.

###### Proof.

Using Proposition 25, take $Y\subseteq X$ such that $Y$ is a nowhere Lindelöf
subspace. Let $\\{\mathcal{Q}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a
$P$-matrix for $Y$ such that if $\alpha<\beta$ and $V\in\mathcal{Q}_{\alpha}$,
then $V$ contains $\omega_{1}$ many elements of $\mathcal{Q}_{\beta}$. The
rest of the proof is a modification of the reasoning of the proof of Lemma 23.
Namely, the family $\mathcal{P}_{0}\subseteq\mathcal{Q}_{0}$ is chosen to be
of cardinality $\omega_{1}$. If $V\in\mathcal{P}_{0}$, then select a point
$E(V)\in V$. Assume that families
$\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$ and points

$\\{E(V)\colon V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\\}$

are defined. If $V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}$, then
we repeat the definition of

$\mathcal{L}_{V}=\\{W\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\colon
E(V)\in W\\}.$

Then a family $\mathcal{R}_{V}\subseteq\mathcal{Q}_{\alpha+1}$ is chosen such
that it consists of $\omega_{1}$ many pairwise disjoint clopen subsets of
$\bigcap\mathcal{L}_{V}\subseteq V$ and $E(V)\in\bigcup\mathcal{R}_{V}$. Let

$\mathcal{P}_{\alpha}=\bigcup\\{\mathcal{R}_{V}\colon\mathcal{R}_{V}\subseteq\mathcal{Q}_{\alpha+1}\mbox{
and }V\in\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\alpha\\}\\}.$

If $W\in\mathcal{R}_{V}$ and $E(V)\notin W$, then choose a point $E(W)\in W$.
Let $Z$ be the set of all points $E(V)$, which are defined above. By the
definition, any base set of $Z\subseteq X$ has a partition consisting of
$\omega_{1}$ clopen subsets, then $Z$ is nowhere Lindelöf.

Since $E\colon\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}\to Z$
is a surjection, then $Z$ has an $\omega_{1}$-thin labeling. ∎

Now, we can prove counterparts of Theorem 13.

######  Theorem 27.

If $\lambda$ is an infinite cardinal number, then any two $P$-spaces which
have $\lambda$-thin labelings are homeomorphic.

###### Proof.

Assume that $X$ and $Y$ have $\lambda$-thin labelings. Let
$\mathfrak{M}_{X}=\\{{Q}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-matrix
of $X$ with a $\lambda$-thin labeling $E\colon\bigcup\mathfrak{M}_{X}\to X$
and let $\mathfrak{M}_{Y}=\\{{R}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a
$P$-matrix of $Y$ with a $\lambda$-thin labeling
$F\colon\bigcup\mathfrak{M}_{Y}\to Y$. Thus, we have two enriched systems
$\mathbb{Q}=\\{Q_{\alpha},q^{\alpha}_{\beta},\iota^{\alpha}_{\beta},\omega_{1}\\}$
and
$\mathbb{R}=\\{R_{\alpha},r^{\alpha}_{\beta},\eta^{\alpha}_{\beta},\omega_{1}\\}$,
where $\iota^{\alpha}_{\beta}$ and $\eta^{\alpha}_{\beta}$ are determined by
$\lambda$-thin labelings $E$ and $F$, respectively, but $q^{\alpha}_{\beta}$
and $r^{\alpha}_{\beta}$ are determined by the inclusion. We shall define a
bijection $s_{\alpha}\colon Q_{\alpha}\to R_{\alpha}$ such that the following
diagram

$\textstyle{Q_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\alpha}_{\beta}}$$\scriptstyle{s_{\beta}}$$\textstyle{Q_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\alpha}_{\beta}}$$\scriptstyle{s_{\alpha}}$$\textstyle{R_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\alpha}_{\beta}}$$\textstyle{R_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\alpha}_{\beta}}$

is commutative, whenever $\beta<\alpha<\omega_{1}$.

If $\alpha=\gamma+1$, then fix $V\in Q_{\gamma}$. Choose an injection

$s^{V}_{\alpha}\colon\\{U\in Q_{\alpha}\colon U\subseteq V\\}\to\\{W\in
R_{\alpha}\colon W\subseteq s_{\gamma}(V)\\}$

such that
$s^{V}_{\alpha}(\iota^{\alpha}_{\gamma}(V))=\eta^{\alpha}_{\gamma}(s_{\gamma}(V))$.
Put $s_{\alpha}(U)=s^{V}_{\alpha}(U)$, where $V$ is a unique element of
$Q_{\gamma}$ containing $U$.

If $\alpha$ is a limit ordinal and $V\in Q_{\alpha}$, then, by condition
$(*)$, it follows that there exists $\gamma<\alpha$ and $U\in Q_{\gamma}$ such
that $E(V)=E(U)$, hence we define $s_{\alpha}(V)=s_{\gamma}(U)$.

By Lemma 5, the inverse limits $\varprojlim\mathbb{Q}$ and
$\varprojlim\mathbb{R}$ are homeomorphic. Condition $(*)$ and Lemma 4 imply
that $X=\varprojlim\mathbb{Q}$ and $Y=\varprojlim\mathbb{R}$. ∎

Because of condition $(*)$, if a $P$-space $X$ has a $\lambda$-thin labeling,
then $X$ is an inverse limit, as it focused on at the end of the above proof.

######  Corollary 28.

If a $P$-space $Y$ has a $\lambda$-thin labeling and a subset $Z\subseteq Y$
is non-empty and clopen, then $Z\subseteq Y$ also has a $\lambda$-thin
labeling.

###### Proof.

Assume that $\\{P_{\alpha}\colon\alpha<\omega_{1}\\}$ is a $P$-matrix and
$E\colon\bigcup\\{P_{\alpha}\colon\alpha<\omega_{1}\\}\to Y$ is a
$\lambda$-thin labeling. Consider the family

$R=\\{V\subseteq Z\colon V\in P_{\alpha}\mbox{ and }0<\alpha<\omega_{1}\\}.$

Let $R_{0}\subseteq R$ be a maximal family of cardinality $\lambda$,
consisting of pairwise disjoint sets. Assume that families
$\\{R_{\beta}\subseteq R\colon\beta<\alpha\\}$ are defined. Fix a maximal
chain $\mathcal{L}\subseteq\bigcup\\{R_{\beta}\colon\beta<\alpha\\}$. Let
$L\subseteq\\{V\subseteq\bigcap\mathcal{L}\colon V\in R\\}$ be the set of all
maximal subsets with respect to inclusion and let $R_{\alpha}$ be the union of
all just defined families $L$. Check that
$E|_{\bigcup\\{R_{\alpha}\colon\alpha<\omega_{1}\\}}$ is a $\lambda$-thin
labeling for $Z$. ∎

Thus we have the following facts about dimensional types of crowded $P$-spaces
of cardinality and weight $\omega_{1}$.

######  Theorem 29.

Assume that a $P$-space $X$ of cardinality and weight $\omega_{1}$ is crowded.
If a $P$-space $Y$ has an $\omega$-thin labeling and a $P$-space $Z$ has an
$\omega_{1}$-thin labeling, then $Y=_{h}Z\subseteq_{h}X.$

###### Proof.

By Lemma 23 and Corollary 26, any crowded $P$-space $X$ of cardinality and
weight $\omega_{1}$ contains copies of a space with an $\omega$-thin labeling
and a space with an $\omega_{1}$-thin labeling. Theorem 27 implies that
$Y=_{h}Z\subseteq_{h}X.$ ∎

######  Corollary 30.

If a $P$-space $X$ has an $\omega$-thin labeling (or an $\omega_{1}$-thin
labeling), then $X$ has the smallest dimensional type in the class of all
crowded $P$-spaces of weight $\omega_{1}$.

###### Proof.

Any crowded $P$-space $X$ of weight $\omega_{1}$ contains a dense subset
$Z\subseteq X$ of cardinality $\omega_{1}$. Since $Z$ is a $T_{1}$-space, it
is crowded and hence $Z$ contains subspaces which have an $\omega$-thin
labeling and an $\omega_{1}$-thin labeling. ∎

##  9\. Remarks on rigid $P$-spaces

K. Kunen showed that there exists a rigid Lindelöf $P$-space of cardinality
and weight $\omega_{1}$, see [11, 2.1. Theorem]. Let us add a few remarks
about rigid Lindelöf $P$-spaces.

######  Proposition 31.

There exist at least $\mathfrak{c}$ many (rigid) $P$-spaces of cardinality and
weight $\omega_{1}$ such that any two of them are not homeomorphic.

###### Proof.

Let $X$ be a rigid Lindelöf $P$-space. Choose an infinite family
$\\{U_{n}\colon n<\omega\\}$ of pairwise disjoint clopen subsets of $X$.
Assign each $A\subseteq\omega$ a subspace $X_{A}=\bigcup\\{U_{n}\colon n\in
A\\}\subseteq X$, which is a clopen subset. If $A\neq B$, then $X_{A}$ and
$X_{B}$ cannot be homeomorphic. Indeed, if $A\setminus B\neq\emptyset$ and
$h\colon X_{A}\to X_{B}$ is a homeomorphism, then $H\colon X\to X$, given by
the formula

$H(x)=\begin{cases}h(x),&\mbox{if }x\in X_{A}\setminus X_{B};\\\
h^{-1}(x),&\mbox{if }x\in h[X_{A}\setminus X_{B}];\\\ x,&\mbox{otherwise},\\\
\end{cases}$

is a non-trivial homeomorphism. So, $X$ is not a rigid space.

Thus, if $X$ is a rigid $P$-space constructed by Kunen [11], then spaces
$\\{X_{A}\colon A\subseteq\omega\\}$ are of cardinality and weight
$\mathfrak{c}$, whenever $A\neq\emptyset$, and also are rigid, Lindelöf and
not homeomorphic. ∎

######  Corollary 32.

If a rigid $P$-space $X$ is of cardinality and weight $\omega_{1}$, then a
closed subset of $X$, which has an $\omega$-thin labeling or an
$\omega_{1}$-thin labeling, is a nowhere dense set.

###### Proof.

Suppose a closed subset $Y\subseteq X$ is not nowhere dense. Choose two
disjoint subsets $U,V\subseteq Y$, which are clopen in $X$. If $Y$ has an
$\omega$-thin labeling ($\omega_{1}$-thin labeling), then $U$ is homeomorphic
to $V$, since Theorem 27 and Corollary 28, which contradicts the rigidity of
$X$. ∎

######  Theorem 33.

If a $P$-space $X$ of cardinality and weight $\omega_{1}$ is rigid and a
$P$-space $Y$ has an $\omega$-thin labeling, then the relation $X\subset_{h}Y$
is not fulfilled.

###### Proof.

It suffices to show that, if there is an embedding of $X$ into a $P$-space
$Y$, which has an $\omega$-thin labeling, then $X$ contains a clopen subset,
which has an $\omega$-thin labeling. Indeed, if $f\colon X\to Y$ is an
embedding, then the image $f[X]$ has a $P$-matrix, which is defined as
follows. Assume that $X=\\{x_{\alpha}\colon\alpha<\omega_{1}\\}$ and let
$\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$ be a $P$-matrix for $Y$,
and let $E\colon\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}\to Y$
be an $\omega$-thin labeling. Choose a maximal and pairwise disjoint family
$\mathcal{R}_{0}\subseteq\bigcup\\{\mathcal{P}_{\alpha}\colon\alpha<\omega_{1}\\}$
such that $f[X]\subseteq\bigcup\mathcal{R}_{0}$ and if $V\in\mathcal{R}_{0}$,
then $E(V)\in f[X]$ and also there exists $V\in\mathcal{R}_{0}$ such that
$E(V)=f(x_{0})$. Suppose that families
$\\{\mathcal{R}_{\beta}\colon\beta<\alpha\\}$ are defined. If
$V\in\bigcup\\{\mathcal{R}_{\beta}\colon\beta<\alpha\\}$, then let

$L_{V}=\bigcap\\{W\in\bigcup\\{\mathcal{R}_{\beta}\colon\beta<\alpha\\}\colon
E(W)=E(V)\\}$

and then choose a family
$R_{V}\subseteq\bigcup\\{\mathcal{P}_{\beta}\colon\beta<\omega_{1}\\}$ such
that

  * •

$R_{V}$ consists of pairwise disjoint sets;

  * •

$R_{V}$ is of the maximal possible cardinality, i.e. $\omega_{1}$ or $\omega$;

  * •

$f[X]\cap L_{V}\subseteq\bigcup R_{V}\subseteq L_{V}$;

  * •

If $f(x_{\alpha})\in V$, then there exists $W\in\mathcal{R}_{V}$ such that
$E(W)=f(x_{\alpha})$;

  * •

If $W\in R_{V}$, then $E(W)\in f[X]$.

Let $\mathcal{R}_{\alpha}$ be the union of above defined families $R_{V}$,
here we assume that if $L_{V}=L_{W}$, then $R_{V}=R_{W}$. For any
$\alpha<\omega_{1}$, put

$\mathcal{Q}_{\alpha}=\\{V\cap f[X]\colon V\in\mathcal{R}_{\alpha}\\}.$

Thus $\\{\mathcal{Q}_{\alpha}\colon\alpha<\omega_{1}\\}$ constitute a
$P$-matrix for $f[X]\subseteq Y$ such that if $E^{*}(V\cap f[X])=E(V)$, then

$E^{*}\colon\bigcup\\{\mathcal{Q}_{\alpha}\colon\alpha<\omega_{1}\\}\colon\to
f[X]$

is a labeling. If all families $R_{V}$ are of cardinality $\omega_{1}$, then
$f[X]$ has an $\omega_{1}$-thin labeling; a contradiction. If there exists
$R_{V}$ of cardinality $\omega$, then $f[X]$ contains the clopen subset (in
the topology inherited on $f[X]$), which has an $\omega$-thin labeling; again
a contradiction. ∎

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