On spaces with a $\pi$-base with elements with compact closure

Abstract

In this paper we show that if X is a $T_1$-space with a $\pi$-base whose elements have compact closure, then $d(X)\le c(X)\cdot 2^{\psi (X)}$ and therefore, for such spaces we have $d(X)^{\psi (X)} = c(X)^{\psi (X)}$. This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition, we show that for such spaces X Šapirovskiĭ’s inequality $d(X)\le \pi \chi (X)^{c(X)}$, which is known to be true for regular Hausdorff spaces, is also valid. In the case when the space X is in addition sequential or radial, we show that $|X|\le 2^{c(X)}$. This result extends two theorems of Arhangel$'$skiĭ to the class of Hausdorff spaces with a $\pi$-base whose elements have compact closures. We also show that spaces with a $\pi$-base with elements with compact closures are $\alpha$-favorable in the Banach–Mazur game, which implies such spaces are Baire. It was shown in Bella et al. (Quaest Math 46(4):745–760, 2023) that if a Hausdorff space X has a $\pi$-base consisting of elements with compact closure, then $|X|\le 2^{wL(X)t(X)\psi _c(X)}$. We give a variation of this result by showing $|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$ for such a space X. Note that since $wL(X)\textrm{ot}(X)\le c(X)$, this result is at least as good as that given by Sun (Proc Am Math Soc 104:313–316, 1988). We also give a possible improvement of the bound in Bella et al. (2023) by showing that $|X|\le 2^{wL(X)wt(X)\psi _c(X)}$ for a Hausdorff space X with a $\pi$-base consisting of elements with compact closure. This uses the weak tightness wt(X) defined in Carlson (Topol Appl 249:103–111, 2018), which has the property $\textrm{ot}(X)\le wt(X)\le t(X)$. We also show that if X is a Hausdorff homogeneous space with a $\pi$-base consisting of elements with compact closure (such spaces are locally compact), then $|X|\le wL(X)^{wt(X)\pi \chi (X)}$. This generalizes the result in Bella and Carlson (Monatsh Math 192(1):39–48, 2020) that if X is a homogeneous compactum, then $|X|\le 2^{wt(X)\pi \chi (X)}$.

1 Introduction

A systematic study of cardinal functions began in the 1960s but fundamental techniques and results were obtained long before then. We refer the reader to the books [23, 29], and the book chapters [28, 30], all from the 1980s, where more information could be found about the history of the subject together with most, now classical, results about cardinal function inequalities for topological spaces.

The quest for finding new, or improving previously obtained cardinal inequalities for different classes of topological spaces continues. The most recent advances (during the last two years) in the theory of cardinal inequalities for topological spaces have been made by Aurichi, Bella, and Spadaro [7], Gotchev and Tkachuk [25], Alas, Gutiérrez-Domínguez, and Wilson [2], Bella, Carlson, and Spadaro [10], Carlson [16] and [17], and Gotchev [24].

Our main goal in this paper is to find new cardinal function inequalities for topological spaces with a $\pi$-base whose elements have compact closure. We will refer to such spaces as spaces with a compact $\pi$-base. This class, among other classes of spaces, contains the class of all (locally) compact Hausdorff spaces and spaces with a dense set of isolated points. Cardinal inequalities about such spaces appeared first in [11] and [12].

In Sect. 2 we give some necessary definitions and make some preliminary observations.

In Sect. 3 we show that if X is a $T_1$-space with a compact $\pi$-base, then $d(X)\le c(X)\cdot 2^{\psi (X)}$ and therefore, $d(X)^{\psi (X)} = c(X)^{\psi (X)}$ for such spaces X. This allowed us in Corollary 3.11 to replace d(X) with c(X) in the upper bounds of the cardinality of a Hausdorff space X given by Pospišil, Willard–Dissanayake, Bella–Cammaroto, Carlson, Gotchev–Tkachenko–Tkachuk, and Gotchev’s inequalities (see Theorem 3.10 for the statements of all these inequalities).

Carlson in [14] showed that Šapirovskiĭ’s inequality $d(X)\le \pi \chi (X)^{c(X)}$, which is known to be true for every regular Hausdorff space X, is also valid for all quasiregular spaces. Since every Hausdorff space X with a compact $\pi$-base is quasiregular (Proposition 3.2), Šapirovskiĭ’s inequality is also true for such spaces X. This allowed us to replace d(X) with $\pi \chi (X)^{c(X)}$ in the above-mentioned inequalities and to obtain some corollaries which are valid for spaces with a compact $\pi$-base (see Corollary 3.13).

We also show in Theorem 3.16 that if X is a Hausdorff space which is sequential or radial and has a compact $\pi$-base, then $|X|\le 2^{c(X)}$. The last inequality could fail consistently if X is a pseudoradial compact Hausdorff space (see the remark after Theorem 3.16). This result extends to the class of Hausdorff spaces with a compact $\pi$-base two results of Arhangel$'$skiĭ (see Theorems 3.18 and 3.19).

In Theorem 3.22 we show that if X is a Hausdorff space with a compact $\pi$-base, then $|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$, where $\textrm{ot}(X)$ is the o-tightness defined in [38]. Since $wL(X)\textrm{ot}(X)\le c(X)$, this upper bound is at least as good as the one given by Sun’s inequality $|X|\le \pi \chi (X)^{c(X)\psi _c(X)}$, which is valid for every Hausdorff space X.

The last result in Sect. 3 is contained in Theorem 3.26 and states that if X is a Hausdorff space with a compact $\pi$-base, then $|X|\le 2^{wL(X)wt(X)\psi _c(X)}$, where wt(X) is the weak tightness defined in [15]. This upper bound is at least as good as the upper bound obtained in [12] where we proved that if X is a Hausdorff space with a compact $\pi$-base, then $|X|\le 2^{wL(X)t(X)\psi _c(X)}$.

Section 4 contains results which are about, or related to, homogeneous spaces. We show in Theorem 4.14 that if X is a locally compact and homogeneous Hausdorff space, then $|X|\le wL(X)^{wt(X)\pi \chi (X)}$, generalizing the result in [13] that if X is a homogeneous compactum, then $|X|\le 2^{wt(X)\pi \chi (X)}$. Since every homogeneous Hausdorff space with a compact $\pi$-base is locally compact (Theorem 4.16), the former inequality is true for such spaces (Corollary 4.18). It is also shown in Theorem 4.29 that if X is a power homogeneous Hausdorff space with a compact $\pi$-base, then $\pi \chi (X)\le t(X)$.

2 Notation and terminology

In this paper all topological spaces are assumed to be at least $T_1$, $\omega$ is (the cardinality of) the set of all non-negative integers, $\alpha$ and $\beta$ are ordinals, while $\kappa$ and $\lambda$ are infinite cardinals. The cardinality of the set X is denoted by |X|, the closure of a subset A of a space X is denoted by $\overline{A}$, and the $\kappa$-closure of A is $cl_\kappa A=\bigcup _{B\in [A]^{\le \kappa }}\overline{B}$, where $[A]^{\le \kappa }$ is the set of all subsets of A with cardinality less than or equal to $\kappa$. $\Delta _X = \{(x,x)\in X^2: x\in X\}$ is the diagonal of X. If ${\mathcal U}$ is a family of subsets of X, $x\in X$, and $G\subset X$, then $\textrm{st}(G,{\mathcal U}) = \bigcup \{U\in {\mathcal U}: U\cap G\ne \emptyset \}$. When $G=\{x\}$ we write $\textrm{st}(x,{\mathcal U})$ instead of $\textrm{st}(\{x\},{\mathcal U})$. If $n\in \omega$, then $\textrm{st}^n(G,{\mathcal U})=\textrm{st}(\textrm{st}^{n-1}(G,{\mathcal U}),{\mathcal U})$ and $\textrm{st}^0(G,{\mathcal U})=G$. A space X has a $G_\delta$-diagonal if there is a family $\{U_k:k<\omega \}$ of open sets in $X\times X$ such that $\Delta _X=\bigcap _{k<\omega } U_k$. Given a space X and $n\in \mathbb {N}$, we say that X has a rank n-diagonal if there exists a family $\{{\mathcal U}_k: k <\omega \}$ of open covers of X such that for any distinct points $x,y\in X$ we have $y\notin {\textrm{st}^n(x,{\mathcal U}_k)}$ for some $k < \omega$. (Recall that a space X has a $G_\delta$-diagonal if and only if X has a rank 1-diagonal [19, Lemma 5.4]).

As usual, d(X), w(X), $\chi (X)$, and $\psi (X)$ denote respectively the density, the wight, the character, and the pseudocharacter of X. The closed pseudocharacter $\psi _c(X)$ (defined only for Hausdorff spaces X) is the smallest infinite cardinal $\kappa$ such that for each $x\in X$, there is a collection $\{V(\alpha ,x):\alpha <\kappa \}$ of open neighborhoods of x such that $\bigcap _{\alpha <\kappa }\overline{V}(\alpha ,x) = \{x\}$. A $\pi$-base for X is a collection ${\mathcal V}$ of non-empty open sets in X such that if U is any non-empty open set in X, then there exists $V\in {\mathcal V}$ such that $V\subset U$. A family ${\mathcal V}$ of non-empty open sets in X is a local $\pi$-base at a point $x\in X$ if for every open neighborhood U of x there is $V\in {\mathcal V}$ such that $V\subset U$. The minimal infinite cardinal $\kappa$ such that for each $x\in X$ there is a collection $\{V(\alpha ,x):\alpha <\kappa \}$ of non-empty open subsets of X which is a local $\pi$-base for x is called the $\pi$-character of X and is denoted by $\pi \chi (X)$. The tightness at $x\in X$ is $t(x,X)=\min \{\kappa :$ for every $Y\subseteq X$ with $x\in \overline{Y}$, there is $A\subset Y$ with $|A|\le \kappa$ such that $x\in \overline{A}\}$ and the tightness of X is $t(X)=\sup \{t(x,X):x\in X\}+\omega$. The weak tightness wt(X) of X is defined as the least infinite cardinal $\kappa$ for which there is a cover $\mathcal {C}$ of X such that $|\mathcal {C}|\le 2^\kappa$ and for every $C\in \mathcal {C}$, $t(C)\le \kappa$ and $X=cl_{2^\kappa }C$. The o-tightness of a space X does not exceed $\kappa$, or $\textrm{ot}(X)\le \kappa$, if for every family ${\mathcal U}$ of open subsets of X and for every point $x\in X$ with $x\in \overline{\bigcup {\mathcal U}}$ there exists a subfamily ${\mathcal V}\subset {\mathcal U}$ such that $|{\mathcal V}|\le \kappa$ and $x\in \overline{\bigcup {\mathcal V}}$. The dense o-tightness of a space X does not exceed $\kappa$, or $\textrm{dot}(X)\le \kappa$, if for every family ${\mathcal U}$ of open subsets of X whose union is dense in X and for every point $x\in X$ there exists a subfamily ${\mathcal V}\subset {\mathcal U}$ such that $|{\mathcal V}|\le \kappa$ and $x\in \overline{\bigcup {\mathcal V}}$ ( [26]).

The Lindelöf number of X is $L(X)=\min \{\kappa :$ every open cover of X has a subcover of cardinality $\le \kappa \}+\omega$. The weak Lindelöf number of X, denoted by wL(X), is the smallest infinite cardinal $\kappa$ such that every open cover of X has a subcollection of cardinality $\le \kappa$ whose union is dense in X. A pairwise disjoint collection of non-empty open sets in X is called a cellular family. The cellularity of X is the cardinal number $c(X)=\sup \{|{\mathcal U}|:{\mathcal U}$ is a cellular family in $X\}+\omega$. The pointwise compactness type, $\textrm{pct}(X)$, of a space X is the least infinite cardinal $\kappa$ such that X can be covered by compact sets K such that $\chi (K, X) \le \kappa$. We say that a space has pointwise countable type if $pct(X)\le \omega$.

In relation to some of the above definitions we would like to note the following:

• (a) If X is compact, then $\textrm{pct}(X) = \omega$ and $\chi (X)=\psi (X)$. Furthermore, it can be shown that $\chi (X) = \psi (X)\textrm{pct}(X)$ for any Hausdorff space X ( [23, 3.1.F.(b)]).

• (b) $\textrm{dot}(X)\le \pi \chi (X)$, $\textrm{dot}(X)\le \textrm{ot}(X)\le t(X)$, and $\textrm{ot}(X)\le c(X)$ for every space X.

• (c) $cl_\kappa (cl_\kappa A)=cl_\kappa A\subseteq \overline{A}$ and if $cl_\kappa A=X$, then A is dense in X.

• (d) $wt(X)\le t(X)$ for every space X, and the relationship between wt(X) and $\textrm{ot}(X)$ is given by Proposition 2.1.

Proposition 2.1

$\textrm{ot}(X)\le wt(X)$ for any space X.

Proof

Let $\kappa =wt(X)$ and let $\mathcal {C}$ be a cover witnessing that $wt(X)=\kappa$. Let $x\in \overline{\bigcup \mathcal {U}}$, where $\mathcal {U}$ is a family of open sets. There exists $C\in \mathcal {C}$ such that $x\in C$. Since C is dense in X, we have $\overline{\bigcup \mathcal {U}}=\overline{\bigcup \mathcal {U}\cap C}$. Therefore $x\in \overline{\bigcup \mathcal {U}\cap C}\cap C=cl_C(\bigcup \mathcal {U}\cap C)$. As $t(C)\le \kappa$, there exists $A\subseteq \bigcup \mathcal {U}\cap C$ such that $|A|\le \kappa$ and $x\in cl_C(A)\subseteq \overline{A}$. Thus, there exists $\mathcal {V}\in [\mathcal {U}]^\kappa$ such that $x\in \overline{\bigcup \mathcal {V}}$. Therefore $\textrm{ot}(X)\le \kappa$. $\square$

We also recall that a space X is called sequential if a set $A\subset X$ is closed if and only if together with any sequence the set A contains all its limits. A transfinite sequence $\{x_\alpha :\alpha <\kappa \}$ is said to converge to a point x in a space X if for every neighborhood U of x there is $\beta <\kappa$ such that $\{x_\alpha :\beta<\alpha <\kappa \}$ is a subset of U. A space X is called pseudoradial if for every non-closed set $A \subset X$ there is a point $x \in \overline{A}{{\setminus }} A$ and a transfinite sequence $\{x_\alpha :\alpha <\kappa \}\subset A$ which converges to x. If for every non-closed set $A \subset X$ and every point $x \in \overline{A}{\setminus } A$ there is a transfinite sequence in A which converges to x, then the space X is called radial.

Finally a space X is homogeneous if for every $x,y\in X$ there exists a homeomorphism $h:X\rightarrow X$ such that $h(x)=y$. X is power homogeneous if there exists a cardinal $\kappa$ such that $X^\kappa$ is homogeneous.

For notions not defined here please see [23, 28, 29].

3 Cardinal inequality results about spaces with a compact $\pi$-base

As we have already mentioned, the class of spaces with a compact $\pi$-base, which we study here, contains the class of all (locally) compact (Hausdorff) spaces and all spaces with a dense set of isolated points. In Proposition 3.2 and Theorem 3.4 below we show that every Hausdorff space of this class is quasiregular and is a Baire space. We recall that a space X is called quasiregular if every non-empty open set in X contains a non-empty regular-closed set. Proposition 3.2 below is a straightforward corollary from that definition and the following lemma.

Lemma 3.1

[12, Lemma 4.10] If X is a space with a compact $\pi$-base $\mathcal {B}$ and $F\subsetneq X$ is closed, then there exists $B\in \mathcal {B}$ such that $\overline{B}\cap F=\varnothing$.

Proposition 3.2

If X is a Hausdorff space with a compact $\pi$-base, then X is quasiregular.

Recall that the Banach-Mazur game BM(X) on the space X is played by two players $\alpha$ and $\beta$ in $\omega$-many innings. At the beginning of the game, $\beta$ chooses a non-empty open set $U_0$ and $\alpha$ responds by choosing a non-empty open set $V_0\subset U_0$. At the n-th inning ($n>0$), $\beta$ chooses a non-empty $U_n\subset V_{n-1}$ and $\beta$ responds by choosing a non-empty open set $V_n\subset U_n$, and so on. The rule is that $\alpha$ wins if and only if $\bigcap \{V_n:n\in \omega \}\ne \emptyset$. See the book by Oxtoby [31] for further reading on the Banach-Mazur game.

It is a theorem of Banach, Mazur and Oxtoby that the space X has the Baire property if and only if $\beta$ does not have a winning strategy in BM(X).

A space X is said $\alpha$-favorable if player $\alpha$ has a winning strategy in BM(X). $\alpha$-favorable spaces are Baire. A Berstein subset of the reals witnesses that $\alpha$-favorable is strictly stronger than Baire.

Theorem 3.3

A Hausdorff space X with a compact $\pi$-base is $\alpha$-favorable.

Proof

Let us describe a winning strategy for $\alpha$ in BM(X). Let $U_0$ be the first move of $\beta$. Since a Hausdorff space with a compact $\pi$-base is quasiregular by Proposition 3.2, player $\alpha$ responds by taking a non-empty open set $V_0$ such that $\overline{V_0}\subset U_0$ and $\overline{V_0}$ is compact. In general at the n-th inning ($n>0$), $\alpha$ responds by taking a non-empty open set $V_n$ with compact closure such that $\overline{V_n}\subset U_n$. The compactness of $\overline{V_0}$ ensures that $\alpha$ wins. $\square$

The property having a compact $\pi$-base is strictly stronger than $\alpha$-favorable: the space X of irrational numbers is $\alpha$-favorable being metric and complete, but it does not have a compact $\pi$-base because it is nowhere locally compact.

Corollary 3.4

Let X be a Hausdorff space with a compact $\pi$-base. Then X is a Baire space.

In [8] the authors showed that if X is a Hausdorff Baire space with a rank 2-diagonal, then $|X|\le wL(X)^\omega$ and if X is a Hausdorff Baire space with a $G_\delta$-diagonal, then $d(X)\le wL(X)^\omega$. Therefore, as corollaries of these results and Corollary 3.4 we have the following:

Corollary 3.5

Let X be a Hausdorff space with a compact $\pi$-base.

• (a) If X has a $G_\delta$-diagonal, then $d(X)\le wL(X)^\omega$;

• (b) If X has a rank 2-diagonal, then $|X|\le wL(X)^\omega$.

The inequality $c(X)\le d(X)$ is true for every space X, while there is no upper bound for d(X) as a function of c(X) even for compact Hausdorff spaces. To see that recall that for every Hausdorff space X we have $|X|\le 2^{2^{d(X)}}$ and that any product of separable spaces has countable celullarity. Therefore, the compact Hausdorff space $X=D^{\kappa }$, where $D=\{0,1\}$ and $\kappa$ is an infinite cardinal, must have density d(X) such that $2^{d(X)}\ge \kappa$ and $c(X)=\omega$. But, as the following straightforward proposition shows, there are classes of spaces where such upper bound for d(X) as a function of c(X) exists.

Proposition 3.6

If X is a space with a dense set of isolated points, then $d(X)=c(X)$.

The class of spaces with a dense set of isolated points is a subclass of the class of all spaces with a compact $\pi$-base, which we study here. Since this larger class contains the class of all compact Hausdorff spaces, we conclude that Proposition 3.6 is not true for this larger class of spaces. The closest equality to the one in Proposition 3.6 we can prove for this larger class of spaces is contained in Corollary 3.9. This corollary follows directly from Theorem 3.8 below which gives an upper bound for the density of a $T_1$-topological space X with a compact $\pi$-base as a function of the cellularity and the pseudocharacter of X. In its proof we use the following theorem that was proved by Gryzlov in [27].

Theorem 3.7

[27] If X is a compact $T_1$-space, then $|X|\le 2^{\psi (X)}$.

Theorem 3.8

If X is a $T_1$-space with a compact $\pi$-base, then

$$\begin{aligned} d(X)\le c(X)\cdot 2^{\psi (X)}. \end{aligned}$$

Proof

Let $\mathcal {B}$ be a $\pi$-base in X of non-empty open sets with compact closures. For each $B\in \mathcal {B}$, if $\psi (\overline{B})$ is the pseudocharacter of the space $\overline{B}$, then we have $\psi (\overline{B})\le \psi (X)$ and since $\overline{B}$ is compact and $T_1$, we have $|\overline{B}|\le 2^{\psi (\overline{B})}$. Using Zorn’s Lemma we can find a maximal cellular family ${\mathcal V}\subset \mathcal {B}$. Clearly, $|{\mathcal V}|\le c(X)$ and $\overline{\bigcup {{\mathcal V}}}=X$. Therefore $d(X)\le |\bigcup {{\mathcal V}}|\le c(X)\cdot 2^{\psi (X)}$. $\square$

Corollary 3.9

If X is a $T_1$-space with a compact $\pi$-base, then $d(X)^{\psi (X)} = c(X)^{\psi (X)}$.

We note that the inequality in Corollary 3.9 is particularly true for every (locally) compact Hausdorff space X and every Hausdorff space X with a dense set of isolated points.

Before we continue, in the following theorem we recall six inequalities about the cardinality of a Hausdorff space X.

Theorem 3.10

If X is a Hausdorff space, then

• (a) $|X|\le d(X)^{\chi (X)}$ (Pospišil, 1937 [32]);

• (b) $|X|\le d(X)^{\pi \chi (X)\psi _c(X)}$ (Willard–Dissanayake, 1984 [39]);

• (c) $|X|\le d(X)^{t(X)\psi _c(X)}$ (Bella–Cammaroto, 1988 [9]);

• (d) $|X|\le d(X)^{wt(X)\psi _c(X)}$ (Carlson, 2018 [15]);

• (e) $|X|\le (\pi \chi (X) d(X))^{\textrm{ot}(X)\psi _c(X)}$ (Gotchev–Tkachenko–Tkachuk, 2016 [26]).

• (f) $|X|\le (\pi \chi (X) d(X))^{\textrm{dot}(X)\psi _c(X)}$ (Gotchev, 2023 [24]).

Notice that in the above theorem (c) follows from (d); (a) follows from (b), (c), (d), (e) and (f); and (b) and (e) follow from (f). We also note that the upper bound in Theorem 3.10(e) could be stricly less than the upper bound in Theorem 3.10(f), as it was shown in [24].

Now using Theorem 3.10 and Corollary 3.9 we immediately obtain the following:

Corollary 3.11

If X is a Hausdorff space with a compact $\pi$-base, then

• (a) $|X|\le c(X)^{\chi (X)}$;

• (b) $|X|\le c(X)^{\pi \chi (X)\psi _c(X)}$;

• (c) $|X|\le c(X)^{t(X)\psi _c(X)}$;

• (d) $|X|\le c(X)^{wt(X)\psi _c(X)}$;

• (e) $|X|\le (\pi \chi (X)c(X))^{\textrm{ot}(X)\psi _c(X)}$;

• (f) $|X|\le (\pi \chi (X)c(X))^{\textrm{dot}(X)\psi _c(X)}$.

Notice that the inequalities in Corollary 3.11 are particularly true for every (locally) compact Hausdorff space X and every Hausdorff space X with a dense set of isolated points. They give the same upper bounds for the cardinality of the Hausdorff space X as those in Theorem 3.10 but as a function of c(X), instead of d(X), and other cardinal functions. Therefore, just as in Theorem 3.10, in Corollary 3.11, (c) follows from (d); (a) follows from (b), (c), (d), (e) and (f); and (b) and (e) follow from (f).

Šapirovskiĭ showed in [36] that if X is a regular Hausdorff space, then $d(X)\le \pi \chi (X)^{c(X)}$. Using different techniques, Charlesworth in [20] established the same result and provided an example due to van Douwen that this inequality is not true for every Hausdorff space X. On the other hand, Carlson in [14] showed that Šapirovskiĭ’s inequality is valid for all quasiregular spaces. Therefore, the following observation is a direct corollary from Carlson’s result and Proposition 3.2.

Proposition 3.12

If X is a Hausdorff space with a compact $\pi$-base, then

$$\begin{aligned} d(X)\le \pi \chi (X)^{c(X)}. \end{aligned}$$

The following corollary is immediate from Proposition 3.12 and Theorem 3.10.

Corollary 3.13

If X is a Hausdorff space with a compact $\pi$-base, then

• (a) $|X|\le \pi \chi (X)^{c(X)\chi (X)}$;

• (b) $|X|\le 2^{c(X)\pi \chi (X)\psi _c(X)}$;

• (c) $|X|\le \pi \chi (X)^{c(X)t(X)\psi _c(X)}$;

• (d) $|X|\le \pi \chi (X)^{c(X)wt(X)\psi _c(X)}$;

• (e) $|X|\le \pi \chi (X)^{c(X)\psi _c(X)}$.

We note that in the above corollary (c) follows from (d); (a) follows from (b), (c), (d) and (e); and (a), (b), (c) and (d) follow from (e), and that (e) is Sun’s inequality which is true for every Hausdorff space X [37].

We also note by passing that it is shown in [25] that Theorem 3.10(e) (and therefore Theorem 3.10(f)) gives better approximation of the cardinality of a Hausdorff space X than Sun’s inequality (Corollary 3.13(e)). The following chain of inequalities confirms that claim.

$$\begin{aligned} (\pi \chi (X)\cdot d(X))^{\textrm{ot}(X)\psi _c(X)}&\le (\pi \chi (X)\cdot |X|)^{c(X)\psi _c(X)}\\&\le (\pi \chi (X)\cdot \pi \chi (X)^{c(X)\psi _c(X)})^{c(X)\psi _c(X)}=\pi \chi (X)^{c(X)\psi _c(X)}. \end{aligned}$$

To see an example where both upper bounds differ, consider a discrete space X with cardinality of the continuum $\mathfrak {c}$. Then Theorem 3.10(e) gives $|X|\le \mathfrak {c}^\omega =\mathfrak {c}$ while Corollary 3.13(e) gives $|X|\le \omega ^\mathfrak {c}=2^\mathfrak {c}$. The paper [25] also contains examples showing that Sun’s inequality (Corollary 3.13(e)) and Willard–Dissanayake inequality (Theorem 3.10(b)) do not follow from each other.

In Theorem 3.16 we establish an upper bound for the cardinality of a Hausdorff space which is either sequential or radial and has a compact $\pi$-base. For its proof we need the inequality in Corollary 3.15 which immediately follows from Arhangel$'$skiĭ’s theorem that if X is a compact Hausdorff pseudoradial space, then $w(X)\le 2^{c(X)}$ ( [5, Theorem 11]). Here we give a different proof, in which we use the following theorem, due to Šapirovskiĭ.

Theorem 3.14

[22] Let X be a compact Hausdorff space and $\kappa$ be an infinite cardinal. X cannot be mapped continuously onto $I^\kappa$, where $I=[0,1]$, if and only if every closed subspace $F \subseteq X$ has a point p such that $\pi \chi (p,F) < \kappa$.

Corollary 3.15

If X is a compact Hausdorff pseudoradial space, then $d(X) \le 2^{c(X)}$.

Proof

A compact Hausdorff pseudoradial space is sequentially compact. Since $I^\mathfrak {c}$ is not sequentially compact, X cannot be mapped continuously onto $I^\mathfrak {c}$. For each non-empty open set U, by Theorem 3.14, there is $p_U\in \overline{U}$ such that $\pi \chi (p_U,\overline{U})<\mathfrak {c}$. If ${\mathcal B}(p_U)$ is a local $\pi$-base for $p_U$ in $\overline{U}$, then $\{B\cap U:B\in {\mathcal B}(p_U)\}$ is a local $\pi$-base for $p_U$ in X. Thus, $\pi \chi (p_U,X)<\mathfrak {c}$. The set $Y=\{p_U:U$ open in $X \}$ is dense in X and therefore $\pi \chi (p_U,Y)<\mathfrak {c}$ for every $p_U\in Y$. Then using Šapirovskiĭ’s inequality $d(X)\le \pi \chi (X)^{c(X)}$ we get $d(X) \le d(Y) \le \pi \chi (Y)^{c(Y)}\le \mathfrak {c}^{c(Y)} = 2^{c(Y)\omega } = 2^{c(X)}$. $\square$

Theorem 3.16

Let X be a Hausdorff space with a compact $\pi$-base. If X is either sequential or radial, then $|X| \le 2^{c(X)}$.

Proof

Let ${\mathcal V}$ be a maximal pairwise disjoint family of open sets with compact closure. If $V \in {\mathcal V}$, then by Corollary 3.15 we have $d(\overline{V}) \le 2^{c(\overline{V})} \le 2^{c(X)}$. Therefore, for each $V \in {\mathcal V}$ we may fix a dense subset $D_V\subset V$ such that $|D_V|\le 2^{c(X)}$. The set $D = \bigcup \{D_V:V\in {\mathcal V}\}$ is dense in X and $|D| \le 2^{c(X)}$.

If X is a sequential space, then $|X| \le |D|^\omega \le 2^{c(X)\omega } = 2^{c(X)}$.

Now, let X be a radial space and $c(X) = \kappa$. If p is an isolated point in X, then $p\in D$. Since $|D| \le 2^{c(X)}$, it suffices to show that for every $p\in X{\setminus } D$ there is a transfinite sequence $S \in [D]^{\le \kappa }$ which converges to p. To see this, notice that since X is Hausdorff, for every $x\in X{{\setminus }}\{p\}$, there exists an open set U such that $x\in U\subset \overline{U}$ and $p\notin \overline{U}$. Let ${\mathcal U}'=\{U: p\notin \overline{U}, U$ open in $X\}$. Since $c(X)\le \kappa$, there is a family ${\mathcal U}\in [{\mathcal U}']^{\le \kappa }$ such that $\bigcup {\mathcal U}$ is dense in $\bigcup {\mathcal U}'$. We may write ${\mathcal U}= \{U_\alpha : \alpha < \kappa \}$. Since X is Hausdorff and p is not isolated, $\bigcup {\mathcal U}$ is dense in X. Let $Y=\bigcup \{U_\alpha \cap D:\alpha <\kappa \}$. Then Y is dense in X and therefore there is a transfinite sequence $S \subset Y$ converging to p. We can assume that S is of minimal cardinality among all transfinite sequences contained in Y which converge to p, so |S| is some regular cardinal $\lambda$. We claim that $\lambda \le \kappa$. If not, the regularity of $\lambda$ would imply the existence of some $\beta < \kappa$ such that $|S \cap U_\beta | = \lambda$. But then, $|S| = |S \cap U_\beta |$ and the convergence of S would imply that $S \cap U_\beta$ also converges to p, hence $p\in \overline{U_\beta }$ – contradiction. Therefore, $|X| \le |D|^\kappa$ $\le 2^{c(X)}$. $\square$

Remark 3.17

The conclusion in Theorem 3.16 may consistently fail for pseudoradial spaces. $X=2^{\omega _1}$ is a compact Hausdorff space with $c(X)=\omega$ and in [6] it is shown that there is a model of ZFC where X is pseudoradial and $|X| = 2^{\omega _1} > 2^\omega = 2^{c(X)}$.

Theorem 3.16 extends to the class of Hausdorff spaces with a compact $\pi$-base the following two theorems of Arhangel$'$skiĭ:

Theorem 3.18

[4] If X is a compact Hausdorff sequential space, then $|X| \le 2^{c(X)}$.

Theorem 3.19

[5] If X is a compact Hausdorff radial space, then $|X| \le 2^{c(X)}$.

Using the same argument as in the proof of Theorem 3.16 we can prove the following theorem.

Theorem 3.20

If X is a quasiregular Hausdorff radial space, then $|X| \le \pi \chi (X)^{c(X)}$.

Proof

Let X be a quasiregular Hausdorff radial space. Then it follows from the inequality $d(X)\le \pi \chi (X)^{c(X)}$, which is valid for every quasiregular space, that there exists a dense set D in X such that $|D|\le \pi \chi (X)^{c(X)}$. Let $c(X) = \kappa$. If p is an isolated point in X, then $p\in D$. Since $|D| \le \pi \chi (X)^{c(X)}$, it suffices to show that for every $p\in X{{\setminus }} D$ there is a transfinite sequence $S \in [D]^{\le \kappa }$ which converges to p. To see this, it suffices to repeat the argument exactly as in the proof of Theorem 3.16. $\square$

In Theorem 3.22 below we show that an inequality at least as good as the one in Corollary 3.13(e) is valid for the cardinality of a Hausdorff space with a compact $\pi$-base. For its proof we need the following observation.

Lemma 3.21

If $\textrm{ot}(X)\le \kappa$ and $\{U_\alpha :\alpha <\kappa ^+\}$ is a non-decreasing chain of open subsets of X, then $\bigcup \{\overline{U_\alpha }:\alpha<\kappa ^+\}=\overline{\bigcup \{U_\alpha :\alpha <\kappa ^+\}}$.

Proof

Let $x\in \overline{\bigcup \{U_\alpha :\alpha <\kappa ^+\}}$. As $\textrm{ot}(X)\le \kappa$, there exists $\mathcal {V}\subseteq \{U_\alpha :\alpha <\kappa ^+\}$ such that $|\mathcal {V}|\le \kappa$ and $x\in \overline{\bigcup \mathcal {V}}$. Since $\{U_\alpha :\alpha <\kappa ^+\}$ is a non-decreasing chain, there exists $\alpha <\kappa ^+$ such that $\bigcup \mathcal {V}\subseteq U_\alpha$, Thus, $x\in \overline{U_\alpha }\subseteq \bigcup \{\overline{U_\alpha }:\alpha <\kappa ^+\}$. $\square$

Theorem 3.22

If X is a Hausdorff space with a compact $\pi$-base, then

$$\begin{aligned} |X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}. \end{aligned}$$

Proof

Let $\kappa =wL(X)\textrm{ot}(X)\psi _c(X)$ and let $\mathcal {B}$ be a $\pi$-base of non-empty open sets with compact closures. Notice that for each $B\in \mathcal {B}$ if $\psi (\overline{B})$ is the pseudocharacter of the space $\overline{B}$, then we have $\psi (\overline{B})\le \psi _c(X)$ and since $\overline{B}$ is compact and Hausdorff, we have $|\overline{B}|\le 2^{\psi (\overline{B})}\le 2^\kappa$. Since $\psi _c(X)\le \kappa$, for each $x\in X$ we can fix a collection $\mathcal {V}_x$ of open neighborhoods of x such that $|\mathcal {V}_x|\le \kappa$ and $\bigcap \{\overline{V}:V\in \mathcal {V}_x\}=\{x\}$. Without loss of generality we may assume that each $\mathcal {V}_x$ is closed under finite intersections.

We will construct by transfinite recursion a non-decreasing chain of open sets $\{U_\alpha : \alpha < \kappa ^+\}$ such that

• (1) $|\overline{U_\alpha }| \le \pi \chi (X)^\kappa$ for every $\alpha < \kappa ^+$, and

• (2) if $X{\setminus }\overline{\bigcup \mathcal {M}} \ne \emptyset$ for some $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$, then there is $B_\mathcal {M}\in \mathcal {B}$ such that $B_\mathcal {M}\subset U_{\alpha +1}{\setminus }\overline{\bigcup \mathcal {M}}$.

Let $B_0\in \mathcal {B}$ be arbitrary. We set $U_0=B_0$. Then $|\overline{U_0}|\le (\pi \chi (X)\cdot |U_0|)^{\textrm{ot}(X)\psi _c(X)}$ by Theorem 3.10(e), and thus $|\overline{U_0}|\le (\pi \chi (X)\cdot 2^\kappa )^\kappa =\pi \chi (X)^\kappa$. If $\beta =\alpha + 1$, for some $\alpha$, then for every $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$ such that $X{\setminus }\overline{\bigcup \mathcal {M}} \ne \emptyset$, we choose $B_{\mathcal {M}} \in \mathcal {B}$ such that $B_{\mathcal {M}} \subseteq X{{\setminus }}\overline{\bigcup \mathcal {M}}$. We define $U_{\beta } = U_\alpha \cup \bigcup \{B_\mathcal {M}: \mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }, X{{\setminus }}\overline{\bigcup \mathcal {M}}\ne \emptyset \}$. Therefore, by Theorem 3.10(e), we have $|\overline{U_{\beta }}| \le (\pi \chi (X)\cdot |U_\beta |)^\kappa \le (\pi \chi (X)\cdot \pi \chi (X)^\kappa )^\kappa =\pi \chi (X)^\kappa$. If $\beta <\kappa ^+$ is a limit ordinal we let $U_\beta = \bigcup _{\alpha <\beta } U_\alpha$. Then clearly $|U_\beta |\le \pi \chi (X)^\kappa$, hence $|\overline{U_\beta }| \le \pi \chi (X)^\kappa$ by Theorem 3.10(e).

Let $F = \bigcup \{\overline{U_\alpha }: \alpha < \kappa ^+\}$. Then $|F| \le \pi \chi (X)^\kappa \cdot \kappa ^+=\pi \chi (X)^\kappa$. Since $\textrm{ot}(X)\le \kappa$, by Lemma 3.21 we have $F = \overline{\bigcup \{U_\alpha : \alpha < \kappa ^+\}}$ and therefore F is a regular-closed set.

We will show that $X = F$. Suppose that $X \ne F$. Since F is closed and $\mathcal {B}$ is a $\pi$-base, there is $B\in \mathcal {B}$ such that $B\subset X{{\setminus }} F$ and $\overline{B}\cap F=\emptyset$ (Lemma 3.1). Also, for every $x\in F$ and $y\in \overline{B}$ there is $V_x(y)\in \mathcal {V}_x$ such that $y\notin \overline{V_x(y)}$. Therefore, using the compactness of $\overline{B}$ and the fact that each $\mathcal {V}_x$ is closed under finite intersections, for every $x\in F$ we can find $V_x \in \mathcal {V}_x$ such that $\overline{V_x} \cap \overline{B}=\emptyset$; hence $\overline{B}\cap V_x=\emptyset$. Clearly $\{V_x: x \in F\}$ is an open cover of F. Since wL(X) is monotone with respect to regular-closed sets, there exists $\mathcal {M} \in \{V_x: x \in F\}^{\le \kappa }$ such that $F \subseteq \overline{\bigcup \mathcal {M}}$. Then there exists $\alpha < \kappa ^+$ such that $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$. As $\overline{B} \cap \bigcup \mathcal {M} = \emptyset$, it follows that $B\subset X{{\setminus }}\overline{\bigcup \mathcal {M}}$, hence $X{\setminus }\overline{\bigcup \mathcal {M}}\ne \emptyset$. Thus, there exists $B_\mathcal {M}\in \mathcal {B}$ such that $\emptyset \ne B_\mathcal {M} \subseteq U_{\alpha +1}{{\setminus }}\overline{\bigcup \mathcal {M}} \subseteq F{{\setminus }}\overline{\bigcup \mathcal {M}} = \emptyset$. Since this is a contradiction, we conclude that $X = F$ and the proof is completed. $\square$

Corollary 3.23

If X is a Hausdorff space with a dense set of isolated points, then $|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$.

Remark 3.24

We note that as a corollary of Theorem 3.22 we obtain that if X is a Hausdorff space with a compact $\pi$-base, then $|X|\le \pi \chi (X)^{aL_c(X)\textrm{ot}(X)\psi _c(X)}$, where $aL_c(X)$ is the almost Lindelöf degree of X with respect to closed sets. But that is a special case of [26, Theorem 5.1] where the authors proved the same inequality for every Hausdorff space X.

We also recall that in [26, Question 5.15] the authors asked if the inequality $|X|\le \pi \chi (X)^{wL_c(X)\textrm{ot}(X)\psi (X)}$ was valid for every regular $T_1$-space. Our Theorem 3.22 shows that even the formally stronger inequality $|X|\le \pi \chi (X)^{wL(X)\textrm{ot}(X)\psi _c(X)}$ is true for every Hausdorff space with a compact $\pi$-base.

In relation to Theorem 3.22 the following question is natural:

Question 3.25

Let X be a Hausdorff space with a compact $\pi$-base. Is it true that

$$\begin{aligned}|X|\le \pi \chi (X)^{wL(X)\textrm{dot}(X)\psi _c(X)}? \end{aligned}$$

Our next theorem should be compared with Theorem 3.22. Since $\textrm{ot}(X)\le wt(X)$ for any space X (Propostion 2.1), we see that Theorem 3.26 increases $\textrm{ot}(X)$ to wt(X) but excludes the use of $\pi \chi (X)$. Thus, Theorems 3.22 and 3.26 are variations of each other.

Theorem 3.26

If X is a Hausdorff space with a compact $\pi$-base, then

$$\begin{aligned}|X|\le 2^{wL(X)wt(X)\psi _c(X)}. \end{aligned}$$

Proof

Let $\kappa =wL(X)wt(X)\psi _c(X)$ and let $\mathcal {B}$ be a $\pi$-base of non-empty open sets with compact closures. Notice that for each $B\in \mathcal {B}$ if $\psi (\overline{B})$ is the pseudocharacter of the space $\overline{B}$, then we have $\psi (\overline{B})\le \psi _c(X)$ and since $\overline{B}$ is compact and Hausdorff, we have $|\overline{B}|\le 2^{\psi (\overline{B})}\le 2^\kappa$. Since $\psi _c(X)\le \kappa$, for each $x\in X$ we can fix a collection $\mathcal {V}_x$ of open neighborhoods of x such that $|\mathcal {V}_x|\le \kappa$ and $\bigcap \{\overline{V}:V\in \mathcal {V}_x\}=\{x\}$. Without loss of generality we may assume that each $\mathcal {V}_x$ is closed under finite intersections.

We will construct by transfinite recursion a non-decreasing chain of open sets $\{U_\alpha : \alpha < \kappa ^+\}$ such that

• (1) $|\overline{U_\alpha }| \le 2^\kappa$ for every $\alpha < \kappa ^+$, and

• (2) if $X{\setminus }\overline{\bigcup \mathcal {M}} \ne \emptyset$ for some $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$, then there is $B_\mathcal {M}\in \mathcal {B}$ such that $B_\mathcal {M}\subset U_{\alpha +1}{\setminus }\overline{\bigcup \mathcal {M}}$.

Let $B_0\in \mathcal {B}$ be arbitrary. We set $U_0=B_0$. Then $|\overline{U_0}|\le 2^\kappa$. If $\beta =\alpha + 1$, for some $\alpha$, then for every $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$ such that $X{\setminus }\overline{\bigcup \mathcal {M}} \ne \emptyset$, we choose $B_{\mathcal {M}} \in \mathcal {B}$ such that $B_{\mathcal {M}} \subseteq X{{\setminus }}\overline{\bigcup \mathcal {M}}$. We define $U_{\beta } = U_\alpha \cup \bigcup \{B_\mathcal {M}: \mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }, X{{\setminus }}\overline{\bigcup \mathcal {M}}\ne \emptyset \}$. Therefore, by Theorem 3.10(d), we have $|\overline{U_{\beta }}| \le |U_\beta |^{wt(X)\psi _c(X)}\le (2^\kappa )^\kappa =2^\kappa$. If $\beta <\kappa ^+$ is a limit ordinal we let $U_\beta = \bigcup _{\alpha <\beta } U_\alpha$. Then clearly $|U_\beta |\le 2^\kappa$, and again by Theorem 3.10(d) we have $|\overline{U_\beta }| \le 2^\kappa$.

Let $F = \bigcup \{\overline{U_\alpha }: \alpha < \kappa ^+\}$. Then $|F| \le 2^\kappa \cdot \kappa ^+=2^\kappa$. Since $\textrm{ot}(X)\le wt(X)\le \kappa$ by Proposition 2.1, we have $F = \overline{\bigcup \{U_\alpha : \alpha < \kappa ^+\}}$ and therefore F is a regular-closed set.

We will show that $X = F$. Suppose that $X \ne F$. Since F is closed and $\mathcal {B}$ is a $\pi$-base, there is $B\in \mathcal {B}$ such that $B\subset X{{\setminus }} F$ and $\overline{B}\cap F=\emptyset$ (Lemma 3.1). Then for every $x\in F$ and $y\in \overline{B}$ there is $V_x(y)\in \mathcal {V}_x$ such that $y\notin \overline{V_x(y)}$. Therefore, using the compactness of $\overline{B}$ and the fact that each $\mathcal {V}_x$ is closed under finite intersections, for every $x\in F$ we can find $V_x \in \mathcal {V}_x$ such that $\overline{V_x} \cap \overline{B}=\emptyset$; hence $\overline{B}\cap V_x=\emptyset$. Clearly $\{V_x: x \in F\}$ is an open cover of F. Since wL(X) is monotone with respect to regular-closed sets, there exists $\mathcal {M} \in \{V_x: x \in F\}^{\le \kappa }$ such that $F \subseteq \overline{\bigcup \mathcal {M}}$. Then there exists $\alpha < \kappa ^+$ such that $\mathcal {M} \in [\bigcup \{\mathcal {V}_x: x \in \overline{U_\alpha }\}]^{\le \kappa }$. As $\overline{B} \cap \bigcup \mathcal {M} = \emptyset$, it follows that $B\subset X{{\setminus }}\overline{\bigcup \mathcal {M}}$, hence $X{\setminus }\overline{\bigcup \mathcal {M}}\ne \emptyset$. Thus, there exists $B_\mathcal {M}\in \mathcal {B}$ such that $\emptyset \ne B_\mathcal {M} \subseteq U_{\alpha +1}{{\setminus }}\overline{\bigcup \mathcal {M}} \subseteq F{{\setminus }}\overline{\bigcup \mathcal {M}} = \emptyset$. Since this is a contradiction, we conclude that $X = F$ and the proof is completed. $\square$

Corollary 3.27

If X is a Hausdorff space with a dense set of isolated points, then $|X|\le 2^{wL(X)wt(X)\psi _c(X)}$.

The previous corollary implies immediately the following result:

Corollary 3.28

[1, Theorem 3] If X is a Hausdorff space with a dense set of isolated points, then $|X|\le 2^{wL_c(X)t(X)\psi _c(X)}$.

We also note that the upper bound in Theorem 3.26 is at least as good as the upper bound in the following theorem from [12]:

Theorem 3.29

If X is a Hausdorff space with a compact $\pi$-base, then

$$\begin{aligned}|X|\le 2^{wL(X)t(X)\psi _c(X)}. \end{aligned}$$

As Theorems 3.22 and 3.26 have similar proofs, one of the referees of this paper asked the following question:

Question 3.30

Let X be a Hausdorff space with a compact $\pi$-base. Is it possible to find an upper bound for |X| such that the upper bounds for |X| in Theorems 3.22 and 3.26 to become corollaries from it?

4 Adding homogeneity

We recall that Ridderbos proved the following result in [33].

Lemma 4.1

If X is a Hausdorff, power homogeneous space, then $|X|\le d(X)^{\pi \chi (X)}$.

Combining Theorem 3.8 with Lemma 4.1 we obtain the following:

Corollary 4.2

If X is a Hausdorff, power homogeneous space with a compact $\pi$-base, then $|X|\le c(X)^{\pi \chi (X)\psi (X)}.$

Corollary 4.3

If X is a power homogeneous Hausdorff space with a dense set of isolated points, then $|X|\le c(X)^{\aleph _0}$.

Proof

By Lemma 4.1 and Proposition 3.6, we have $|X|\le d(X)^{\pi \chi (X)}=c(X)^{\pi \chi (X)}$. But a power homogeneous space with a dense set of isolated points has countable $\pi$-character (Lemma 2.2 in [35]). Thus $|X|\le c(X)^{\aleph _0}$. $\square$

Definition 4.4

Given a cover $\mathcal {C}$ of X, a subset $A\subseteq X$ is $\mathcal {C}$-saturated if $A\cap C$ is dense in A for every $C\in \mathcal {C}$.

It is clear that the union of $\mathcal {C}$-saturated subsets is $\mathcal {C}$-saturated.

Lemma 4.5

Let X be a space and $wt(X)\le \kappa$. Let also $\mathcal {C}$ be a cover of X witnessing that $wt(X)\le \kappa$. If S is $\mathcal {C}$-saturated and $S\cap U\ne \varnothing$, where U is an open set in X, then $S\cap U$ is $\mathcal {C}$-saturated.

Proof

Let $C\in \mathcal {C}$. As S is $\mathcal {C}$-saturated, $S=cl_S(S\cap C)\subseteq cl_X(S\cap C)$. We want to show that $S\cap U=cl_{S\cap U}(S\cap U\cap C)=cl_X(S\cap U\cap C)\cap S\cap U$. Let $x\in S\cap U$ and let V be an open set containing x. Then $U\cap V$ is an open set containing x, and since $x\in S\subseteq cl_X(S\cap C)$, we have $U\cap V\cap S\cap C\ne \varnothing$. Thus, $x\in cl_X(S\cap U\cap C)$. This shows $S\cap U\subseteq cl_X(S\cap U\cap C)$ and that $S\cap U=cl_{S\cap U}(S\cap U\cap C)$. Thus, $S\cap U$ is $\mathcal {C}$-saturated. $\square$

Definition 4.6

Let X be a space and $\kappa$ be an infinite cardinal. A $G_\kappa ^c$-set is a set $G\subseteq X$ such that there exists a $\kappa$-family of open sets $\mathcal {U}$ such that $G=\bigcap \mathcal {U}=\bigcap _{U\in \mathcal {U}}\overline{U}$. (Some authors use the term regular $G_\kappa$-set instead of $G_\kappa ^c$-set.)

Lemma 4.7

[Lemma 3.4 in [15]] Let X be a space, $D\subseteq X$, $wt(X)\le \kappa$, and let $\mathcal {G}$ be a cover of $\overline{D}$ consisting of $G^c_\kappa$-sets of X. Then there exists $\mathcal {G}^\prime \subseteq \mathcal {G}$ such that $|\mathcal {G}^\prime |\le |D|^\kappa$ and $\mathcal {G}^\prime$ covers $\overline{D}$.

Proposition 4.8

[Proposition 3.2 in [15]] Let X be a space, $wt(X)=\kappa$, and let $\mathcal {C}$ be a cover witnessing that $wt(X)=\kappa$. Then for every $x\in X$ there exists $S(x)\in [X]^{\le 2^\kappa }$ such that $x\in S(x)$ and S(x) is $\mathcal {C}$-saturated.

Lemma 4.9

[Lemma 2.7 in [15]] Let X be a space, $wt(X)\le \kappa$, and $\mathcal {C}$ be a cover of X witnessing that $wt(X)\le \kappa$. If $\mathcal {B}$ is an increasing chain of $\kappa ^+$-many $\mathcal {C}$-saturated subsets of X, then

$$\begin{aligned}\overline{\bigcup \mathcal {B}}=\bigcup _{B\in \mathcal {B}}\overline{B}. \end{aligned}$$

Theorem 4.10

Let X be a space, W be a non-empty open set, $L(\overline{W})wt(X)\le \kappa$, and suppose $\mathcal {G}$ is a cover of $\overline{W}$ consisting of $G^c_\kappa$-sets of X. Then there exists $\mathcal {G}^\prime \subseteq \mathcal {G}$ such that $|\mathcal {G}^\prime |\le 2^\kappa$ and $\overline{W}\subseteq \overline{\bigcup \mathcal {G}^\prime }$.

Proof

Let $\mathcal {C}$ be a cover of X witnessing that $wt(X)\le \kappa$. By Proposition 4.8, for every $x\in X$ there exists a $\mathcal {C}$-saturated set S(x) such that $|S(x)|\le 2^\kappa$ and $x\in S(x)$.

For every $G\in \mathcal {G}$ fix a family $\mathcal {U}(G)$ of open sets such that $G=\bigcap \mathcal {U}(G)=\bigcap _{U\in \mathcal {U}(G)}\overline{U}$. If $\mathcal {G}^\prime \subseteq \mathcal {G}$, let $\mathcal {U}(\mathcal {G}^\prime )=\bigcup \{\mathcal {U}(G):G\in \mathcal {G}^\prime \}$.

We build a non-decreasing chain $\{D_\alpha :\alpha <\kappa ^+\}$ of subsets of X and a non-decreasing chain $\{\mathcal {G}_\alpha :\alpha <\kappa ^+\}$ of subsets of $\mathcal {G}$ such that

• (1) $|\mathcal {G}_\alpha |\le 2^\kappa$, $|D_\alpha |\le 2^\kappa$, and $D_\alpha \subseteq W$,

• (2) $\overline{D_\alpha }\subseteq \bigcup \mathcal {G}_\alpha$, and each $D_\alpha$ is $\mathcal {C}$-saturated,

• (3) if $\mathcal {V}\in [\mathcal {U}(\mathcal {G}_\alpha )]^{\le \kappa }$ is such that $W\backslash \bigcup \mathcal {V}\ne \varnothing$, then $D_{\alpha +1}\backslash \bigcup \mathcal {V}\ne \varnothing$.

Let $z\in W$ and set $D_0=S(z)\cap W$. Note that $z\in D_0$ and therefore $D_0$ is non-empty. By Lemma 4.5, $D_0$ is $\mathcal {C}$-saturated and $|D_0|\le |S(z)|\le 2^\kappa$. For limit ordinals $\beta <\kappa ^+$, let $D_\beta =\bigcup _{\alpha <\beta }D_\alpha$. Then $|D_\beta |\le 2^\kappa$, $D_\beta$ is $\mathcal {C}$-saturated as a union of $\mathcal {C}$-saturated sets, and $D_\beta \subseteq W$. As $\overline{D_\beta }\subseteq \overline{W}\subseteq \bigcup \mathcal {G}$, by Lemma 4.7, we obtain $\mathcal {G}_\beta \subseteq \mathcal {G}$ such that $\overline{D_\beta }\subseteq \bigcup \mathcal {G}_\beta$ and $|\mathcal {G}_\beta |\le |D_\beta |^\kappa =(2^\kappa )^\kappa =2^\kappa$.

For successor ordinals $\beta +1$, for every $\mathcal {V}\in [\mathcal {U}(\mathcal {G}_\beta )]^{\le \kappa }$ with $W\backslash \bigcup \mathcal {V}\ne \varnothing$, let $x_\mathcal {V}\in W\backslash \bigcup \mathcal {V}$. Define

$$\begin{aligned}D_{\beta +1}=D_\beta \cup \bigcup \left\{ S(x_\mathcal {V})\cap W:\mathcal {V}\in [\mathcal {U}(\mathcal {G}_\beta )]^{\le \kappa } and W\backslash \bigcup \mathcal {V}\ne \varnothing \right\} .\end{aligned}$$

Note that $x_\mathcal {V}\in S(x_\mathcal {V})\cap W$ and thus $S(x_\mathcal {V})\cap W$ is $\mathcal {C}$-saturated for each $\mathcal {V}$ by Lemma 4.5. Therefore $D_{\beta +1}$ is $\mathcal {C}$-saturated as a union of $\mathcal {C}$-saturated sets. Note also that $D_{\beta +1}\subseteq W$. As $|S(x_\mathcal {V})\cap W|\le |S(x_\mathcal {V})|\le 2^\kappa$ and $|\mathcal {U}(\mathcal {G}_\beta )|\le 2^\kappa$, we have $|D_{\beta +1}|\le 2^\kappa$. As $\overline{D_{\beta +1}}\subseteq \overline{W}\subseteq \bigcup \mathcal {G}$, by Lemma 4.7 we obtain $\mathcal {G}_{\beta +1}\subseteq \mathcal {G}$ such that $\overline{D_{\beta +1}}\subseteq \bigcup \mathcal {G}_{\beta +1}$ and $|\mathcal {G}_{\beta +1}|\le |D_{\beta +1}|^\kappa =(2^\kappa )^\kappa =2^\kappa$.

Let $D=\bigcup \{D_\alpha :\alpha <\kappa ^+\}$ and note that $D\subseteq W$. It follows from Lemma 4.9 that $\overline{D}=\bigcup \{\overline{D_\alpha }:\alpha <\kappa ^+\}$. Let $\mathcal {G}^\prime =\bigcup \{G_\alpha :\alpha <\kappa ^+\}$. Note that $\overline{D}\subseteq \bigcup \mathcal {G}^\prime$. We will show now that $W\subseteq \bigcup \mathcal {G}^\prime$. Suppose there exists $x\in W\backslash \bigcup \mathcal {G}^\prime$. Then $x\notin G$ for every $G\in \mathcal {G}^\prime$. For each $G\in \mathcal {G}^\prime$ there exists $U(G)\in \mathcal {U}(G)$ such that $x\notin \overline{U(G)}$. Now, $\mathcal {U}=\{U(G):G\in \mathcal {G}^\prime \}$ covers $\bigcup \mathcal {G}^\prime$ and hence covers $\overline{D}$. As $D\subseteq W$ we have $\overline{D}\subseteq \overline{W}$ and thus $L(\overline{D})\le L(\overline{W})\le \kappa$. There exists $\mathcal {V}\subseteq \mathcal {U}$ such that $|\mathcal {V}|\le \kappa$ and $\overline{D}\subseteq \bigcup \mathcal {V}$. Now, $\mathcal {V}\in [\mathcal {U}(\mathcal {G}_\alpha )]^{\le \kappa }$ for some $\alpha <\kappa ^+$. Since $x\in W\backslash \bigcup \mathcal {V}$, we have $D_{\alpha +1}\backslash \bigcup \mathcal {V}\ne \varnothing$ and

$$\begin{aligned} \varnothing \ne D_{\alpha +1}\backslash \bigcup \mathcal {V}\subseteq D\subseteq \overline{D}\subseteq \bigcup \mathcal {V}, \end{aligned}$$

a contradiction. Thus $W\subseteq \bigcup \mathcal {G}^\prime$ and $\overline{W}\subseteq \overline{\bigcup \mathcal {G}^\prime }$. $\square$

Theorem 4.11

[Theorem 3.15 in [16]] If X is a Hausdorff space and $wt(X)\textrm{pct}(X)\le \kappa$, then there exists a non-empty compact set $G\subseteq X$ and $H\subseteq X$ such that $\chi (G,X)\le \kappa$, $G\subseteq \overline{H}$, $|H|\le 2^\kappa$, and H is $\mathcal {C}$-saturated in any cover $\mathcal {C}$ witnessing $wt(X)\le \kappa$.

Lemma 4.12

[Lemma 3.5 in [11]] Let X be a power homogeneous Hausdorff space. If $D\subseteq X$ and U is an open set such that $U\subseteq \overline{D}$, then $|U|\le |D|^{\pi \chi (X)}$.

Theorem 4.13

If X is a homogeneous Hausdorff space and $U\subseteq X$ is a non-empty open set, then $|U|\le 2^{L(\overline{U})wt(X)\pi \chi (X)\textrm{pct}(X)}$.

Proof

Let $\kappa =L(\overline{U})wt(X)\pi \chi (X)\textrm{pct}(X)$. As $wt(X)\textrm{pct}(X)\le \kappa$, by Theorem 4.11, there exists a non-empty compact set G and $H\subseteq X$ such that $G\subseteq \overline{H}$, $\chi (G,X)\le \kappa$, and $|H|\le 2^\kappa$. Fix $p\in G$. For every $x\in X$ there exists a homeomorphism $h_x:X\rightarrow X$ such that $h_x(p)=x$. Note that G is a $G_\kappa ^c$-set as X is Hausdorff and G is compact.

Let $\mathcal {G}=\{h_x[G]:x\in \overline{U}\}$. Then $\mathcal {G}$ is a cover of $\overline{U}$ by $G^c_\kappa$-sets of X. By Theorem 4.10, there exists $\mathcal {G}^\prime \subseteq \mathcal {G}$ such that $|\mathcal {G}^\prime |\le 2^\kappa$ and $\overline{U}\subseteq \overline{\bigcup \mathcal {G}^\prime }$. Also, for every $G\in \mathcal {G}^\prime$ there exists $H_G\subseteq X$ such that $G\subseteq \overline{H_G}$ and $|H_G|\le 2^\kappa$.

Let $D=\bigcup \{H_G:G\in \mathcal {G}^\prime \}$ and note that $|D|\le 2^\kappa \cdot 2^\kappa =2^\kappa$. We will show that $U\subseteq \overline{D}$. Let $x\in U$, and let V be an open set containing x. Then there exists $G\in \mathcal {G}^\prime$ such that $U\cap V\cap G\ne \varnothing$. As $G\subseteq \overline{H_G}$, we have $U\cap V\cap H_G\ne \varnothing$ and thus $x\in \overline{D}$. This shows that $U\subseteq \overline{D}$.

As X is homogeneous, by Lemma 4.12 we have $|U|\le |D|^{\pi \chi (X)}\le (2^\kappa )^\kappa =2^\kappa$. $\square$

Theorem 4.14

If X is a locally compact and homogeneous Hausdorff space, then $|X|\le wL(X)^{wt(X)\pi \chi (X)}.$

Proof

As X is locally compact, for every $x\in X$ there exists an open set $U_x$ such that $x\in U_x$ and $\overline{U_x}$ is compact. Also, note that $\textrm{pct}(X)$ is countable. By Theorem 4.13, $|U_x|\le 2^{wt(X)\pi \chi (X)}$ for each $x\in X$. Then $\mathcal {U}=\{U_x:x\in X\}$ is an open cover of X. Hence, there exists $\mathcal {V}\subseteq \mathcal {U}$ such that $|\mathcal {V}|\le wL(X)$ and $X=\overline{\bigcup \mathcal {V}}$. Thus, $d(X)\le |\bigcup \mathcal {V}|\le wL(X)\cdot 2^{wt(X)\pi \chi (X)}\le wL(X)^{wt(X)\pi \chi (X)}$.

As X is homogeneous and Hausdorff, we have

$$\begin{aligned} |X|\le d(X)^{\pi \chi (X)}\le \left( wL(X)^{wt(X)\pi \chi (X)}\right) ^{\pi \chi (X)}=wL(X)^{wt(X)\pi \chi (X)}. \end{aligned}$$

$\square$

Before we state some corollaries of Theorem 4.14 let us make the following observation.

Proposition 4.15

Let X be a homogeneous Hausdorff space with a non-empty open set U such that $\overline{U}$ is compact. Then X is locally compact.

Proof

Let $p\in U$. For each $x\in X$, let $h_x:X\rightarrow X$ be a homeomorphism such that $h_x(p)=x$. Then for each $x\in X$, we have $x\in h_x[U]\subseteq \overline{h_x[U]}$, where $h_x[U]$ is open and $h_x[\overline{U}]=\overline{h_x[U]}$ is compact. Thus, every point has a compact neighborhood, implying X is locally compact. $\square$

As immediate corollary of the above proposition we obtain the following result:

Theorem 4.16

If X is a homogeneous Hausdorff space with a compact $\pi$-base, then X is locally compact.

Corollary 4.17

If X is a homogeneous Hausdorff space with a non-empty open set U such that $\overline{U}$ is compact, then $|X|\le wL(X)^{wt(X)\pi \chi (X)}$.

Proof

This follows from Proposition 4.15 and Theorem 4.14. $\square$

Corollary 4.18

If X is a homogeneous Hausdorff space with a compact $\pi$-base, then $|X|\le wL(X)^{wt(X)\pi \chi (X)}$.

Theorem 4.14 should be compared with the following two results from [12].

Theorem 4.19

[12] If X is a locally compact Hausdorff space, then $|X|\le wL(X)^{\psi (X)}.$

Theorem 4.20

[12] If X is a locally compact power homogeneous Hausdorff space, then $|X|\le wL(X)^{t(X)}.$

To show that $\pi \chi (X)\le t(X)$ for every locally compact Hausdorff space X, all we need is a corollary of the following observation, which extends to the class of Hausdorff spaces with pointwise countable type Šapirovskiĭ’s result that $\pi \chi (X)\le t(X)$ for every compact Hausdorff space X [36, Corollary 4].

Proposition 4.21

If X is a Hausdorff space with $\textrm{pct}(X)\le \omega$, then $\pi \chi (X)\le t(X)$.

Proof

Let X be a Hausdorff space with $\textrm{pct}(X)\le \omega$. Since $\pi \chi (X) \le t(X)\textrm{pct}(X)$ for any Hausdorff space X [36, Theorem 4], it follows that $\pi \chi (X)\le t(X)$ for every Hausdorff space X with $\textrm{pct}(X)\le \omega$. $\square$

Lemma 4.22

[3] If X is a locally compact Hausdorff space, then $\textrm{pct}(X)\le \omega$.

As a consequence of Proposition 4.21 and Lemma 4.22, we have the following.

Corollary 4.23

If X is a locally compact Hausdorff space, then $\pi \chi (X)\le t(X)$.

It follows from Corollary 4.23 that the upper bound $wL(X)^{wt(X)\pi \chi (X)}$ in Theorem 4.14, for locally compact homogeneous Hausdorff spaces, is at least as good as the upper bound $wL(X)^{t(X)}$ in Theorem 4.20.

Lemma 4.25 below is an obvious generalization of the following Proposition 3.5 from [3].

Proposition 4.24

[3] Let X be a compact Hausdorff space, U be an open set in X and $F\subseteq U\subseteq X$ be a closed set in X. Then there exists a compact Hausdorff space G with $\chi (G, X)\le \omega$ such that $F\subset G\subseteq U$.

Lemma 4.25

Let X be a Hausdorff space and suppose U is a non-empty open set of X such that $\overline{U}$ is compact. Then for every closed subset F of X with $F\subseteq U$ there exists a compact set G such that $F\subseteq G\subseteq U$ and $\chi (G, X)\le \omega$.

Proof

Let U be a non-empty open set in X and $F\subseteq U$ be a closed subset in X. As $\overline{U}$ is compact and Hausdorff, it follows from Proposition 4.24 that there exists a compact Hausdorff subspace G of X such that $F\subseteq G\subseteq U$ and $\chi (G,\overline{U})\le \omega$. Since $G\subseteq U\subseteq \overline{U}$, we have $\chi (G,U)\le \omega$ and hence $\chi (G,X)\le \omega$, for U is open in X. $\square$

Observe that Lemma 4.22 follows from Lemma 4.25.

Čoban showed in [21] that if X is a Hausdorff space with pointwise countable type, then $\psi (x,X)=\chi (x,X)$ for every $x\in X$. Therefore, if X is a locally compact Hausdorff space, then $\psi (X) = \chi (X)$. Thus, $wt(X)\pi \chi (X)\le t(X) \le \chi (X) = \psi (X)$. (The same conclusion could be deducted from the fact that $\chi (X)=\psi (X)\textrm{pct}(X)$ for any Hausdorff space X [23, 3.1.F.(b)].) Therefore, the upper bound $wL(X)^{wt(X)\pi \chi (X)}$ in Theorem 4.14 is at least as good as the upper bound $wL(X)^{\psi (X)}$ in Theorem 4.19.

We know from Theorem 4.16 that every homogeneous Hausdorff space X with a compact $\pi$-base is locally compact and therefore $\pi \chi (X)\le t(X)$ (Corollary 4.23). In Theorem 4.29 we will show that the same inequality is true for every power homogeneous Hausdorff space X with a compact $\pi$-base. For its proof we need some preliminary results.

Lemma 4.26

[34, Lemma 2.4.3] Suppose F is a compact subset of a space X and $x\in F$. Then $\pi \chi (x,X)\le \pi \chi (x,F)\chi (F,X)$.

Theorem 4.27

Let X be a Hausdorff space with a compact $\pi$-base $\mathcal {B}$. Then X has a dense, open set D such that for every $x\in D$,

• (a) x is contained in a compact set G such that $\chi (G,X)\le \omega$, and

• (b) $\pi \chi (x,X)\le t(X)$.

Proof

Let $D=\bigcup \mathcal {B}$. Then D is open and dense as $\mathcal {B}$ is a $\pi$-base. Let $x\in D$. There exists $B\in \mathcal {B}$ such that $x\in B$ and $\overline{B}$ is compact. By Lemma 4.25, there exists a compact set G such that $x\in G\subseteq B$ and $\chi (G,X)\le \omega$. This establishes (a). As G is compact (and Hausdorff), by Proposition 4.21 we have $\pi \chi (x,G)\le t(G)\le t(X)$. Then, again using the compactness of G and Lemma 4.26, we have $\pi \chi (x,X)\le \pi \chi (x,G)\chi (G,X)\le t(X)\cdot \omega =t(X)$. This establishes (b). $\square$

Lemma 4.28

[35, Lemma 2.2] Let X be a power homogeneous Hausdorff space and let $\kappa$ be an infinite cardinal. If the set $\{x\in X:\pi \chi (x,X)\le \kappa \}$ is dense in X, then $\pi \chi (X)\le \kappa$.

Theorem 4.29

If X is a power homogeneous Hausdorff space with a compact $\pi$-base, then $\pi \chi (X)\le t(X)$.

Proof

Let $\kappa =t(X)$. By Theorem 4.27, X has a dense, open set D such that $\pi \chi (x,X)\le t(X)=\kappa$ for every $x\in D$. Thus, $D\subseteq \{x\in X:\pi \chi (x,X)\le \kappa \}$ and $\{x\in X:\pi \chi (x,X)\le \kappa \}$ is dense in X. As X is power homogeneous, by Lemma 4.28 it follows that $\pi \chi (X)$ $\le \kappa$. $\square$

Theorem 4.30

If X is compact, pseudoradial, and power homogeneous, then $\pi \chi (X)\le \mathfrak {c}$ and $|X|\le 2^{c(X)\cdot \mathfrak {c}}$.

Proof

A compact pseudoradial space is sequentially compact. Since $I^\mathfrak {c}$ is not sequentially compact, X cannot be continuously mapped onto $I^\mathfrak {c}$. By Theorem 3.14, for each non-empty open set U there exists $p_U\in \overline{U}$ such that $\pi \chi (p_U,\overline{U})<\mathfrak {c}$. If $\mathcal {B}(p_U)$ is a local $\pi$-base for $p_U$ in $\overline{U}$ such that $|\mathcal {B}(p_U)|<\mathfrak {c}$, then $\{B\cap U:B\in \mathcal {B}(p_U)\}$ is a local $\pi$-base for $p_U$ in X and hence $\pi \chi (p_U,X)<\mathfrak {c}$. Now $Y=\{p_U:U is open in X\}$ is dense in X. Since $Y\subseteq \{x\in X:\pi \chi (x,X)<\mathfrak {c}\}\subseteq \{x\in X:\pi \chi (x,X)\le \mathfrak {c}\}$, we see that $\{x\in X:\pi \chi (x,X)\le \mathfrak {c}\}$ is dense in X. Thus $\pi \chi (X)\le \mathfrak {c}$ by Lemma 4.28.

Finally, as X is power homogeneous we have that $|X|\le 2^{c(X)\pi \chi (X)}$ as shown in [18]. This implies $|X|\le 2^{c(X)\cdot \mathfrak {c}}$. $\square$