Pseudocompact $\varDelta$-spaces are often scattered

Abstract

Given a pseudocompact $\varDelta$-space X, we establish that countable subsets of X must be scattered. This implies that pseudocompact $\varDelta$-spaces of countable tightness are scattered. If a pseudocompact $\varDelta$-space has the Souslin property, then it is separable and has a dense set of isolated points. It is shown that adding a countable subspace to a pseudocompact $\varDelta$-space can destroy the $\varDelta$-property. However, if X is countably compact and $Y\subset X$ is a $\varDelta$-space for some $Y\subset X$ such that $|X\backslash Y|\le \omega$, then X is a $\varDelta$-space. We also show that monotonically normal $\varDelta$-spaces must be hereditarily paracompact. Besides, if X is a subspace of an ordinal with its order topology, then X is hereditarily paracompact if and only if it has the $\varDelta$-property.

1 Introduction

In 1975, Reed defined a set $D\subset \mathbb {R}$ to be a $\varDelta$ -set if D is uncountable and, for any decreasing sequence $\{H_n: n\in \omega \}$ of subsets of D, if $\bigcap _{n\in \omega }H_n= \emptyset$, then there exists a sequence $\{V_n: n\in \omega \}$ of $G_\delta$-subsets of D such that $H_n\subset V_n$ for each $n\in \omega$ and $\bigcap _{n\in \omega }V_n=\emptyset$. Przymusinski proved in [13] that existence of a $\varDelta$-set is equivalent to existence of a countably paracompact, separable non-normal Moore space. Reed mentions in his paper [14], that, in the definition of a $\varDelta$-set, the term “$G_\delta$-sets” can be replaced with “open sets”; this was pointed out by van Douwen.

In [9], the essence of the definition of a $\varDelta$-set was the base for introducing general $\varDelta$-spaces by saying that X is a $\varDelta$-space if, for any decreasing sequence $\mathcal {S}= \{X_n: n\in \omega \}$ of subsets of X with empty intersection, there exists a sequence $\{U_n: n\in \omega \}$ of open subsets of X with empty intersection such that $X_n\subset U_n$ for each $n\in \omega$. According to this definition, a $\varDelta$-space need not be uncountable and a $\varDelta$-set is an uncountable $\varDelta$-subspace of $\mathbb {R}$.

The paper [9] features a systematic study of general $\varDelta$-spaces. One of its results states that X is a $\varDelta$-space if and only if it has the following $\varDelta$ -property: for any disjoint family $\mathcal {A}= \{A_n: n\in \omega \}$ of subsets of X, there is a point-finite open expansion of $\mathcal {A}$, i.e., there exists a point-finite family $\{U_n: n\in \omega \}$ of open subsets of X such that $A_n\subset U_n$ for each $n\in \omega$. It was also proved in [9] that Čech-complete $\varDelta$-spaces must be scattered and every scattered Eberlein compact space must have the $\varDelta$-property. Besides, a Tychonoff space X has the $\varDelta$-property if and only if $C_p(X)$ is distinguished.

In [10] the authors proved that any $\sigma$-product of Eberlein compact spaces must be a $\varDelta$-space and established that each countably compact $\varDelta$-space is scattered. As a natural attempt to look for a generalization of this result, they asked whether every pseudocompact $\varDelta$-space is scattered; this question was an inspiration for the authors of this paper to study the $\varDelta$-property in compact-like spaces.

In this article, we show that, in a pseudocompact $\varDelta$-space, all countable subsets must be scattered. Therefore pseudocompact $\varDelta$-spaces of countable tightness are scattered. We also establish that adding a countable set to a pseudocompact $\varDelta$-space can destroy its $\varDelta$-property. On the other hand, a countably compact space X must be a $\varDelta$-space if it has a $\varDelta$-subspace Y such that $X\backslash Y$ is countable. It turns out that monotonically normal $\varDelta$-spaces are hereditarily paracompact while a subspace X of an ordinal has the $\varDelta$-property if and only if X is hereditarily paracompact. A by-product of this study is the fact that any pseudocompact crowded space of countable tightness must be $\omega$-resolvable.

2 Notation and terminology

All spaces are assumed to be Tychonoff. Given a space X, the family $\tau (X)$ is its topology and $\tau ^*(X) =\tau (X) \backslash \{\emptyset \}$; let $\tau (x,X) =\{U\in \tau (X): x\in U\}$ for any point $x\in X$. The set $\mathbb {R}$ is the real line with its usual topology while $\mathbb {I}=[0,1]\subset \mathbb {R}$ and $\mathbb {D}$ is the two-point set $\{0,1\}\subset \mathbb {R}$. If $\kappa$ is a cardinal, then $[X]^{\le \kappa }=\{ A\subset X:|A|\le \kappa \}$. Given a family $\mathcal {F}$ of subsets of a space X, an open expansion of $\mathcal {F}$ is a family $\{U_F: F\in \mathcal {F}\}\subset \tau (X)$ such that $F\subset U_F$ for each $F\in \mathcal {F}$. If $A\subset X$, then a family $\mathcal {U}\subset \tau (X)$ is an open expansion of the set A if it is an open expansion of the family $\{ \{ x\}: x\in A\}$. Recall that $S\subset X$ is a free $\omega _1$ -sequence in a space X if $S=\{x_\alpha : \alpha <\omega _1\}$ and $\overline{\{x_\gamma : \gamma <\alpha \}} \cap \overline{\{x_\gamma : \gamma \ge \alpha \}}=\emptyset$ for any $\alpha <\omega _1$.

The space X is Fréchet–Urysohn provided that, for any $A\subset X$, if $x\in \overline{A}$, there is a sequence $\{a_n: n\in \omega \}\subset A$ that converges to x. If, for any non-closed set $A\subset X$, there exists a sequence $\{a_n: n\in \omega \}\subset A$ such that $a_n\rightarrow x\notin A$, then the space X is called sequential.

The cardinal $t(X)=\min \{ \kappa :\overline{A}=\bigcup \{ \overline{B}:B\in [A]^{\le \kappa }\}$ for every $A\subset X\}$ is the tightness of the space X. A family $\mathcal {U}$ of subsets of X is point-finite if every $x\in X$ belongs only to finitely many elements of $\mathcal {U}$. If $x\in X$, then the cardinal $\psi (x,X)=min\{|\mathcal {U}|:\mathcal {U}\subset \tau (X)$ and $\bigcap \mathcal {U}=\{x\}\}$ is called the pseudocharacter of x in X and $\psi (X)=sup\{ \psi (x,X):x\in X\}$ is the pseudocharacter of X. The minimal cardinality of a local base at a point $x\in X$ is called the character of X at x; it is denoted by $\chi (x,X)$ and $\chi (X)=\sup \{\chi (x,X):x\in X\}$.

A space X is called monotonically normal if it admits an operator O (called the monotone normality operator) that assigns to any point $x\in X$ and any $U\in \tau (x,X)$ a set $O(x,U)\in \tau (x,X)$ such that $O(x,U)\subset U$ and for any points $x,y\in X$ and sets $U,V\in \tau (X)$ such that $x\in U$ and $y\in V$, it follows from $O(x,U) \cap O(y,V)\ne \emptyset$ that $x\in V$ or $y\in U$. Recall that X is a GO space if it embeds into a linearly ordered space. Given a cardinal $\kappa$, a space X is said to be $\kappa$ -resolvable if it is possible to find $\kappa$-many disjoint dense subspaces in X. Furthermore, X is a P-space if every $G_\delta$-subset of X is open.

A space X is crowded if X has no isolated points; if every non-empty subspace of a space X has an isolated point, the space X is called scattered.

We recall some basic facts about Eberlein compact spaces. A space K is an Eberlein compact space if K is homeomorphic to a weakly compact subset of a Banach space. Equivalently, a compact space K is Eberlein if and only if K can be homeomorphically embedded into

$$\begin{aligned} c_0(T) = \left\{ x\in \prod _{t\in T} I_t:\,\,\text{ the } \text{ set }\,\,\{t: |x(t)|> \epsilon \}\,\,\text{ is } \text{ finite } \text{ for } \text{ every }\,\,\epsilon > 0\right\} , \end{aligned}$$

where $I_t$ denotes a copy of the closed united interval [0, 1], for each $t\in T$.

A compact space K is a scattered Eberlein compact space if and only if K can be homeomorphically embedded into

$$\begin{aligned} \sigma (T) = \left\{ x\in \prod _{t\in T} D_t:\,\,\text{ the } \text{ set }\,\,\{t: x(t) \ne 0\}\,\,\text{ is } \text{ finite }\right\} , \end{aligned}$$

where $D_t$ denotes a copy of the two-points set $\{0,1\}$, for each $t\in T$.

More generally, let $\{M_t: t\in T\}$ be any family of topological spaces, and $M=\prod _{t\in T}M_t$ be a topological product. For any fixed point $a\in M$ we denote the $\sigma$-product

$$\begin{aligned} \sigma (M,a)=\{x\in M:\,\,\text{ the } \text{ set }\,\,\{t\in T: x(t)\ne a(t)\}\,\,\text{ is } \text{ finite }\}. \end{aligned}$$

Denote also $\sigma _n(M,a)=\{x\in M: |\{t\in T: x(t)\ne a(t)\}| \le n\}$, for every $n\in \omega$. Evidently, $\sigma (M,a) = \bigcup \{\sigma _n(M,a): n\in \omega \}$.

The rest of our notation is standard and follows the book [5]. All relevant information on cardinal invariants can be found in the paper of Hodel [8].

3 Compactness-like properties in $\varDelta$-spaces

We will show that pseudocompact $\varDelta$-spaces of countable tightness are scattered and monotonically normal $\varDelta$-spaces must be hereditarily paracompact.

Theorem 3.1

If X is a pseudocompact $\varDelta$-space, then every countable subspace of X is scattered.

Proof

If some countable subset of X is not scattered, then there is a countable crowded set $A\subset X$; let $\{a_n: n\in \omega \}$ be a faithful enumeration of A. By the $\varDelta$-property of X, there exists a point-finite open expansion $\mathcal {U}=\{U_n: n\in \omega \}$ of the set A. Let $k_0=0$ and pick a set $V_0\in \tau (a_{k_0},X)$ such that $\overline{V}_0 \subset U_{k_0}$.

Proceeding by induction, assume that, for some $n\in \omega$, we have open sets $V_0, \ldots , V_n$ in the space X together with $k_0, \ldots , k_n\in \omega$ with the following properties:

(1) $\overline{V}_{i+1}\subset V_i$ and $k_i<k_{i+1}$ whenever $0\le i<n$;

(2) $a_{k_i}\in V_i \subset \overline{V}_{i}\subset U_{k_0} \cap \ldots \cap U_{k_i}$ for every $i\le n$.

The set $V_n\cap A$ being infinite, we can find a number $k_{n+1}\in \omega$ such that $k_{n+1}>k_n$ and $a_{k_{n+1}}\in V_n$. There exists a set $V_{n+1}\in \tau (a_{k_{n+1}},X)$ such that $\overline{V}_{n+1}\subset V_n \cap U_{k_{n+1}}$. It is straightforward that (1) and (2) now hold if replace n with $n+1$ so our inductive procedure can be continued to construct a family $\{V_n: n\in \omega \}$ and a sequence $\{k_n: n\in \omega \}$ such that the conditions (1) and (2) are satisfied for each $n\in \omega$.

The property (1), together with pseudocompactness of the space X implies that $F=\bigcap _{n\in \omega }V_n\ne \emptyset$; take a point $x\in F$. The property (2) shows that $x\in U_{k_n}$ for every $n\in \omega$ and hence the family $\mathcal {U}$ is not point-finite. This contradiction proves that every countable subset of X is scattered. $\square$

It is conjectured in [10, Problem 3.21] that every pseudocompact $\varDelta$-space X is scattered. We will show that this is true if X has countable tightness.

Corollary 3.2

Any pseudocompact $\varDelta$-space of countable tightness must be scattered.

Proof

Let X be a pseudocompact $\varDelta$-space with $t(X)\le \omega$. If $A\subset X$ and $x\in \overline{A}$, then take a countable $B\subset A$ such that $x\in \overline{B}$. The set B is scattered by Theorem 3.1 so there is a discrete set $D\subset B$ such that $B\subset \overline{D}$ and, in particular, $x\in \overline{D}$. This shows that

(3) if $A\subset X$ and $x\in \overline{A}$, then there exists a countable discrete set $D \subset A$ such that $x\in \overline{D}$.

If X is not scattered, then fix a crowded subset $Y\subset X$. It is standard that there exists a countably infinite discrete subset $D_0\subset Y$. Proceeding by induction, assume that $n\in \omega$ and we have disjoint countable discrete subsets $D_0,\ldots , D_n$ in the space Y such that

(4) $D_0\cup \ldots \cup D_i\subset \overline{D}_{i+1}$ for any $i<n$.

The set $D'=D_0\cup \ldots \cup D_n$ is nowhere dense in Y so $Y\backslash D'$ is dense in Y. Let $\{O_x: x\in D_n\}$ be a disjoint open expansion of $D_n$ in the space Y. The property (3) shows that there exists a countable discrete set $E_x\subset O_x\backslash D'$ such that $x\in \overline{E}_x$ for every $x\in D_n$. The set $D_{n+1}= \bigcup \{E_x: x\in D_n\}$ is discrete, disjoint from $D'$ and $D_n\subset \overline{D}_{n+1}$. This shows that our inductive procedure can be continued to construct a disjoint family $\{D_n: n\in \omega \}$ of countably infinite discrete subsets of Y such that the condition (4) is satisfied for all $n\in \omega$. The same condition (4) easily implies that $D=\bigcup \{D_n: n\in \omega \}$ is a countable crowded subspace of X; this contradiction with Theorem 3.1 proves that X is scattered. $\square$

Proposition 2.3 of [9] states that the $\varDelta$-property is preserved if we add a finite set to a $\varDelta$-space. However, the same conclusion cannot be made if we add a countable set to a space because Example 59 of the paper [6] shows that Michael line is not a $\varDelta$-space and hence the $\varDelta$-property can be destroyed by adding a countable set to a discrete space. Our next example shows that even the $\varDelta$-property of a pseudocompact space can be lost if we add a countable set to the space.

Example 3.3

Recall that M is a Mrowka space if it is pseudocompact, the set I of isolated points of M is countable and dense in M and, additionally, $D=M\backslash I$ is closed, discrete and uncountable. Let M be a Mrowka space for which there is a continuous onto map $f:M\rightarrow \mathbb {I}$ (see [16, Fact 2 of S.154]). If $K=\beta M$, then there exists a countable crowded set $A\subset K\backslash M$ and hence $X=M\cup A$ is not a $\varDelta$-space. Since M is a $\varDelta$-space by Corollary 3.9 of [9], it is possible to destroy the $\varDelta$-property of a pseudocompact space by adding a countable set.

Proof

Take a continuous map $g:K\rightarrow \mathbb {I}$ such that $g|M=f$; it is clear that $g(K)=\mathbb {I}$ and hence K is not scattered by Problem 129 of [17]. Take a closed crowded subspace $Z\subset K$. Then $Z\subset K\backslash I$ because every point of I is isolated in K. The subspace $D\cap Z$ is discrete and hence nowhere dense in Z so we can find a crowded compact set $Z_0\subset Z\backslash D \subset K\backslash M$. It is standard that every compact crowded space contains a countable crowded space so pick a countable set $A\subset Z_0$ which is dense in itself. We already saw that M is a $\varDelta$-space so all is left is to note that the space $X=M\cup A$ does not have the $\varDelta$-property by Theorem 3.1. $\square$

Our next step is to show that there are many situations where adding a countable set to a $\varDelta$-space preserves the $\varDelta$-property.

Proposition 3.4

Given a space X, assume that Y is a $\varDelta$-subspace of X, the set $A=X\backslash Y$ is countable and has a point-finite open expansion in X. Then X is a $\varDelta$-space.

Proof

If the set A is finite, then X is a $\varDelta$-space by Proposition 2.3 of [9] so we can assume that A is infinite; let $\{a_n: n\in \omega \}$ be a faithful enumeration of the set A. By our hypothesis, there exists a point-finite expansion $\{O_n: n\in \omega \}$ of the set A.

Take any disjoint collection $\mathcal {H}= \{X_n: n\in \omega \}$ of subsets of X and apply the $\varDelta$-property of Y to find a point-finite open expansion $\{U'_n: n\in \omega \}$ of the family $\mathcal {G}= \{X_n\cap Y: n\in \omega \}$ in the space Y. Pick a set $U_n\in \tau (X)$ such that $U_n\cap Y= U'_n$ and consider the set $V_n=U_n\backslash \{a_0, \ldots ,a_{n}\}$ for every $n\in \omega$. It is immediate that the family $\{V_n: n\in \omega \}$ is a point-finite open expansion of $\mathcal {G}$ in the space X.

Given any $n\in \omega$, let $W_n=\bigcup \{O_i: a_i\in X_n\}$; we omit a straightforward proof of the fact that $\{W_n: n\in \omega \}$ is a point-finite open expansion of the family $\{X_n\cap A: n\in \omega \}$. As a consequence, $\{V_n\cup W_n: n\in \omega \}$ is a point-finite open expansion of $\mathcal {H}$ so X is a $\varDelta$-space. $\square$

Corollary 3.5

Suppose that Y is a $\varDelta$-subspace of a space X such that $X\backslash Y$ is countable and scattered. Then X is a $\varDelta$-space.

Proof

By [19, Theorem 3.1], the set $X\backslash Y$ has a point-finite open expansion in X; Proposition 3.4 does the rest. $\square$

Proposition 3.6

If X is a countably compact $\varDelta$-space and $f:X\rightarrow Y$ is a continuous onto map of X onto a sequential space Y, then Y is a $\varDelta$-space.

Proof

It is well known that any continuous map of a countably compact space onto a sequential space is closed so f is closed and hence Y is a $\varDelta$-space by Theorem 2.1 of [10]. $\square$

Corollary 3.7

If X is a countably compact $\varDelta$-space and $f:X\rightarrow M$ is a continuous onto map of X onto a second countable space M, then M is countable.

Proof

Just note that M is a second countable compact space which has the $\varDelta$-property by Proposition 3.6. Therefore M is countable by Proposition 3.5 of [9]. $\square$

There is an important case when a countable complement of a $\varDelta$-subspace is scattered automatically.

Theorem 3.8

If X is a countably compact space and Y is a $\varDelta$-subspace of X such that $A=X\backslash Y$ is countable, then X is a $\varDelta$-space.

Proof

Assume that the space X is not scattered. Then there is a continuous onto map $f:X\rightarrow \mathbb {I}$ (see Problem 133 of [17]). Since $f(A)\subset \mathbb {I}$ is countable, we can find an uncountable compact set $K\subset \mathbb {I}\backslash f(A)$. Then $L= f^{-1}(K)$ is a closed subset of X contained in Y so L is a countably compact $\varDelta$-space that maps continuously onto K which is impossible by Corollary 3.7; this contradiction shows that X must be scattered and hence so is A. Finally, apply Corollary 3.5 to conclude that X is a $\varDelta$-space. $\square$

It was proved in [11] that there exists a compact space X that fails to be a $\varDelta$-space, but there is a discrete uncountable subset $Y\subset X$ such that $A=X\backslash Y$ is a scattered Eberlein compact. So, in the above Corollary 3.5 one cannot assume that $A=X\backslash Y$ is a compact $\varDelta$-space.

Proposition 2.12 of the paper [10] states that the $\varDelta$-property is preserved by inverse images of continuous maps with finite fibers. It turns out that if the respective map is open, then the $\varDelta$-property is preserved in both directions.

Proposition 3.9

Given spaces X and Y, let $f:X\rightarrow Y$ be a continuous open onto map with finite fibers. Then X is a $\varDelta$-space if and only if so is Y.

Proof

If Y is a $\varDelta$-space, then so is X by Proposition 2.12 of [10]; here we don’t even need the map f to be open.

Now, if X is a $\varDelta$-space, then take any disjoint family $\mathcal {F}=\{P_n: n\in \omega \}$ of subsets of Y. Then $\{f^{-1}(P_n): n\in \omega \}$ is a disjoint family of subsets of X so it has a point-finite open expansion $\{U_n:n\in \omega \}$. Then $\mathcal {G}= \{f(U_n): n\in \omega \}$ is an open expansion of the family $\mathcal {F}$ and it is standard to deduce that $\mathcal {G}$ is point-finite from the fact that $f^{-1}(y)$ is finite for every $y\in Y$. $\square$

It was proved in [10, Proposition 2.10] that the unions of $\sigma$-locally finite families of closed $\varDelta$-subspaces have the $\varDelta$-property. The result that follows shows when the $\varDelta$-property is preserved by the unions of families of open sets.

Corollary 3.10

If a space X has a point-finite open cover $\mathcal {U}$ such that every $U\in \mathcal {U}$ is a $\varDelta$-space, then X is a $\varDelta$-space.

Proof

Let $Z=\bigoplus \{U: U\in \mathcal {U}\}$; if $x\in U\in \mathcal {U}$, then, letting $\varphi (x)=x\in X$, we obtain a continuous open surjective map $\varphi :Z\rightarrow X$. The family $\mathcal {U}$ being point-finite, every fiber of the map $\varphi$ is finite. It is an easy exercise that direct sums preserve the $\varDelta$-property so Z is a $\varDelta$-space and hence so is the space X by Proposition 3.9. $\square$

Theorem 3.11

Assume that $M_t$ is a space of countable pseudocharacter for every $t\in T$ and $a\in M=\prod _{t\in T}M_t$. Then every Lindelöf scattered subspace X of the $\sigma$-product $\sigma (M,a)$ is the union of countably many scattered Eberlein compact subspaces and, in particular, X has the $\varDelta$-property.

Proof

Let $p_t:M\rightarrow M_t$ be the projection map for every $t\in T$ and denote by $X_{\omega }$ the set X with the topology generated by all $G_\delta$-subsets of the space X. Then $X_{\omega }$ is a Lindelöf P-space (see Problem 128 of [17]) and X is a continuous image of $X_{\omega }$. Therefore $p_t(X)$ is also a continuous image of $X_{\omega }$. The Lindelöf P-property of $X_{\omega }$ together with countable pseudocharacter of $p_t(X)$ imply that $p_t(X)$ is countable and hence we can assume, without loss of generality, that $M_t$ is countable for each $t\in T$.

As we have noted in Sect. 2, $\sigma (M,a) = \bigcup \{\sigma _n(M,a): n\in \omega \}$. Hence, it suffices to show that every $X\cap \sigma _n(M,a)$ is the union of countably many Eberlein compact subspaces. Observe that if every space $M_t$ is a finite space with the same size $m\in \omega$, then every $\sigma _n(M,a)$ is a scattered Eberlein compact space (see, for example, [2]). Now choose an enumeration $\{q^t_n: n\in \omega \}$ of the set $M_t$ such that $a(t)=q^t_0$ and let $M^n_t=\{q_i^t: i\le n\}$ for every $t\in T$. If $Q_n=\prod _{t\in T}M^n_t$, then $a\in Q_n$ for each $n\in \omega$ and $\sigma (M,a)= \bigcup \{\sigma (Q_k,a): k\in \omega \}$.

All finite sets $M^n_t$ have the same size, this fact implies that every $\sigma (Q_k,a) =\bigcup \{\sigma _n(Q_k,a): n\in \omega \}$ is represenatble as the union of countably many Eberlein compact subspaces.

Write $\sigma (Q_n,a) = \bigcup \{K_m: m\in \omega \}$, where each $K_m$ is a scattered Eberlein compact space. Theorem 3.7 of [18] implies that $X\cap K_m$ is $\sigma$-compact for every $m\in \omega$ and therefore $X_n= X\cap \sigma (Q_n,a)$ is $\sigma$-compact as well for every $n\in \omega$. Thus, $X=\bigcup _{n\in \omega } X_n$ is the union of countably many scattered Eberlein compact subspaces. Finally, recalling that every scattered Eberlein compact space has the $\varDelta$-property by Theorem 49 of [6] and Theorem 2.1 of [9], we conclude that X is a $\varDelta$-space by Proposition 2.2 of [10]. $\square$

Corollary 3.12

Suppose that X is a scattered compact space. If X embeds in a $\sigma$-product of spaces of countable pseudocharacter, then X is a $\varDelta$-space.

Corollary 3.13

Suppose that X is a scattered Lindelöf space. If X embeds in a $\sigma$-product of real lines, then X is a $\varDelta$-space.

It was proved in [9] that the space $\omega _1+1$ does not have the $\varDelta$-property. This result was strengthened in [11] where it was established that a set $X\subset \omega _1$ is a $\varDelta$-space if and only if it is not stationary. Our next group of results describe the behavior of the $\varDelta$-property in a more general context.

Proposition 3.14

If $S\subset \kappa$ is a stationary subset of an uncountable regular cardinal $\kappa$, then S is not a $\varDelta$-space.

Proof

It follows from Theorem 9 of [15] that there exists a disjoint partition $\mathcal {F}= \{S_n: n\in \omega \}$ of the set S such that every $S_n$ is also stationary. If $\{U_n: n\in \omega \}$ is an open expansion of $\mathcal {F}$ in the space $\kappa$, then $|\kappa \backslash U_n|<\kappa$ for every $n\in \omega$ so there is $\alpha <\kappa$ such that $[\alpha ,\kappa )\subset \bigcap _{n\in \omega } U_n$ and therefore $S\cap \bigcap _{n\in \omega }U_n\ne \emptyset$ which in turn implies that the partition $\mathcal {F}$ has no point-finite open expansion in the space S. $\square$

Corollary 3.15

If X is a monotonically normal $\varDelta$-space, then X must be hereditarily paracompact.

Proof

Assume that a subspace $Y\subset X$ is not paracompact. Since Y is also monotonically normal, we can apply Theorem 4.5 of [3] to see that there is a closed subset F in the space Y which is homeomorphic to a stationary set in an uncountable regular cardinal. Then F is a $\varDelta$-space which is a contradiction with Proposition 3.14. $\square$

Corollary 3.16

If X is a pseudocompact monotonically normal $\varDelta$-space, then X is a compact Fréchet–Urysohn space and $\overline{A}$ is countable whenever A is a countable subset of X.

Proof

The space X must be compact because it is paracompact according to Corollary 3.15. If $A\subset X$ is countable, then $\overline{A}$ is hereditarily Lindelöf being a separable monotonically normal space (see Theorem A of [7]). This, together with compactness of $\overline{A}$, shows that $\chi (\overline{A})\le \omega$. Recalling that $\overline{A}$ is also a $\varDelta$-space, we conclude that $\overline{A}$ is countable (see Proposition 3.5 of [9]).

As X is a compact $\varDelta$-space, it has countable tightness by a result obtained in [10]. If $A\subset X$ and $x\in \overline{A}$, then there is a countable set $B\subset A$ such that $x\in \overline{B}$. But $\chi (\overline{B})\le \omega$ so there exists a sequence $S=\{a_n: n\in \omega \}\subset B$ that converges to x. Therefore S witnesses the Fréchet-Urysohn property of X. $\square$

It is worth noting that the converse of Corollary 3.15 is not true even for linearly ordered spaces: the real line $\mathbb {R}$ is a counterexample. We will show that, if a topology of a space X is generated by a well-order, then the $\varDelta$-property of X is equivalent to it hereditary paracompactness.

Theorem 3.17

If X is a subspace of an ordinal with its order topology, then X has the $\varDelta$-property if and only if it is hereditarily paracompact.

Proof

Since necessity is an immediate consequence of Corollary 3.15, assume that the space X is hereditarily paracompact. Our proof will be by induction on the order type ot(X) of the ordered set X. Observe that any countable space has the $\varDelta$-property so there is nothing to prove if $ot(X)<\omega _1$. Now, assume that $ot(X)=\beta$ and any subspace Y of an ordinal has the $\varDelta$-property whenever $ot(Y)<\beta$. Given any $x\in X$ let $L_x=\{y\in X: y\le x\}$. It is easy to see that $L_x$ is an open neighborhood of the point x and if $Y=\{x\in X: ot(L_x)<ot(X)\}$, then $X\backslash Y$ contains at most one point.

Thus, $\mathcal {L}= \{L_x: x\in Y\}$ is a cover of Y by open subsets with the $\varDelta$-property. By paracompactness of Y, there exists a locally finite open refinement $\mathcal {U}$ of the cover $\mathcal {L}$. Since every element of $\mathcal {U}$ has the $\varDelta$-property, Corollary 3.10 is applicable to conclude that Y is a $\varDelta$-space. Finally, observe that $|X\backslash Y|\le 1$ so X is a $\varDelta$-space by Proposition 2.3 of [9]. $\square$

Corollary 3.18

If X is a pseudocompact GO space with the $\varDelta$-property, then X is countable.

Proof

Observe that every GO space is monotonically normal so X is compact and $t(X)\le \omega$ by Corollary 3.16. Since tightness and character coincide in GO spaces (see Theorem 1.3.1 of [1]), we conclude that $\chi (X) \le \omega$ and hence X is countable by Proposition 3.5 of [9]. $\square$

Any compact $\varDelta$-space must have countable tightness [10]. It is still an open question whether compactness can be replaced with pseudocompactness in this result. However, the situation is quite different if we consider $\sigma$-compact $\varDelta$-spaces.

Example 3.19

For any uncountable cardinal $\kappa$, there exists a $\sigma$-compact $\varDelta$-space X such that $t(X)=\kappa$.

Proof

Let $S=\{x\in \mathbb {D}^\kappa : |x^{-1}(1)|<\omega \}$ be the $\sigma$-product in $\mathbb {D}^\kappa$. It is a consequence of Corollary 2.5 of [10] that S is a $\sigma$-compact $\varDelta$-space. Let $u(\alpha )=1$ for all $\alpha <\kappa$. Then $X=S\cup \{u\}$ is a $\sigma$-compact $\varDelta$-space by Proposition 2.3 of [9]. It is easy to see that the set S and the point u witness that $t(X)=\kappa$. $\square$

The following fact was established in [11].

Proposition 3.20

Let X be a crowded Baire space. If X is $\omega$-resolvable, then it does not have the $\varDelta$-property.

Corollary 3.21

Let X be a pseudocompact $\varDelta$-space. If $c(X)=\omega$, then X is separable and has a dense set of isolated points.

Proof

Let D be the set of isolated points of X; if D is not dense in X, then take a non-empty open set $U\subset X\backslash D$. The space X is $\mathfrak {c}$-resolvable and hence $\omega$-resolvable—this was proved in [12]. As an immediate consequence, the set U is $\omega$-resolvable as well. Since U is a crowded Baire space, it does not have the $\varDelta$-property by Proposition 3.20. However, the $\varDelta$-property is hereditary so we have obtained a contradiction. Therefore D is dense in X and it follows from the Souslin property of X that D is countable so the space X is separable. $\square$

Our last result was originally obtained as an auxiliary step in the proof that any pseudocompact $\varDelta$-space of countable tightness is scattered. After we discovered a different proof for a stronger fact, the theorem below is not needed to study the $\varDelta$-property but we decided to keep it here anyway because it seems to be interesting in itself.

Theorem 3.22

Any pseudocompact crowded space of countable tightness must be $\omega$-resolvable.

Proof

Let X be a pseudocompact crowded space with $t(X)\le \omega$. It is standard to see that X has no countable open sets and hence

(5) if $U\in \tau ^*(X)$ and $A\subset U$ is countable, then $U\backslash A$ is dense in U.

Given any $U\in \tau ^*(X)$ take a countably infinite set $D_0\subset U$. Using the property (5) and countable tightness of X, it is easy to construct by induction a sequence $\{D_n: n\in \omega \}$ of disjoint countable subsets of U such that

(6) $D_0\cup \ldots \cup D_n\subset \overline{D}_{n+1}$ for any $n\in \omega$.

Let $D=\bigcup _{n\in \omega }D_n$ and choose a disjoint family $\{A_n: n\in \omega \}$ of infinite subsets of $\omega$; it is an easy consequence of (6) that $E_n=\bigcup _{i\in A_n}D_n$ is dense in D for any $n\in \omega$. Therefore

(7) every non-empty open subset of X contains an $\omega$-resolvable subspace.

Take a maximal disjoint subfamily $\mathcal {F}$ of the family of all $\omega$-resolvable subsets of X. It is an immediate consequence of (7) that $Y=\bigcup \mathcal {F}$ is dense in X. It is standard that Y is $\omega$-resolvable and hence X is $\omega$-resolvable as well. $\square$

4 Open questions

The $\varDelta$-property is a classical notion that has important applications both in descriptive set theory and functional analysis. The study of $\varDelta$-property in general topological spaces is comparatively recent so there are still numerous interesting open questions about its behavior as can be seen for the list below.

Question 4.1

Suppose that X is a pseudocompact $\varDelta$-space. Is it true that $t(X)\le \omega$?

Dow and Vaughan showed in [4] that for every ordinal $\gamma < {\mathfrak t}^{+}$ (where $\mathfrak t$ is the tower number), there is a maximal almost disjoint family $\mathcal {A}$ on $\omega$ such that the Stone-Čech remainder of the corresponding Mrowka $\Psi$-space $X=\omega \cup \mathcal {A}$ is homeomorphic to $\gamma +1$ with the order topology. Let p be the last point in $\gamma +1$, where $\gamma =\omega _1$. We do not know if the space $X\cup \{p\}$, with the topology inherited from $\beta {X}$, has uncountable tightness at the point p. If this is true, then $X\cup \{p\}$ would be a counterexample to Question 4.1.

Question 4.2

Is it true in ZFC that every countably compact $\varDelta$-space has countable tightness?

Question 4.3

Suppose that X is a compact $\varDelta$-space. Is it true in ZFC that X is sequential?

Question 4.4

Suppose that X is a separable pseudocompact $\varDelta$-space. Must X be scattered?

Question 4.5

Assume that X is a Lindelöf scattered subspace of an Eberlein compact space. Must X be a $\varDelta$-space?

Question 4.6

Is it true that every compact $\varDelta$-space is the countable union of Eberlein compact spaces?

Question 4.7

Is it true in ZFC that every countably compact $\varDelta$-space must be compact?

Question 4.8

Suppose that X is a countably compact $\varDelta$-space. Is it true that $X\times X$ is countably compact?

Question 4.9

Suppose that X is a countably compact $\varDelta$-space. Is it true that every continuous image of X is a $\varDelta$-space?

Question 4.10

Suppose that X is a space and there is a family $\{U_n: n\in \omega \}$ of open $\varDelta$-subspaces of X such that $X=\bigcup _{n\in \omega }U_n$. Is it true that X must be a $\varDelta$-space?