Question

There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.

Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $G_\delta$-diagonal?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. More on star compactness see here.

My question is this: Is always the cardinality of such regular (Tychonoff) star compact space less than $2^{\omega_0}$? See the related link here.

Answer 1

We shall show that every metrizable space of cardinality greater than $2^{\aleph_{0}}$ is not star-compact. Assume that $(X,d)$ is a metric space with $|X|>2^{\aleph_{0}}$. Color the collection of pairs $\{{x,y}|x,y\in X,x\neq y\}$ where ${x,y}$ has color $n$ for some natural number $n$ with $d(x,y)>\frac{1}{n}$. Then by the Erdos-Rado theorem there is a uncountable subset $A$ and some $n$ such that if $a,b\in A,a\neq b$, then ${a,b}$ has color $n$ (i.e. $A$ is homogeneous). Therefore $d(a,b)>\frac{1}{n}$ for distinct $a,b\in A$. We may therefore assume that $d(a,b)>8$ for $a,b\in A,a\neq b$. Now consider the open cover $\mathcal{U}=\{B_{2}(a)|a\in A\}\cup(\bigcup_{a\in A}\overline{B_{1}(a)})^{c}$. Let $K\subseteq X$ be compact. Then since $K$ is totally bounded, the set $K$ can intersect only finitely many balls $B_{2}(a)$ for $a\in A$. Therefore $St(K,\mathcal{U})$ can only contain finitely many points in $A$, so $St(K,\mathcal{U})\neq X$. Therefore $X$ is not star-compact.