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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 $X = \operatorname{St}(K,\mathscr{U})$.

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

We know that there are some topological spaces which are star countable but not star compact. However I don't know whether star compact implies star countable. Is there a topological space which is star compact but not star countable?

Added: $St(K, \mathscr{U})=\cup\{u\in \mathscr{U}: u \cap K \neq \emptyset\}$

t.b.
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Paul
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  • Could you provide a definition of $\operatorname{St}(K,\mathscr{U})$, please? –  Jan 01 '12 at 09:25
  • $St(K, \mathscr{U})=\cup{u\in \mathscr{U}: u \cap K \neq \emptyset}$ – Paul Jan 01 '12 at 09:49
  • Such a space (if it exists) cannot be normal, because for $T_4$ spaces star compact is equivalent to star finite, so certainly star countable. – Henno Brandsma Jan 01 '12 at 10:46
  • It also hasn't countable extent, and hence it can't be countably compact. However, every star compact space is pseducompact. So the space must be pseudocompact but not countably compact! – Paul Jan 01 '12 at 10:58
  • The definition of $\operatorname{St}(K, \mathscr{U})$ added ad the end of your post does not say what you believe it says. You define a set $V\subseteq\mathscr{U}$ by $V={u\in \mathscr{U}: u \cap K \neq \emptyset}$, then what is $\cup V$... I guess you mean something like $\operatorname{St}(K, \mathscr{U})=\bigcup\limits_{u\in V}u$. – Did Jan 01 '12 at 12:28
  • @Didier Piau Right. However we always define that in topological language ：） – Paul Jan 01 '12 at 12:47
  • In set theory such a notation is widely used – Norbert Jan 01 '12 at 12:48
  • @Norbert: Any reference? – Did Jan 01 '12 at 12:53
  • @Didier: A reference shouldn’t be necessary: that $\cup x=\cup{y:y\in x}$ is bog-standard, at least in set theory. However, Ken Kunen’s Set Theory (North Holland, 1980) defines $\bigcup\mathscr{F}={x:\exists Y\in\mathscr{F}(x\in Y)}$ at the foot of p. 12. – Brian M. Scott Jan 01 '12 at 13:19
  • @Didier: Essentially the same definition is found on p. 11 of Hrbacek & Jech, Introduction to Set Theory (Marcel Dekker, 1978) and on p. 34 of Judy Roitman’s Introduction to Modern Set Theory (Wiley, 1990). Willard’s General Topology notes the usage on p. 2, though he tends to use indices instead. – Brian M. Scott Jan 01 '12 at 13:33
  • @Brian: Thanks. – Did Jan 01 '12 at 15:26

1 Answers1

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Let $X=\Big(\beta\omega_1\times(\omega_2+1)\Big)\setminus\Big((\beta\omega_1\setminus \omega_1)\times\{\omega_2\}\Big)$ as a subspace of $\beta\omega_1\times(\omega_2+1)$; I claim that $X$ is star compact.

Let $\mathscr{U}$ be an open cover of $X$. For each $\xi\in\omega_1$ there are $U_\xi\in\mathscr{U}$ and $\alpha_n\in\omega_2$ such that $$\langle \xi,\omega_2\rangle\in \{\xi\}\times(\alpha_n,\omega_2]\subseteq U_\xi\;.$$ Let $\alpha=\sup_\xi\alpha_\xi<\omega_2$, and let $K=\beta\omega_1\times\{\alpha+1\}$; $K$ is compact, and $$\omega_1\times \{\omega_2\}\subseteq \operatorname{st}(K,\mathscr{U})\;,$$ since $U_\xi\subseteq \operatorname{st}(K,\mathscr{U})$ for each $\xi\in\omega_1$. The ordinal space $\omega_2$ is countably compact, so $\beta\omega_1\times\omega_2$ is countably compact and therefore star finite, and there is a finite $F\subseteq \beta\omega_1\times\omega_2$ such that $\beta\omega_1\times\omega_2\subseteq\operatorname{st}(F,\mathscr{U})$. But then $K\cup F$ is compact, and $\operatorname{st}(K\cup F,\mathscr{U})=X$, as desired.

However, $X$ is not star countable. To see this, let $$\mathscr{U}=\{\beta\omega_1\times\omega_2\}\cup\Big\{\{\xi\}\times(\omega_2+1):\xi\in\omega_1\Big\}\;;$$ $\mathscr{U}$ is certainly an open cover of $X$, but if $C$ is any countable subset of $X$, we can choose $\xi\in\omega_1$ such that $C\cap\big(\{\xi\}\times (\omega_2+1)\big)=\varnothing$, and then $\langle \xi,\omega_2\rangle\notin\operatorname{st}(C,\mathscr{U})$.

This is a modification of Example 2.1 of Yan-Kui Song, On $\mathcal{K}$-Starcompact Spaces, Bull. Malays. Math. Soc. (2) 30(1) (2007), 59-64, which is available as a PDF here.

Brian M. Scott
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