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How to prove that every pseudocompact, paralindelöf Tychonoff space is compact?

  • pseudocompact = every continuous real-valued function is bounded.
  • paralindelöf = every open cover has a locally countable open refinement.

Thanks for any help:)

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    I'd try playing around here http://austinmohr.com/home/?page_id=1462012-06-27
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    I can't open it.2012-06-27
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    I have no problem with the site. Unfortunately, paralindelof is not an option, however they suggest the cocountable topology (http://en.wikipedia.org/wiki/Cocountable_topology) as a non-compact, pseudo compact, Lindelof space. Does that help?2012-06-27
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    It's not Tychonoff:)2012-06-27
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    Sorry I only read the title not the body of your problem.2012-06-27

1 Answers 1

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Let $X$ be a para-Lindelöf, pseudocompact Tikhonov space. To show that $X$ is compact, it suffices to show that $X$ is Lindelöf, since a Lindelöf Tikhonov space is normal, a normal pseudocompact space is countable compact, and a countably compact Lindelöf space is compact.

Let $\mathscr{U}$ be an open cover of $X$, and let $\mathscr{R}$ be a locally countable open refinement of $\mathscr{U}$; I’ll show that $\mathscr{R}$ is countable, from which it follows immediately that $\mathscr{U}$ has a countable subcover.

Let $\mathscr{V}$ be an open cover of $X$ such that each $V\in\mathscr{V}$ meets only countably many members of $\mathscr{R}$, and let $\mathscr{W}$ be a locally countable open refinement of $\mathscr{V}$. Note that since $\mathscr{W}$ refines $\mathscr{V}$, each $W\in\mathscr{W}$ meets only countably many members of $\mathscr{R}$.

Suppose that $\langle G_n:n\in\omega\rangle$ is an infinite sequence of non-empty open sets such that $$G_n\subseteq X\setminus\bigcup_{k for each $n\in\omega$. (Here $\operatorname{st}(V_k,\mathscr{W})=\bigcup\{W\in\mathscr{W}:W\cap V_k\ne\varnothing\}$.) For $n\in\omega$ let $$H_n=\bigcup_{k\ge n}G_k\;,$$ and let $\mathscr{H}=\{H_n:n\in\omega\}$. $\mathscr{H}$ is an open filterbase in a pseudocompact Tikhonov space, so it clusters at some $x\in X$. Pick any $W\in\mathscr{W}$ such that $x\in W$; $x\in\bigcap_{n\in\omega}\operatorname{cl}H_n$, so $W\cap H_n\ne\varnothing$ for each $n\in\omega$. This implies that there are $m,n\in\omega$ such that $m

It follows that any attempt to construct such a sequence recursively must halt after only finitely many $G_n$ have been chosen. Thus, there is a finite sequence $\langle G_0,\dots,G_m\rangle$ of open sets such that $(1)$ holds for $n$\mathscr{R}$, so $\mathscr{R}$ must be countable.

This is a result of Dennis Burke and Sheldon Davis. Note that para-Lindelöf cannot be weakened to meta-Lindelöf: a bit earlier I constructed an example (assuming CH) of a meta-Lindelöf, pseudocompact Tikhonov space that is not compact.

Dennis K. Burke and Sheldon Davis, Pseudocompact paralindelöf spaces are compact, Abstracts Amer. Math. Soc. 3 (1982), 213.

Brian M. Scott, Pseudocompact, metacompact spaces are compact, Top. Procs. 4 (1979), 577-587. link

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    Congratulations on your gold specialist badge in [general topology](http://math.stackexchange.com/badges/213/general-topology).2012-06-30
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    Another realted paper: W.Stephen Watson: A pseudocompact meta-lindeöf space which is not compact, Topology and its Applications 20 (1985) 237-243; DOI: [10.1016/0166-8641(85)90091-4](http://dx.doi.org/10.1016/0166-8641(85)90091-4). *We settle the remaining question of whether pseudocompact is compatible with meta-Lindelöf by constructing, in ZFC, a pseudocompact meta-Lindelöf space which is not compact.*2012-06-30