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:)
How to prove that every pseudocompact, paralindelöf Tychonoff space is compact?
Thanks for any help:)
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
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
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