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This one is probably simple, but I don't see it yet.

Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?

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    $\begingroup$ If you sometimes have questions like this, get a copy of the book COUNTEREXAMPLES IN TOPOLOGY by Steen and Seebach. There are tables in the back to look up examples with various combinations of properties. $\endgroup$
    – Gerald Edgar
    Commented May 3, 2012 at 12:26
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    $\begingroup$ @AustinMohr's link referenced above seems now to have moved to π-base. $\endgroup$
    – LSpice
    Commented Mar 11 at 17:05
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    $\begingroup$ Specifically to this question: topology.pi-base.org/spaces/… $\endgroup$
    – Steven Clontz
    Commented Mar 11 at 22:31

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Any pseudocompact paracompact Hausdorff space is compact. So $\omega_1$ with the order topology is a counterexample to your question since it is locally compact, pseudocompact (any real valued continuous function is eventually constant) and not compact.

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The long line is one of the standard examples of a space that is not paracompact. It is however Hausdorff, locally compact, and pseudocompact because maps from it to $\mathbb R$ are eventually constant.

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  • $\begingroup$ Thanks. This seem correct, but the example below $\omega_1$ is a bit simpler. $\endgroup$
    – Fred Dashiell
    Commented May 3, 2012 at 12:28

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