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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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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. –  Gerald Edgar May 3 '12 at 12:26
    
There is a searchable version of the table in the book here: www.austinmohr.com/spacebook. –  Austin Mohr May 15 '12 at 19:09
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2 Answers

up vote 1 down vote accepted

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