# Does locally compact plus pseudocompact imply paracompact?

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