MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?

share|cite|improve this question
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: – Austin Mohr May 15 '12 at 19:09
up vote 2 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.

share|cite|improve this answer

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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.