Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge 2^{\aleph_0}$.

But I haven't been able to find anything sensible about whether dependent choice is needed to prove the weaker statement that $|X| \not\le \aleph_0$. seems to try to answer the question, but it falls short. Choosing points is not a problem, but choosing open sets is.

share|cite|improve this question
Oh good! This is better than sleep! gets out his pen and paper... – Asaf Karagila May 7 '13 at 0:59
(In my deleted answer, to those who can read it, there are a couple of mistakes that I have yet to correct. When those are cleared up, I'll fix and undelete it.) – Asaf Karagila May 7 '13 at 20:33
I avoid using the built in chat system. Feel free to send me an email, my address is not very difficult to find... – Asaf Karagila May 19 '13 at 17:14

Your Answer


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

Browse other questions tagged or ask your own question.