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

Suppose that $\{X_n\}_{n\in \mathbb{N}}$ is an increasing sequence of compact Hausdorff spaces and $X$ is their union, equipped with the finest topology under which all the inclusion maps are continuous. Is it well-known that $X$ is realcompact? Is there a reference or an easy argument for this fact? Thank you very much.

share|cite|improve this question
After chapter 4, Rings of Continuous Functions' apparently assumes all spaces are completely regular. In particular the general claim that all Lindelof spaces are realcompact' appears incorrect. For example every non Hausdorff finite space is Lindelof, but no such space is realcompact. Given the hypothesis of the orginal question, can anyone clarify the following: 1) Must X be Hausdorff? 2) If X is Hausdorff, must X be regular? – Paul Fabel Oct 17 '12 at 15:58

Clearly your $X$ is Lindelöf, so it follows from Theorem 8.2 in Gillman & Jerison, Rings of Continuous Functions: Every Lindelöf space is realcompact.

share|cite|improve this answer

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.