Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$

Is $(X,\mathcal D)$ compact?

share|cite|improve this question
up vote 4 down vote accepted


The following are equivalent for a Tychonov space $X$:

  1. $X$ is locally compact.
  2. There is a minimal uniformity on $X$.
  3. There is a minimal totally bounded uniformity.
  4. The uniformities form a complete lattice.
  5. The totally bounded uniformities form a complete lattice.

See Shirota On systems of structures of a completely regular space Osaka Math. J. Volume 2, Number 2 (1950), 131-143.

share|cite|improve this answer

Your Answer


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