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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?

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up vote 4 down vote accepted

No.

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.

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