TITLE: Is a minimal Hausdorff uniformity compact?
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: No.
The following are equivalent for a Tychonov space $X$:

$X$ is locally compact.
There is a minimal uniformity on $X$.
There is a minimal totally bounded uniformity.
The uniformities form a complete lattice.
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.