Take the 2-minute tour ×  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|improve this question
add comment  1 Answer  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|improve this answer add
comment  Your Answer