# Are the three notions of compactness equivalent for uniform spaces?

A topological space is compact if every open cover has a finite subcover. A topological space is sequentially compact if every sequence has a convergence subsequence. And a topological space is limit point compact if every infinite subspace has a limit point.

For metric spaces, compactness, sequential compactness, and limit point compactness are equivalent. My question is, is the same true for uniform spaces? It's possible since a lot of the properties of metric spaces are really properties of their uniform structure.

Now a topological space is uniformizable if and only if it is completely regular. (Uniformizable means that there exists a uniform structure on the space which induces the topology, and completely regular means that a point and a closed set can be separated using a continuous function.) So another way of asking my question is, are these three notions equivalent for completely regular spaces?

$$\omega_1$$ (countably compact, sequentially compact, not compact).
$$\{0,1\}^\mathbb{R}$$ (compact, countably compact, not sequentially compact)
Of course, compact always implies countably compact in all spaces, we have a "free" implication there. To get the equivalence of sequential and countable compactness (both essentially "countable" notions) we need something like sequentiality of $$X$$ (so first countability of metric spaces is a reason), so that sequentially closed and closed sets are the same. $$T_1$$-ness is also handy there (but this is included in Tychonoffness/uniformisability, at least in my definition).
• @KeshavSrinivasan in $T_1$ sequential spaces we have equivalence between sequential and countable compactness. To get equivalence with full compactness we must add Lindelöfness, e.g. – Henno Brandsma Nov 2 '18 at 5:20