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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?

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No, they're not. The standard examples are all completely regular, so uniformisable. e.g.

$\omega_1$ (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).

I don't think that uniform structure has much to do with things like sequential compactness (being a sequential space is independent of that). To go from countable compactness to compactness, Lindelöfness is necessary and sufficient, trivially. This also has nothing to with uniform structure. There we'd have to go to properties like "uniform boundedness" and "completeness", two uniform space notions that together are equivalent to compactness (where completeness is meant in the nets or filter sense, not sequential completeness as in metric spaces). You might want to look into that direction for equivalences.

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  • $\begingroup$ So then is there a class of uniformizable spaces which contain the metrizable spaces as a subset for which three notions are equivalent? $\endgroup$ – Keshav Srinivasan Nov 2 '18 at 4:50
  • $\begingroup$ @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. $\endgroup$ – Henno Brandsma Nov 2 '18 at 5:20

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