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?