Does every net has a convergent subnet in a compact uniform space?

Question

We know that

Result 1: In a metric space compactness implies sequential compactness, that is, if $(X,d)$ is a compact metric space then every sequence $\{x_n\}$ in $X$ has a convergent subsequence.

In my proof of Result 1, I utilized Cantor Intersection Theorem which is not available for Nets.

I wonder whether an analogous result hold in an uniform space for nets, that is,

If $(X,\mathcal U)$ is an compact uniform space and $(S_n)_{n\in D},D$ being a directed set, is a net in it then can we conclude that there exists a convergent subnet of $(S_n)_{n\in D}?$

Please help!

Answer 1

The answer is positive, see compactness in https://en.wikipedia.org/wiki/Net_(mathematics)#Characterizations_of_topological_properties

Answer 2

I add this as an additional answer, since I guess the OP meant by subnet something different than the meaning of the Wikipedia article cited by Yuval Peres.

Please check here for the definition of subnet as meant by Wikipedia. For instance, as is mentioned in this article, a subnet of a sequence need not be a sequence. I guess the OP meant by subnet a net indexed by a cofinal subset of the index set. Is this correct?

Indeed, the Stone-Cech compactification of the integers does not contain any non-trivial sequence. In particular, the sequence $1, 2, ...$ does not contain a convergent subsequence, which gives a negative answer to the question, if my guess was correct.

BTW, convergence and compactness are topological concepts. Furthermore, each compact space has a uniform structure, which induces its topology. Hence, the question should be asking for compact topological spaces rather than for uniform spaces.