Compactness

I heard that the defn of compactness by filter basis of closed sets,
or the convergence of ultrafilters, is related to the following in some
way, which
I don't know:
Compactness of anything, not only of topological spaces, means " it is
without holes",
where holes are infinite process without end.

Kindly explain if there is at all such concepts, and if there is,
explain it.

.

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