Approximation by finite sets

Question

I'm reading the book "Topology and Order" by L.Nachbin.

In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes:

[A]lthough these spaces may be infinite, they admit approximation by finite sets (in the sense expressed in precise terms by the covering property of Borel-Lebesgue).

What is the exact formulation of this property? I couldn't find anything relevant when I tried searching these terms (approximation by finite sets and covering property) online.

Any formulations and/or references would be of great help!

Thank you.

Answer 1

For compact metric spaces there is a very precise formulation:

If $X$ is a compact metric space, then for every $\epsilon>0$ there is a finite set $A\subset X$ such that every point of $X$ is within distance less than $\epsilon$ from some point of $A$.

The proof is what Etienne wrote in a comment: take a finite subcover of the open cover $$\{B_\epsilon(x): x\in X\}$$ where $B_\epsilon(x) = \{y\in X: d(y,x)<\epsilon\}$.

For general compact Hausdorff spaces the notion of "approximation" is necessarily vague, since approximation means being "close" to something, and we don't have a metric to quantify the concept of "close". I'd say the following:

If for every point $x\in X$ we designate some neighborhood of $x$ as "small" (based on whatever criterion), then the whole space can be covered by finitely many "small" neighborhoods.

The proof is essentially the same as in the metric space.