The following maximization result makes it very explicit how one can use compactness to transfer results about finite sets to compact sets.

Let X be a compact topological space and P be a (strict) partial order on X. Assume that $L_x=${$y\in X:y P x$} is open for all x. Then there exists a P-maximal element.

Proof: Suppose not. Then the family of all $L_x$ covers X. By compactness, there is a finite subcover, so X is covered by ${L_x}_1,{L_x}_2,\ldots,{L_x}_n$. So the finite set containing $x_1,\ldots,x_n$ has no P-maximal element, which is impossible.