In my earlier post on Tychonoff’s theorem, I talked about the original proof, based on the following characterization of compactness which is due to Kuratowski.
Definition Let \(S\) be a subset of a topological space \(X\). A point \(x\in X\) is a point of perfection of \(S\) if, for every neighborhood \(U\) of \(x\), the set \(U\cap S\) has the same cardinality as \(S\).
Lemma (Kuratowski) A topological space \(X\) is compact if and only if every infinite subset has a point of perfection.
Now I will give the proof of this characterization (again following Wright).