Theorem

An infinite subsetof a discrete spaceis not compact.

Proof

Letbe a discrete topological space and letbe an infinite subset ofConsider the set of singleton sets

It is an open cover ofbecause andis open inbecause every singleton set is open in the discrete topology.

No proper subset ofis an open cover of A. Since A is infinite so isand the open cover contains no fininite subcover andis not compact.

Obviously ifis a finite subset of a discrete space it is compact.