Compact subsets of CW complexes lie in finite subcomplexes

A compact subset of a CW complex is contained in a finite CW complex that is a subcomplex of the original complex. Otherwise choose one point in each of infinitely many cells met by the subset. The closure-finiteness axiom makes each closed cell meet the chosen set in finitely many points. The weak topology axiom then makes the chosen set, and each of its subsets, closed. It is consequently an infinite closed discrete subspace of the compact subset, a contradiction. Only finitely many cells are met; their closures form a finite subcomplex containing the subset.