Alexander’s subbase lemma

Abstract

Lemma (Alexander). Let (X ,T) be a topological space and 3 a subbase of closed sets. If every family of closed sets in 5^ with the finite intersection property has nonempty intersection, then (X , T) is compact. Proof. Recall that (X, T) is compact iff *X \JxeXn(x) [2, Theorem (2.9), Chapter III] and that the monad of x is p(x) f){ *G\x G G, X G G S?} [2, Proposition (1.4) of Chapter III]. Let ae*X. Consider &~ {F\F eS, a G *F}. Then &~ has the finite intersection property and, by assumption, there is a point x such that x G f]{F\F G y}. We show that a g p(x) : if x G G and X G G 5?, then a $. *(X G), by our choice of x, hence a G * G, as required.