The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

Abstract

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.