Almost homeomorphic mappings of compact spaces

A continuous surjection from a compact space to itself is an almost homeomorphism if it is one-to-one except on a finite subset. It is shown that every almost homeomorphism on a simply connected compact space is a homeomorphism. Similar questions are examined for totally disconnected and related spaces and for products. The question of characterizing those spaces on which every almost homeomorphism is a homeomorphism is motivated by the study of the Fredholm spectrum of composition operators in spaces of continuous functions.