The structure of (exactly) 2-to-1 maps on metric compacta

Abstract

It is shown that the domain of a 2 2 -to- 1 1 continuous map f f contains two disjoint open sets V V and V ∧ {V^ \wedge } such that f ( V ) = f ( V ∧ ) f\left ( V \right ) = f\left ( {{V^ \wedge }} \right ) and f ↾ V f \upharpoonright V is a homeomorphism from V V onto a dense open subset of the image of f f . The restriction of f f to V ∪ V ∧ V \cup {V^ \wedge } is the first "fold", and f f is shown to be the union of a finite or transfinite sequence of similar folds.