Homeomorphisms between Cantor sets

Abstract

By a Cantor set we mean a compact metrizable space in which every point is a limit point, and which is totally disconnected, in the sense that the only connected subsets are formed by single points. (The prototype is the “middle-third” Cantor set in R. See Problem set 10). In the following section we shall show that if C 1 and C 2 are Cantor sets in R 2, then every homeomorphism h: C 1↔C 2 can be extended to give a homeomorphism R 2↔R 2. This is a very strong homogeneity property of R 2. More generally, a topological space [X, O] is homogeneous if for every two points P, Q of X there is a homeomorphism X↔X, P↦ Q. (This means that every trivial homeomorphism of the type h: {P}↔{Q} can be extended.)