Homeomorphisms between Cantor sets

  • Edwin E. Moise
Part of the Graduate Texts in Mathematics book series (GTM, volume 47)

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 1C 2 can be extended to give a homeomorphism R 2R 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 XX, PQ. (This means that every trivial homeomorphism of the type h: {P}↔{Q} can be extended.)

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Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Edwin E. Moise
    • 1
  1. 1.Department of MathematicsQueens College, CUNYFlushingUSA

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