Geometric Topology in Dimensions 2 and 3 pp 83-90 | Cite as

# 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.)

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