On Continuous Surjections from Cantor Set

Abstract

It is a famous result of Alexandroff and Urysohn [1] that every compact metric space is a continuous image of Cantor set ∆. In this short note we complement this result by showing that a certain “uniqueness ” property holds. Given topological spaces X and Y, let C(X,Y) denote the collection of all continuous mappings from X to Y. If Y is metrized by d, one can endow C(X,Y) with the uniform metric dist(f, g) = sup x∈X d(f(x), g(x)). It is clear that if ϕ is a homeomorphism of X, the map f 7 → ϕ ◦ f is an isometry of C(X,Y). Our result is the following. Theorem. Let K be a compact metric space and let f and g be two continuous mappings from ∆ onto K. For every ε> 0 there exists a homeo-morphism ϕ of ∆ such that dist(g, f ◦ ϕ) < ε. Before going into the proof, let us fix some notations. We regard the points of ∆ = {0, 1}N as functions x: N → {0, 1} and we write ∆(n) for the set of all two-valued functions y: {1, 2,..., n} → {0, 1}. Given y ∈ ∆(n), put ∆y = {x ∈ ∆ : x(k) = y(k), 1 ≤ k ≤ n}. Obviously ∆y is homeomorphic to ∆. The n-th standard decomposition of ∆ is the partition of ∆ into the 2n clopen sets

Cite this paper

@article{sanchez2004continuous,
  title = {{On Continuous Surjections from Cantor Set}},
  author = {Sánchez, Félix Cabello},
  journal = {Extracta Mathematicae},
  volume = {19},
  number = {3},
  pages = {335--338},
  year = {2004}
}

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