On continuous surjections from Cantor set

Abstract

It is a famous result of Alexandroff and Urysohn 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. Namely, if (K, d) is a compact metric space and f and g are two continuous mappings from ∆ onto K, then, for every ε > 0 there exists a homeomorphism φ of ∆ such that d(g(x), f (φ(x)) < ε for all x∆.