Range-preserving AE(0)-spaces

Abstract

All spaces here are Tychonoff spaces. The class AE(0) consists of those spaces which are absolute extensors for compact zero-dimensional spaces. We define and study here the subclass AE(0) rp , consisting of those spaces for which extensions of continuous functions can be chosen to have the same range. We prove these results. If each point of T ∈ AE(0) is a G δ-point of T , then T ∈ AE(0) rp. These are equivalent: (a) T ∈ AE(0) rp ; (b) every compact subspace of T is metrizable; (c) every compact subspace of T is dyadic; and (d) every subspace of T is AE(0). Thus in particular, every metrizable space is an AE(0) rp-space.