On uniformly continuous maps between function spaces

Abstract

In this paper we develop a technique of constructing uniformly continuous maps between function spaces C p (X) endowed with the pointwise topology. We prove that if a space X is compact metrizable and strongly countable-dimensional, then there exists a uniformly continuous surjection from C p ([0, 1]) onto C p (X). We provide a partial result concerning the reverse implication. We also show that, for every infinite Polish zero-dimensional space X, the spaces C p (X) and C p (X) × C p (X) are uniformly homeomorphic. This partially answers two questions posed by Krupski and Marciszewski.