On uniformly continuous maps between function spaces
Volume 246 / 2019
Fundamenta Mathematicae 246 (2019), 257-274
MSC: Primary 54C35; Secondary 54E15.
DOI: 10.4064/fm647-10-2018
Published online: 17 May 2019
Abstract
We develop a technique of constructing uniformly continuous maps between function spaces $C_p(X)$ endowed with the pointwise topology. We prove that if $X$ 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 converse. We also show that, for every infinite Polish zero-dimensional space $X$, the spaces $C_p(X)$ and $C_p(X) \times C_p(X)$ are uniformly homeomorphic.
