On uniformly continuous maps between function spaces

TL;DR

This paper develops methods for constructing uniformly continuous maps between function spaces, proving surjections from Cp([0,1]) onto Cp(X) for certain spaces, and exploring uniform homeomorphisms in zero-dimensional spaces, addressing open questions.

Contribution

It introduces techniques for creating uniformly continuous maps between function spaces and provides new results on their surjectivity and homeomorphism properties, partially answering existing open questions.

Findings

Existence of uniformly continuous surjections from Cp([0,1]) onto Cp(X) for compact metrizable strongly countable-dimensional spaces.

Spaces Cp(X) and Cp(X) x Cp(X) are uniformly homeomorphic for infinite Polish zero-dimensional spaces.

Partial results on the reverse implication of the main surjection theorem.

Abstract

In this paper we develop a technique of constructing uni- formly continuous maps between function spaces Cp(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 contin- uous surjection from Cp([0,1]) onto Cp(X). We provide a partial result concerning the reverse implication. We also show that, for every infinite Polish zero-dimensional space X, the spaces Cp(X) and Cp(X) x Cp(X) are uniformly homeomorphic. This partially answers two questions posed by Krupski and Marciszewski.