On uniformly continuous maps between function spaces

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.