On uniformly continuous maps between function spaces
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.