On finite-dimensional maps

Abstract

Let $f$ : $X\rightarrow Y$ be a perfect surjective map of paracompact spaces. It is shown that if $Y$ is a C-space (resp., $\dim Y\leq n$ and $\dim f\leq m)$ , then the function space $C(X, I^{\infty})$ (resp., $C(X,$ $I^{2n+1+m})$) equipped with the source limitation topology contains a dense $G_{\delta}$-set $\mathscr{H}$ such that $f\times g$ embeds $X$ into $Y\times I^{\infty}$ (resp., into $Y\times I^{2n+1+m}$) for every $g\in \mathscr{H}$. Some applications of this result are also given.