Extension dimensional approximation theorem

Abstract

Let $L$ be a countable CW-complex and $F:X\to Y$ be upper semicontinuous $U{V}^{[L]}$-valued mapping of a paracompact space $X$ to a complete metric space $Y$. We prove that if $X$ is a C-space of extension dimension $\ed X \le [L]$, then $F$ admits single-valued graph approximations. For $L=S^n$ our result implies well-known approximation theorem for $U{V}^{n-1}$-valued mappings of $n$-dimensional spaces. And for $L=\{\mathrm{point}\}$ our theorem implies a theorem of Ancel on approximations of $U{V}^{\infty }$-valued mappings of C-spaces.

7 pages, final version, minor corrections