Universal Cell-Like Maps

Abstract

The main results of the paper are the following: Theorem. Suppose $n \leq \infty$. There is a cell-like map $f:X \to Y$ of complete and separable metric spaces such that $\dim X \leq n$, and for any cell-like map $f’ :X’ \to Y’$ of (complete) separable metric spaces with $\dim X’ \leq n$ there exist (closed) embeddings $i:Y’ \to Y$ and $j:X’ \to {f^{ - 1}}(i(Y’ ))$ such that $fj = if’$. Corollary. Suppose $n < \infty$. There is a complete and separable metric space Y such that ${\dim _{\mathbf {Z}}}Y \leq n$, and any (complete) separable metric space $Y’$ with ${\dim _{\mathbf {Z}}}Y’ \leq n$ embeds as a (closed) subset of Y.