Another approach to parametric Bing and Krasinkiewicz maps

Abstract

Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f:X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g:X\to {\mathbb{I}}^{{\aleph }_0}$ embedding each $f^{-1}(y)$ into ${\mathbb{I}}^{{\aleph }_0}$. Then for every $n\ge 1$ the space $C^{\ast }(X,{\mathbb{R}}^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map.

5 pages