Approximation by light maps and parametric Lelek maps

Taras Banakh; Vesko Valov

paper

Approximation by light maps and parametric Lelek maps

arXiv:0801.3107

Abstract

The class of metrizable spaces $M$ with the following approximation property is introduced and investigated: $M \in A P (n, 0)$ if for every $\e>0$ and a map $g\colon\I^n\to M$ there exists a 0-dimensional map $g'\colon\I^n\to M$ which is $\e$-homotopic to $g$ . It is shown that this class has very nice properties. For example, if $M_{i} \in A P (n_{i}, 0)$ , $i = 1, 2$ , then $M_{1} \times M_{2} \in A P (n_{1} + n_{2}, 0)$ . Moreover, $M \in A P (n, 0)$ if and only if each point of $M$ has a local base of neighborhoods $U$ with $U \in A P (n, 0)$ . Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of $n$ -dimensional Lelek maps.

34 pages