Approximation by light maps and parametric Lelek maps

TL;DR

This paper introduces and studies a class of metrizable spaces with an approximation property related to 0-dimensional maps, exploring their properties and implications for Lelek maps.

Contribution

It defines the AP(n,0) class of spaces, investigates its properties, and generalizes results on residual sets of n-dimensional Lelek maps.

Findings

AP(n,0) spaces are closed under products.

Local base neighborhoods in AP(n,0) spaces also belong to AP(n,0).

Results extend to the existence of residual sets of n-dimensional Lelek maps.

Abstract

The class of metrizable spaces $M$ with the following approximation property is introduced and investigated: $M\in AP(n,0)$ 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 $M_i\in AP(n_i,0)$, $i=1,2$, then $M_1\times M_2\in AP(n_1+n_2,0)$. Moreover, $M\in AP(n,0)$ if and only if each point of $M$ has a local base of neighborhoods $U$ with $U\in AP(n,0)$. 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.