Approximation by light maps and parametric Lelek maps
Abstract
The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈ AP(n,0) if for every >0 and a map gn M there exists a 0-dimensional map g'n M which is -homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈ AP(ni,0), i=1,2, then M1× M2∈ AP(n1+n2,0). Moreover, M∈ AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈ 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.
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