Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M ∈ AP ( n , 0 ) if for every ε > 0 and a map g : I n → M there exists a 0-dimensional map g ′ : I n → M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if M i ∈ AP ( n i , 0 ) , i = 1 , 2 , then M 1 × M 2 ∈ AP ( n 1 + n 2 , 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.