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 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 ndimensional Lelek maps.

Spaces with fibered approximation property in dimension n

“…Proof. The lemma follows from the proof of [2,Proposition 3.3]. For completeness, we provide the arguments.…”

“…Proof. We follow the construction from the proof of [2,Proposition 3.4]. Fix a simplicially factorizable map g ∈ C(X, M) and ǫ ∈ C(X, (0, 1]).…”

“…The class of AP (n, 0)-spaces was introduced by the authors in [2]: we say that a metrizable space M has the AP(n, 0)-approximation property (br., 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. Next proposition provides a wide class of spaces with the FAP(n)-property. Proposition 4.1.…”

“…Denote by π the projection π : I m × I n → I m . Since M 1 has the AP(n, 0)-property, the map g 1 : I m ×I n → M 1 can be homotopically approximated by a map h 1 : I m ×I n → M 1 such that all restrictions h 1 |({z} × I n ), z ∈ I m , have 0-dimensional fibers, see [2,Theorem 1.1]. This means that the diagonal product π△h 1 : I m × I n → I m × M 1 is a 0-dimensional map.…”