Parametric set-wise injective maps

Abstract

We introduce the notion of set-wise injective maps and provide results about fiber embeddings. Our results improve some previous results in this area. IntroductionAll spaces in the paper are assumed to be metrizable and all maps continuous. Unless stated otherwise, any function space C(X, M) is endowed with the source limitation topology. This topology, known also as the fine topology, was introduced in [17] and has a base at a given g ∈ C(X, M) consisting of the setswhere ̺ is a fixed compatible metric on M and ε : X → (0, 1] runs over continuous functions into (0, 1]. The symbol ̺(h, g) < ε means that ̺ h(x), g(x) < ε(x) for all x ∈ X. The source limitation topology doesn't depend on the metric ̺ [7] and has the Baire property provided M is completely metrizable [8]. Obviously, this topology coincides with the uniform convergence topology when X is compact.We say that a space M has the m-DD {n,k} -property if any two maps f : I m × I n → M, g : I m × I k → M can be approximated by maps f ′ : I m × I n → M and g ′ : I m × I k → M, respectively, such that f ′ ({z} × I n ) ∩ g ′ ({z} × I k ) = ∅ for all z ∈ I m . Obviously, if M has the m-DD {n,k} -property, then it also has the m ′ -DD {n ′ ,k ′ } -property for all m ′ ≤ m, n ′ ≤ n and k ′ ≤ k. The 0-DD {n,k} -property coincides with the well known disjoint (n, k)-cells property. The m-DD {n,k} -property is very similar to the m-DD {n,k} -property introduced in [1, Definition 5.1], where it is required for any open cover U of M the maps f, g to be approximated by maps f ′ , g ′ such that f ′ , g ′ are U-homotopic to f and g, respectively and f ′ ({z} × I n ) ∩ g ′ ({z} × I k ) = ∅ for all z ∈ I m . For example, it follows from [1, Proposition 5.6 and Theorem 1991 Mathematics Subject Classification. Primary 54F15; Secondary 54F45, 54C10.