On dimensionally restricted maps

Topology and its Applications

2006

Self Cite

Section: Some Preliminary Resultsmentioning

confidence: 99%

Extraordinary dimension of maps

Chigogidze

Topology and its Applications

2006

Self Cite

“…Suppose first that k ≤ n − 1. Since f is σ-perfect and dim f ≤ n, there exist closed subsets F i ⊂ X, i = 1, 2, .., such that dim F i ≤ k for each i and the restriction f [14,Theorem 1.4]. Because each f i is a perfect map, by [9, Proposition 9.1], there exist maps h i : X i → I ℵ 0 embedding all fibers of f i , i ≥ 1.…”

Section: Proofsmentioning

confidence: 99%

Parametric set-wise injective maps

Matsuhashi

Topology and its Applications

2017

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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.

“…Choose a sequence {γ k } k≥1 of open covers of I ℵ 0 with mesh(γ k ) ≤ 1/k, and let ω k = λ −1 (γ k ) for all k. We denote by C (ω k ,0) (X, I n , f ) the set of all maps g ∈ C(X, I n ) with the following property: every z ∈ (f △g)(X) has a neighborhood V z in Y × I n such that (f △g) −1 (V z ) can be represented as the union of a disjoint open in X family refining the cover ω k . According to [17,Lemma 2.5], each of the sets C (ω k ,0) (X, I n , f ), k ≥ 1, is open in C(X, I n ) with respect to the source limitation topology. It follows from the definition of the covers ω k that k≥1 C (ω k ,0) (X, I n , f ) consists of maps g with dim f △g ≤ 0.…”

Section: Proof Of Theorem 13 and Corollary 15mentioning

confidence: 99%

Maps with dimensionally restricted fibers

2011

Colloq. Math.

Self Cite

Abstract. We prove that if f : X → Y is a closed surjective map between metric spaces such that every fiber f −1 (y) belongs to a class of space S, then there exists an F σ -set A ⊂ X such that A ∈ S and dim f −1 (y)\A = 0 for all y ∈ Y . Here, S can be one of the following classes: (i) {M : e − dimM ≤ K} for some CW -complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = {M : dim M ≤ n}, then dim f △g ≤ 0 for almost all g ∈ C(X, I n+1 ).