Lelek maps and n-dimensional maps from compacta to polyhedra

Chain recurrent sets of generic mappings on compact spaces

“…This fact also follows from a more general result on Lelek maps [16, 2.7]. It was proved in [16] that class 0-DIM contains all Menger manifolds and n-dimensional ANR-compacta with piecewise embedding dimension ped = n. We will not work with ped here. Instead, extending methods from [18], we are going to show that finite products of graphs are in 0-DIM.…”

Approximation by light maps and parametric Lelek maps

“…Let H be the set from the proof of Theorem 3.1. To show that H is dense in C(X, M) equipped with the uniform convergence topology, we used an idea from the proof of [12,Corollary 2.8]. According to [4], for any ε > 0 there exists an n-dimensional polyhedron P ⊂ M of piecewise embedding dimension n and maps u : M → P and v : P → M such that u is a retraction, v is 0-dimensional, v • u is ε/2-close to the identity id M .…”

“…According to [4], for any ε > 0 there exists an n-dimensional polyhedron P ⊂ M of piecewise embedding dimension n and maps u : M → P and v : P → M such that u is a retraction, v is 0-dimensional, v • u is ε/2-close to the identity id M . Since every ANR of piecewise embedding dimension n has the AP (n, 0)-property (see [12,Propsition 2.1]), according to Theorem 3.1, for every g ∈ C(X, M) there is g ′ : X → P such that g ′ is δ-close to u • g and g ′ |f −1 (y) is an (m − n)-dimensional Lelek map for all y ∈ Y . Here δ > 0 is chosen such that dist(v(x), v(y)) < ε/2 for any x, y ∈ P which are δ-close.…”

“…There is a compact X ∈ AR with X ∈ AP (∞, 0) such that X × I is homeomorphic to the Hilbert cube but X contains no proper ANR-subspace of dimension ≥ 2. Now, we consider the spaces with piecewise embedding dimension n. According to [12], a map h : P → M from a finite polyhedron P is said to be a piecewise embedding if there is a triangulation T of P such that h embeds each simplex σ ∈ T and h(P ) is an ANR. For a space M, the piecewise embedding dimension ped(M) is the maximum k such that for any ε > 0 and any map g : P → M from a finite polyhedron P with dim P ≤ k there exists a piecewise embedding g ′ : P → M which is ε-close to g. If y ∈ M, then ped y (M) is the maximum of all ped(U), where U is a neighborhood of y in M. Obviously, ped(M) ≤ min{ped y (M) : y ∈ m}.…”

“…Lelek [15] constructed such a map from I n+1 onto a dendrite. Levin [17] improved this result by showing that the space C(X, I n ) of all maps from X into I n contains a dense G δ -subset consisting of (m − n)dimensional Lelek maps for any compactum X with dim X = m ≤ n. Recently Kato and Matsuhashi [12] established a version of the Levin result with I n replaced by more general class of spaces. This class consists of complete metric ANR-spaces having a piecewise embedding dimension ≥ n.…”