A selection theorem for C-spaces

On dimensionally restricted maps

“…He also proved that if f : X → Y is a k-dimensional map between metric compacta, then the following two conditions are equivalent: (i) there exists a map g : X → I k with dim(f g) = 0; (ii) f admits an approximation by k-dimensional simplicial maps (see [32] for this notion).…”

Continuous selections and $C$-spaces

“…We conclude this section with the following special case of a result actually proved by Uspenskij [16]. …”

“…A space S is called aspherical [16] if every continuous map f : P → S from a compact polyhedron P can be continuously extended over the cone of P. Note that S is aspherical if and only if any continuous image of an n-sphere, n ∈ N, in S is contractible in S, i.e. when S is C n for every n ∈ N. Finally, we say that a set-valued map Φ :…”

“…We apply a recent characterization of paracompact C-spaces given by Uspenskij [16] to obtain a selection theorem of Michael's type characterizing this class of spaces. More precisely, the following theorem will be proved.…”

Open maps having the Bula property

“…It is based on an application of Uspenskij's selection theorem [23] that under the assumptions of Theorem 1.1 there exists a continuous function g : X → [0, 1] which is nonconstant on each fiber of f (see Lemma 2.1). The proof is then accomplished in Section 3 relying on a "parametric" version of an idea in the proof of [13, Theorem 1.3].…”