Another approach to parametric Bing and Krasinkiewicz maps

TL;DR

This paper presents a new proof for the existence and density of parametric Bing and Krasinkiewicz maps using Pasynkov's factorization theorem, with implications for maps between paracompact spaces.

Contribution

It introduces a novel approach to establishing the density of parametric Bing and Krasinkiewicz maps via a short proof leveraging Pasynkov's factorization theorem.

Findings

Dense set of maps with Bing and Krasinkiewicz properties in function spaces

Applicable to surjective maps with compact fibers and embeddings into leph_0 spaces

Extends understanding of the structure of function spaces with specific topological properties

Abstract

Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g\colon X\to\mathbb I^{\aleph_0}$ embedding each $f^{-1}(y)$ into $\mathbb I^{\aleph_0}$. Then for every $n\geq 1$ the space $C^*(X,\mathbb R^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map.