Parametric Bing and Krasinkiewicz maps: revisited

TL;DR

This paper investigates the properties of certain metric spaces related to Bing and Krasinkiewicz maps, showing how these properties are preserved under perfect surjections and applying the results to extensional dimension theory.

Contribution

It establishes that if a metric ANR-space has dense Bing or Krasinkiewicz maps in its function space, then this property is preserved under perfect surjections, with applications to extensional dimension.

Findings

Dense sets of Bing and Krasinkiewicz maps exist in the function space of the ANR-space.

The property of having dense Bing or Krasinkiewicz maps is preserved under perfect surjections.

Applications to theorems concerning extensional dimension are demonstrated.

Abstract

Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitation topology contains a dense $G_\delta$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension.