Skeletally Dugundji spaces

TL;DR

This paper introduces skeletally Dugundji spaces, a new class extending Dugundji spaces, and characterizes them through compactifications, embeddings, and skeletal maps, enriching the understanding of space structures in topology.

Contribution

It defines skeletally Dugundji spaces and establishes their equivalence with several key properties, providing a new framework in topological space theory.

Findings

Skeletally Dugundji spaces are characterized by multiple equivalent conditions.

Every compactification of such a space is co-absolute to a Dugundji space.

They possess a multiplicative lattice of skeletal maps.

Abstract

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. The main result states that the following conditions are equivalent for a given space $X$: (i) $X$ is skeletally Dugundji; (ii) Every compactification of $X$ is co-absolute to a Dugundji space; (iii) Every $C^*$-embedding of the absolute $p(X)$ in another space is strongly $\pi$-regular; (iv) $X$ has a multiplicative lattice in the sense of Shchepin \cite{s76} consisting of skeletal maps.