The structure of k_{\omega} spaces and a question of Arhangel'skii's

TL;DR

This paper addresses a longstanding open question about the structure of k_{}-spaces by extending existing theorems and introducing pure quotient maps, ultimately providing a more complete understanding of these spaces.

Contribution

It introduces pure quotient maps, extends Morita's theorem to these, and uses Fell's topology to clarify the structure of k_{}-spaces, filling a gap in the existing theory.

Findings

Every quotient map onto a k_{}-space can be 'purified'

Extended Morita's theorem to pure quotient maps

Provided a fuller answer to Arhangel'skii's question

Abstract

In 2010 a question of Arhangel'skii's highlighted a gap in the knowledge of k_{\omega}-spaces. His specific question had in fact been answered by Siwiec in 1976, but the highlighted gap still remains. We introduce the simple idea of pure quotient maps, extend Morita's theorem to these, and use Fell's topology to show that every quotient map onto a k_{\omega}-space can be 'purified'; and thus fill the gap, elucidate the structure of k_{\omega}-spaces, and obtain a fuller answer to Arhangel'skii's question.