Pure quotients and Morita's theorem for $k_{\omega}-spaces

TL;DR

This paper extends Morita's theorem to pure quotient maps in $k_$-spaces, clarifying their structure and addressing a longstanding open question about non-closed quotient maps.

Contribution

The authors introduce pure quotient maps, extend Morita's theorem to these, and demonstrate that every $k_$-space arises from such a map, providing a comprehensive structural understanding.

Findings

Extended Morita's theorem to pure quotient maps

Every $k_$-space is the image of a pure quotient map

Clarified the structure of arbitrary $k_$-spaces

Abstract

A $k_\omega$-space $X$ is a Hausdorff quotient of a locally compact, $\sigma$-compact Hausdorff space. A theorem of Morita's describes the structure of $X$ when the quotient map is closed, but in 2010 a question of Arkhangel'skii's highlighted the lack of a corresponding theorem for non-closed quotient maps (even from subsets of $\mathbb{R}^n$). Arkhangel'skii's specific question had in fact been answered by Siwiec in 1976, but a general structure theorem for $k_\omega$-spaces is still lacking. We introduce pure quotient maps, extend Morita's theorem to these, and use Fell's topology to show that every quotient map can be 'purified' (and thus every $k_\omega$-space is the image of a pure quotient map). This clarifies the structure of arbitrary $k_\omega$-spaces and gives a fuller answer to Arkhangel'skii's question.