Pure quotients and Morita’s theorem for -spaces

Abstract

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 nonclosed 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.