Geodesic
Pure quotients and Morita’s theorem for $k_{\omega }$-spaces
Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 582-597

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DOI

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.
DOI : 10.4153/S0008439521000515
Mots-clés : kω-space, Morita’s theorem, pure quotient 
Lazar, Aldo J.; Somerset, Douglas W.B. Pure quotients and Morita’s theorem for $k_{\omega }$-spaces. Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 582-597. doi: 10.4153/S0008439521000515
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     title = {Pure quotients and {Morita{\textquoteright}s} theorem for $k_{\omega }$-spaces},
     journal = {Canadian mathematical bulletin},
     pages = {582--597},
     year = {2022},
     volume = {65},
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     doi = {10.4153/S0008439521000515},
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