Abstract
AbstractA $k_{\omega }$ -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.
Highlights
A kω-space—or a hemicompact k-space—is a Hausdorff space which is the image of a locally compact, σ-compact Hausdorff space under a quotient map
The class of kω-spaces was extensively studied from the 1940s and the literature of the subject contains some famous names: R
When a kω-space X is the image of a general quotient map q, the set L can still be characterized by the biquotient property and L ∪ F by pseudoopenness, but there is no explicit extension of Morita’s theorem relating these properties to q being locally closed or attempting to describe the sets F and N
Summary
A kω-space—or a hemicompact k-space—is a Hausdorff space which is the image of a locally compact, σ-compact Hausdorff space under a quotient map. L is obviously an open set, and Morita and Arkhangel’skii (Theorems 2.1 and 2.3) showed that if X is the image of a closed quotient map L is dense, F is discrete, and N is empty. When a kω-space X is the image of a general (nonclosed) quotient map q, the set L can still be characterized by the biquotient property and L ∪ F by pseudoopenness (definitions below), but there is no explicit extension of Morita’s theorem relating these properties to q being locally closed or attempting to describe the sets F and N. The authors encountered pure quotient maps in work on Glimm spaces of C∗algebras, where they occur naturally They look less obvious when translated into a topological context, but perhaps that is part of their wider interest
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