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Bijections of scattered spaces onto compact Hausdorff. (English) Zbl 0676.54027

In 1935, S. Banach asked when a metric space can have a continuous bijection onto a compact metric space [Colloq. Math. 1, p. 150 (1948)]. M. Katětov [Colloq. Math. 2, 30-33 (1949; Zbl 0039.186)] gave a partial answer: a countable regular space has a continuous bijection onto a compact metric space iff it is scattered; i.e., iff every non-empty subset of the space has an isolated point. The authors generalize Katětov’s result by showing that every scattered and hereditarily paracompact space has a continuous bijection onto a compact Hausdorff space.
Reviewer: D.L.Grant

MSC:

54D30 Compactness
54F99 Special properties of topological spaces
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C05 Continuous maps
54E45 Compact (locally compact) metric spaces

Citations:

Zbl 0039.186
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