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Selectors and scattered spaces. (English) Zbl 0973.54010
The paper develops the research area initiated in [R. Engelking, R. W. Heath, and E. Michael, Invent. Math. 6, 150-158 (1968; Zbl 0167.20504)]. Let \(X\) be a non-Archimedean space in which every countable subset is closed. Then the following conditions are proved to be equivalent: (i) \(X\) is scattered; (ii) \(X\) is topologically well-orderable; (iii) there exists a continuous selector on the hyperspace of nonempty closed subsets of \(X\).
The existence of continuous selectors for the hyperspaces of ultracompact scattered spaces and of spaces with a unique accumulation point is also investigated.

MSC:
54B20 Hyperspaces in general topology
54G12 Scattered spaces
54C65 Selections in general topology
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