First countable hyperspaces
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- by R. E. Smithson PDF
- Proc. Amer. Math. Soc. 56 (1976), 325-328 Request permission
Abstract:
An example is given which shows that the space of compact subsets $\mathcal {K}(X)$ of a first countable space $X$ need not be first countable in the finite topology. Further, it is shown that if $\mathcal {K}(X)$ is first countable then each compact subset of $X$ is separable. Finally a characterization of $\mathcal {K}(X)$ first countable in terms of a weak second countability condition is derived.References
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Lynn A. Steen and J. Arthur Seebach Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0266131
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 56 (1976), 325-328
- MSC: Primary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-1976-0402667-7
- MathSciNet review: 0402667