First countable hyperspaces
Abstract: An example is given which shows that the space of compact subsets of a first countable space need not be first countable in the finite topology. Further, it is shown that if is first countable then each compact subset of is separable. Finally a characterization of first countable in terms of a weak second countability condition is derived.
-  Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 0042109, https://doi.org/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
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Keywords: First countable, separable, hyperspaces, the finite topology, second countability
Article copyright: © Copyright 1976 American Mathematical Society