First Countable Hyperspaces

R. E. Smithson
1976 Proceedings of the American Mathematical Society  
An example is given which shows that the space of compact subsets %(X) of a first countable space X need not be first countable in the finite topology. Further, it is shown that if 5C(A") is first countable then each compact subset of X is separable. Finally a characterization of %(X) first countable in terms of a weak second countability condition is derived.
doi:10.2307/2041629 fatcat:lnmsjfdpgvckneshy5at5f4swe