First Countable Hyperspaces
R. E. Smithson · Proceedings of the American Mathematical Society · 1976
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.