| Abstrakt: |
Let CL(X) be the collection of all non-empty closed subsets of X, and Δ any subfamily of CL(X). By τ Δ = τ V − ∨ τ Δ + , we denote the hit-and-miss topology on CL(X). In 1995, Arhangel'skii focused his attention on the following general problem was formulated: given any subspace Y of X, study the subspace Y+ of CL(X) in the Vietoris topology and the other natural topologies, where Y+ is the subspace of CL(X) consisting of all non-empty closed subsets of X which are contained in Y. In this paper, we investigate when Y+ is first (second) countable in (CL(X), τ), for τ ∈ { τ Δ , τ V − , τ Δ + }. All content of this work extend some results of the theory of hyperspaces, when Y=X or when Δ coincides with CL(X) or when is equal to the collection of all non-empty compact subsets of X. [ABSTRACT FROM AUTHOR] |
