Relative versions of first and second countability in hyperspaces

Abstract Let CL( X ) be the collection of all non-empty closed subsets of X , and Δ any subfamily of CL( X ). By <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>τ</m:mi> <m:mi mathvariant="normal">Δ</m:mi> </m:msub> <m:mo>=</m:mo> <m:msubsup> <m:mi>τ</m:mi> <m:mi>V</m:mi> <m:mo>−</m:mo> </m:msubsup> <m:mo>∨</m:mo> <m:msubsup> <m:mi>τ</m:mi> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> $\tau_\Delta = \tau_V^- \lor \tau_\Delta^+$ , 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>τ</m:mi> <m:mo>∈</m:mo> <m:mo fence="false" stretchy="false">{</m:mo> <m:msub> <m:mi>τ</m:mi> <m:mi mathvariant="normal">Δ</m:mi> </m:msub> <m:mo>,</m:mo> <m:msubsup> <m:mi>τ</m:mi> <m:mi>V</m:mi> <m:mo>−</m:mo> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mi>τ</m:mi> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>+</m:mo> </m:msubsup> <m:mo fence="false" stretchy="false">}</m:mo> </m:math> $ \tau\in \{\tau_\Delta,\tau_V^- ,\tau_\Delta^+\} $ . 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 .