Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

On separation axioms in hyperspaces

It is shown that the well-known characterizations of separation axiomsT ₂ andT ₃, respectively of hit-and-miss hyperspace topologies withT ₁ base space (cf. [9], [10]) are valid with no preliminary conditions on the base space.     

On (strong) α-favorability of the Vietoris hyperspace

For a normal space X, α (i.e. the nonempty player) having a winning strategy (resp. winning tactic) in the strong Choquet game Ch(X) played on X is equivalent to α having a winning strategy (resp. winning tactic) in the strong Choquet game played on the hyperspace CL(X) of nonempty closed subsets endowed with the Vietoris topology τ _V . It is shown that for a non-normal X where α has a winning strategy (resp. winning tactic) in Ch(X), α may or may not have a winning strategy (resp. winning tactic) in the strong Choquet game played on the Vietoris hyperspace. If X is quasi-regular, then having a winning strategy (resp. winning tactic) for α in the Banach-Mazur game BM(X) played on X is sufficient for α having a winning strategy (resp. winning tactic) in BM(CL(X), τ _V ), but not necessary, not even for a separable metric X. In the absence of quasi-regularity of a space X where α has a winning strategy in BM(X), α may or may not have a winning strategy in the Banach-Mazur game played on the Vietoris hyperspace.     

Completeness properties of the generalized compact-open topology on partial functions with closed domains

The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τ_C on the space of continuous partial functions f :A→Y with a closed domain AX. Various sufficient and necessary conditions are given. It is shown, e.g., that is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τ_C, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results.     

Baire spaces and hyperspace topologies

Sufficient conditions for abstract (proximal) hit-and-miss hyperspace topologies and the Wijsman hyperspace topology, respectively, are given to be Baire spaces, thus extending results of McCoy, Beer, and Costantini. Further the quasi-regularity of (proximal) hit-and-miss topologies is investigated.

     

Polishness of the Wijsman topology revisited

Let be a completely metrizable space. Then the space of nonempty closed subsets of endowed with the Wijsman topology is -favorable in the strong Choquet game. As a consequence, a short proof of the Beer-Costantini Theorem on Polishness of the Wijsman topology is given.

     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).