László Zsilinszky

Baire Spaces and Hyperspace Topologies

Sufficient conditions for abstract (proximal) hit-and-miss hyperspace topologies and the Wisjman hyperspace topology, respectively are given to be Baire spaces, thus extending results of [MC],[B1],[C]. Further the quasi-regularity of (proximal)hit-and-miss topologies is investigated.

Polishness of the Wijsman Topology Revisited

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

Topological Games and Hyperspace Topologies

The paper proposes a unified description of hypertopologies, i.e. topologies on the nonempty closed subsets of a topological space, based on the notion of approach spaces introduced by R. Lowen. As a special case of this description we obtain the abstract hit-and-miss, proximal hit-and-miss and a big portion of weak hypertopologies generated by gap and excess functionals (including the Wijsman topology and the finite Hausdorff topology), respectively. We also give characterizations of separation axioms T0, T1, T2, T3 and complete regularity as well as of metrizability of hypertopologies in this general setting requiring no additional conditions. All this is done to provide a background for proving the main results in Section 4, where we apply topological games (the Banach–Mazur and the strong Choquet game, respectively) to establish various properties of hypertopologies; in particular we characterize Polishness of hypertopologies in this general setting.

On hereditary Baireness of the Vietoris topology

It is shown that a metrizable space X, with completely metrizable separable closed subspaces, has a hereditarily Baire hyperspace K(X) of nonempty compact subsets of X endowed with the Vietoris topology tv. In particular, making use of a construction of Saint Raymond, we show in ZFC that there exists a non-completely metrizable, metrizable space X with hereditarily Baire hyperspace (K(X),tv); thus settling a problem of Bouziad. Hereditary Baireness of (K(X),tv) for a Moore space X is also characterized in terms of an auxiliary product space and the strong Choquet game. Finally, using a result of Kunen, a non-consonant metrizable space having completely metrizable separable closed subspaces is constructed under CH. ? 2001 Elsevier Science B.V. All rights reserved.

On separation axioms in hyperspaces

It is shown that the well-known characterizations of separation axiomsT2 andT3, respectively of hit-and-miss hyperspace topologies withT1 base space (cf. [9], [10]) are valid with no preliminary conditions on the base space.

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 a-favorability for the generalized compact-open topology tC on the space P of continuous partial functions f :A ? Y with a closed domain A ? X. Various sufficient and necessary conditions are given. It is shown, e.g.,that (P, tC) is weakly a-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 a-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 tC, 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.

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-R0 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].

Čech-completeness and related properties of the generalized compact-open topology

Abstract The generalized compact-open topology τ C on partial continuous functions with closed domains in X and values in Y is studied. If Y is a non-countably compact Čech-complete space with a Gδ -diagonal, then τ C is Čech-complete, sieve complete and satisfies the p-space property of Arhangelskiǐ, respectively, if and only if X is Lindelöf and locally compact. Lindelöfness, paracompactness and normality of τ C is also investigated. New results are obtained on Čech-completeness, sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y.

More on products of Baire spaces

Abstract New results on the Baire product problem are presented. It is shown that an arbitrary product of almost locally ccc Baire spaces is Baire; moreover, the product of a Baire space and a 1st countable space which is β-unfavorable in the strong Choquet game is Baire.

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 τV. 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), τV), 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.