L'ubica Holá

Topologies on the space of continuous functions

Abstract Let X and Y be Tychonoff spaces and C(X, Y) be the space of all continuous functions from X to Y. The coincidence of the fine topology with other function space topologies on C(X, Y) is discussed. Also cardinal invariants of the fine topology on C(X, R ) , where R is the space of reals, are studied. To answer some questions of Di Maio and Naimpally (1992) other function space topologies are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence, proximal graph topology.

Decomposition Properties of Hyperspace Topologies

Let X be a T1 topological space and Δ a nonempty family of closed subsets of X. We study the hit-and-miss hyperspace topology generated by Δ in terms of its upper and lower parts, focusing on first and second countability and quasi-uniformization. We also obtain some new results on the Vietoris and Fell topologies.

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and Dk(X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: Dk(X,Y) is metrizable iff Dk(X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then Dk(X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.

Normality and paracompactness of the Fell topology

Let X be a Hausdorff topological space and CL(X) the hyperspace of all closed nonempty subsets of X. We show that the Fell topology on CL(X) is normal if and only if the space X is Lindelöf and locally compact. For the Fell topology normality, paracompactness and Lindelöfness are equivalent. Throughout the paper all spaces are assumed to be Hausdorff. By X we always denote a space, while CL(X) (resp. K(X)) is the set of all nonempty closed (compact) subsets of X . We quote [En] and [Be1] for the basic notions. One of the most important and well-studied hyperspace topologies on CL(X) is the Fell topology [At], [Be1], [Be2], [Fe], [Fl], [Po]. The Fell topology can be considered a classical one, as it has found numerous applications in different fields of mathematics ([Ma], [At]). To describe this topology, we need to introduce some notation. For E a subset of X , we associate the following subsets of CL(X): E− = {A ∈ CL(X) : A ∩ E 6= ∅}, E = {A ∈ CL(X) : A ⊂ E}. The Fell topology τF on CL(X) has as a subbase all sets of the form V −, where V is an open subset of X plus all sets of the form (K), where K ∈ K(X) and K is the complement of K. In locally compact spaces, convergence with respect to the Fell topology is Kuratowski convergence of nets of sets. If compact subsets in the above definition are replaced by closed sets, we obtain the stronger Vietoris topology, also called the finite topology [Mi]. The normality of the Vietoris topology on CL(X) is equivalent to the compactness of X as was shown by Veličko in [Ve]. We refer also to Keesling’s deep study of normality of the Vietoris topology [Ke1], [Ke2]. The regularity and Hausdorffness of the Fell topology was studied by Poppe [Po] and the complete regularity by Beer and Tamaki in [BT]. We can summarize here these results as follows: Received by the editors February 12, 1997 and, in revised form, October 7, 1997. 1991 Mathematics Subject Classification. Primary 54B20.

Complete metrizability of generalized compact-open topology

Abstract Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y; a partial map is a pair (B, f), where B ϵ CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to Y. Denote by τc the generalized compact-open topology on P . We show that if X is a hemicompact metrizable space and Y is a Frechet space, then ( P , τ C ) is completely metrizable and homeomorphic to a closed subspace of (CL(X), τF) × (C(X,Y), τCO), where τF is the Fell topology on CL(X) and τCO is the compact-open topology on C(X,Y).

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.

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.

Compactness in the fine and related topologies

Abstract Let X be a Tychonoff space, Y a metrizable space and C(X,Y) be the space of continuous functions from X to Y . For a paracompact, locally hemicompact k-space X we characterize compact subsets of C(X,Y) topologized with the fine, graph and Krikorian topologies. Our results concerning compactness in the fine topology greatly generalized those of Spring [Topology Appl. 18 (1984) 87].

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.