Completeness properties of the generalized compact-open topology on partial functions with closed domains (original) (raw)

Completeness and related properties of the graph topology on function spaces

2013

The graph topology τ Γ is the topology on the space C(X) of all continuous functions defined on a Tychonoff space X inherited from the Vietoris topology on X × R after identifying continuous functions with their graphs. It is shown that all completeness properties between complete metrizability and hereditary Baireness coincide for the graph topology if and only if X is countably compact; however, the graph topology is α-favorable in the strong Choquet game, regardless of X. Analogous results are obtained for the fine topology on C(X). Pseudocompleteness, along with properties related to 1st and 2nd countability of (C(X), τ Γ) are also investigated.

A Compact-Open Topology on Partial Maps with Open Domain

Journal of the London Mathematical Society, 1980

In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X × Y using various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X × Y. We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.

Čech-Completeness and Related Properties of the Generalized Compact-Open Topology

Journal of Applied Analysis, 2010

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 noncountably compactČech-complete space with a G δ-diagonal, then τ C isČechcomplete, sieve complete and satisfies the p-space property of Arhangel'skiǐ, 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 onČech-completeness, sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y .

Complete metrizability of generalized compact-open topology

Topology and its Applications, 1999

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 E CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to El'. Denote by 7~ the generalized compact-open topology on P. We show that if X is a hemicompact metrizable space and Y is a FrCchet space. then (P. TC) is completely metrizable and homeomorphic to a closed subspace of (CL(X), TF) x (C(X. Y). T~,cJ), where T,T is the Felt topology on CL(X) and 71'0 is the compact-open topology on C(X, Y).

Completeness Properties in the Compact-Open Topology on Fans

2013

It is an open problem to characterize those spaces X for which Ck(X), the space of real-valued continuous functions on X with the compactopen topology, has various completeness properties, in particular, the Baire property. We investigate completeness properties of Ck(X) for a class of spaces X having intermediate topologies between the metric and sequential fans. We obtain necessary and sufficient conditions on these X for Ck(X) to be Baire, and show that, except for the sequential fan whose function space is completely metrizable, these Ck(X), while they can be Baire, are never hereditarily Baire or Choquet (a property also known as weakly α-favorable).

Wijsman topology on function spaces

Rendiconti del Circolo Matematico di Palermo, 1997

Let C(X, Y) be the space of continuous functions from a metric space (X,d) to a'metric space (Y,e). C(X, Y) can be thought as a subset of the hyperspace CL(X x Y) of closed and nonempty subsets of X• Y by identifying each element of C(X, Y) with its graph. We consider C(X, Y) with the topology inherited from the Wijsman topology induced on CL(X • Y) by the box metric of ~/ and e. We study the relationships between the Wijsman topology and the compact-open topology on C(X, Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology on C(X, Y) is weaker than the Wijsman topology are given (If Y is totally bounded, then for every metric space X the compact-

The compact-open topology: A new perspective

Topology and its Applications, 2009

This paper studies the compact-open topology on the set KC (X) of all real-valued functions defined on a Tychonoff space, which are continuous on compact subsets of X. In addition to metrizability, separability and second countability of this topology on KC (X), various kinds of topological properties of this topology are studied in detail. Actually the motivation for studying the compact-open topology on KC (X) lies in the attempt of having a simpler proof for the characterization of a completeness property of the compact-open topology on C (X) ...

Some topologies on the spaces of USCO maps and densely continuous forms

Acta Mathematica Hungarica, 2008

We study the completeness of three (metrizable) uniformities on the sets D(X, Y ) and U (X, Y ) of densely continuous forms and USCO maps from X to Y : the uniformity of uniform convergence on bounded sets, the Hausdor metric uniformity and the uniformity U B . We also prove that if X is a nondiscrete space, then the Hausdor metric on real-valued densely continuous forms D(X, R) (identied with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y ) equipped with the Hausdor metric is dense equicontinuity introduced by Hammer and McCoy in [7].