Minimal Usco Maps Research Articles

Minimal usco maps and cardinal invariants of the topology of uniform convergence on compacta

Minimal usco maps and cardinal invariants of the topology of uniform convergence on compacta

Topological Properties of the Space of Convex Minimal Usco Maps

We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete metrizability of the space of convex minimal usco maps equipped with the upper Vietoris topology. We also study first countability, second countability and other properties of the (upper) Vietoris topology on this space.

Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

Abstract In [HOLÁ, Ľ.—HOLÝ, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLÁ, Ľ.—HOLÝ, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.

Minimal Usco Maps, Densely Continuous Forms and Upper Semi-Continuous Functions

Minimal Usco Maps, Densely Continuous Forms and Upper Semi-Continuous Functions

The convergence space of minimal USCO mappings

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.

Some notes on densely continuous forms

Abstract We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLÁ, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLÝ, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLÝ, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D p*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D p*(X) and on real-valued continuous functions C p(X) and a generalization of a sufficient condition for the countable cellularity of D p*(X).

Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

For metric spaces (X, dx) and (Y, dy) we consider the Hausdorff metric topology \( \tau _{H_\rho } \) on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology \( \tau _{\delta _\rho } \) defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τuc on the set F(X, 2Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),\( H_{d_y } \)) of closed subsets of Y metrized by the Hausdorff metric \( H_{d_y } \). We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms.

Cardinal Invariants of the Topology of Uniform Convergence on Compact Sets on the Space of Minimal USCO Maps

For a Baire space X the set of all minimal USCO real-valued maps on X coincides with the space D∗(X) of locally bounded densely continuous real-valued forms on X. When X is a locally compact space, the space D∗ k(X) of locally bounded densely continuous real-valued forms on X, under the topology of uniform convergence on compact sets, is a locally convex linear topological space. This paper gives characterizations and bounds for the cardinal function properties on D∗ k(X) of character, pseudocharacter, density, weight, netweight and cellularity. Examples are given to show how these properties can be the same or different. We answer also some questions posed in [17].

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.