Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

Abstract

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.

Cardinal invariants and special maps of quasicontinuous functions with the topology of pointwise convergence

“…In this section, we discuss first countability, metrizability and cardinal functions of the space Q p (X, Y ). Before generalizing some results obtained in [8,9], first we recall definitions of the cardinal functions for a topological space [8,14].…”

“…The set of all real-valued quasicontinuous maps on a topological space X with the topology of pointwise convergence, denoted by Q p (X, R), is studied in [7,8,9]. The pointwise convergence of real-valued quasicontinuous maps defined on a Baire space is examined in [7].…”

“…The pointwise convergence of real-valued quasicontinuous maps defined on a Baire space is examined in [7]. In [9] metrizability, first countability, closed and compacts subsets of the space Q p (X, R) are discussed. The cardinal functions of the space Q p (X, R) are studied in [8].…”

Fréchet-Urysohn property of quasicontinuous functions

“…The condition of quasicontinuity can be found in the paper of R. Baire [2] in study of continuity point of separately continuous functions from R 2 into R. The formal definition of quasicontinuity were introduced by Kempisty in 1932 in [7]. Quasicontinuous functions were studied in many papers, see for examples [3,13,14,15,16,17], [19,25,21] and other. They found applications in the study of topological groups [4,22,24], in the study of dynamical systems [5], in the the study of CHART groups [23] and also used in the study of extensions of densely defined continuous functions [18] and of extensions to separately continuous functions on the product of pseudocompact spaces [26], etc.…”

Upper quasi continuous maps and quasi continuous selections

“…For example, a compact valued multifunction F acting from a Baire space to a metric one has the Baire property if and only if F is u-E I -continuous except for a set of first category, where I is the ideal of all sets of first category (see [10]). Now we give a definition which is a natural generalization of a minimal multifunction ( [2], [6], [8]) and in this form has been studied in [11]. Definition 3.…”