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

Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

“…In paper [18] the following characterization of minimal USCO maps is given: Ò Ø ÓÒ 3.1º ([19]) Let f λ : λ ∈ Λ be a net of real-valued functions defined on a topological space X. We say that the net f λ : λ ∈ Λ is equiquasicontinuous at x ∈ X if for every ε > 0 and every U ∈ U (x) there is λ 0 ∈ Λ and a nonempty open set W ⊂ U such that |f λ (z) − f λ (x)| < ε for every z ∈ W and for every λ ≥ λ 0 .…”

“…The notion of quasicontinuity recently turned out to be instrumental in the proof that some semitopological groups are actually topological ones (see Bouziad [8]), in the proof of some generalizations of Michael's selection theorem (see Giles, Bartlett [13]) and in characterizations of minimal usco maps via their selections (see Holá, Holý [18]). In the paper of Matejdes [28] a characterization is given for the minimality of maps via their quasicontinuous selections.…”

“…In the paper [18] a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is given. Using results of papers [18] and [19] we 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.…”

“…Using results of papers [18] and [19] we 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. We also study closed and compact subsets of the space of minimal USCO maps with topology of pointwise convergence.…”

Topological properties of the multifunction space L(X) of cusco maps

“…There has been interest in studying extensions of natural topologies on the space of continuous functions to the space of densely continuous forms, and to the spaces of usco and minimal usco maps ( [15], [14], [17], [20], [25]). Hyperspace topologies on set-valued maps with closed graphs were studied in [12], [18], [27], [29], [26], in which multifunctions are identified with their graphs and are considered as elements of a hyperspace.…”