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ABSTRACT. We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff F is USCO. A uniform space Y is complete iff for every topological space X and for every net {F a }, F a ⊂ X × Y , of multifunctions subcontinuous at x ∈ X, uniformly convergent to F , F is subcontinuous at x. A Tychonoff space Y isČech-complete (resp. G m -space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G δ -subset (resp. G m -subset) of X. There is a very natural extension of subcontinuity to multifunctions, introduced by Hrycay [22]1 . There is a large number of publications dealing with subcontinuity of either functions or multifunctions. Subcontinuity is often treated as a supplementary property, like before mentioned "upgrade" of closed graph property 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 54C60, 54A25, 54E15. K e y w o r d s: subcontinuous, locally totally bounded, USCO multifunction, Vietoris topology, Hausdorff uniformity,Čech complete space. Both authors were supported by VEGA 2/0047/10. 1 This is the first reference we were able to find. Probably Smithson [34] did it independently in 1975, his definition is cited more frequently.

“…There has been interest in studying extensions of natural topologies on the spaces of functions to the space of multifunctions [11], [14], [18], [21].…”

Section: Convergence Of Subcontinuous Multifunctionsmentioning

confidence: 99%

Subcontinuity

Holá

Novotný

2012

“…Denote by F (X, 2 Y ) the set of all maps from a topological space X to a topological space Y with closed values. Following the paper [17] we will define the topology τ p of pointwise convergence on F (X, 2 R ). We use the symbols τ p (U p ), also for the topology (uniformity) of pointwise convergence on the space F (X, 2 R ).…”

Section: Remark 21ºmentioning

confidence: 99%

“…+ be a sequence in B, which is pointwise convergent At last we will mention the so-called densely continuous forms introduced by McCoy and Hammer in [17] and then studied by Holá, McCoy, Holý, Vadovič in their papers [16,20,22,23].…”

Section: Minimal Usco Maps and Topology Of Pointwise Convergencementioning

confidence: 99%

“…There are many books and papers concerning topologies and convergences on spaces of set-valued maps: Attouch [1], Aubin, Frankowska [2], Rockafellar, Wets [37], Hammer, McCoy [17], Holá [15], [16], Holý [14], Holý, Vadovič [23]. In particular, graph convergence has found many applications to variational and optimization problems, differential equations and approximation theory.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

Holý

Matejíčka

2010

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Holá

Jain

McCoy

2008

ABSTRACT. A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F (x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × R such that F is the graph of a cusco map from the space X to the real line R. In this paper, we study topological properties of L(X) endowed with the Vietoris topology.