Ascoli-type theorems for locally bounded quasicontinuous functions, minimal usco and minimal cusco maps

Holý, Dušan

doi:10.15352/afa/06-3-3

Cited by 10 publications

(5 citation statements)

References 14 publications

(31 reference statements)

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“…In [11] it is proved the Ascoli-type theorem for quasicontinuous locally bounded functions, in [9] we proved Ascoli-type theorems for quasicontinuous subcontinuous functions and in [10] we proved Ascoli-type theorems for quasicontinuous functions. To present our characterizations of compact subsets of Q(X, Y ), τ UC we need the following definition, which was introduced in [11] in the context of locally bounded functions from F (X, Y ).…”

Section: From This Follows That D R G(xmentioning

confidence: 80%

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Quasicontinuous Functions, Densely Continuous Forms and Compactness

Holá

Holý

2017

Tatra Mountains Mathematical Publications

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ABSTRACT. Let X be a locally compact space. A subfamily F of the space D (X, R) of densely continuous forms with nonempty compact values from X to R equipped with the topology τ UC of uniform convergence on compact sets is compact if and only if sup(F ) : F ∈ F is compact in the space Q(X, R) of quasicontinuous functions from X to R equipped with the topology τ UC . IntroductionQuasicontinuous functions were introduced by K e m p i s t y in 1932 in [14]. They are important in many areas of mathematics. They found applications in the study of minimal USCO and minimal CUSCO maps [7], [8], in the study of topological groups [3], [16], [18], in proofs of some generalizations of Michael's selection theorem [5], in the study of extensions of densely defined continuous functions [6], in the study of dynamical systems [4]. The quasicontinuity is also used in the study of CHART groups [17].Densely continuous forms were introduced by H a m m e r and M c C o y in [12]. Densely continuous forms can be considered as set-valued mappings from a topological space X into a topological space Y which have a kind of minimality property found in the theory of minimal USCO mappings. In particular, every minimal USCO mapping from a Baire space into a metric space is a densely continuous form. There is also a connection between differentiability properties of convex functions and densely continuous forms as expressed via the subdifferentials of convex functions, which are a kind of convexification of minimal USCO mappings [12].

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Section: From This Follows That D R G(xmentioning

confidence: 80%

“…To present our characterizations of compact subsets of Q(X, Y ), τ UC we need the following definition, which was introduced in [11] in the context of locally bounded functions from F (X, Y ).…”

Section: From This Follows That D R G(xmentioning

confidence: 99%

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Holá

Holý

2017

Tatra Mountains Mathematical Publications

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“…Notice that in the study of Ascoli-type theorem for quasicontinuous functions in [15,16,21], the notion of a densely equiquasicontinuous family of functions was used. Proposition 5.…”

Section: Ascoli-type Theorem For Locally Bounded Baire 1 Functionsmentioning

confidence: 99%

“…In [15], Ascoli-type theorem for quasicontinuous locally bounded functions was studied, and in [16], Ascoli-type theorems for quasicontinuous subcontinuous functions were proved. In [17], the authors studied compact subsets of quasicontinuous functions, and in [18], they studied compact subsets of minimal usco and minimal cusco maps equipped with the topology of uniform convergence on compact sets.…”

Section: Introductionmentioning

confidence: 99%

Ascoli-Type Theorem for Baire 1 Functions

Holá

2023

Mathematics

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“…Let X be a Hausdorff topological space, (Y , d) be a metric space and MU (X , Y ) be the space of all minimal usco maps from X to (Y , d). The topology τ UC of uniform convergence on compacta on MU (X , Y ) was studied in many papers [17][18][19][21][22][23][24]. A nice generalization of the Arzela-Ascoli Theorem from continuous functions to minimal usco maps into metric spaces was proved in [19].…”

Section: Introductionmentioning

confidence: 99%

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

Holá

Holý

2021

RACSAM

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Minimal usco maps have applications in functional analysis, in optimization, they appear also in selection theorems, in the definition of Stegall spaces, etc. Therefore it is important to know their topological properties. One of the most important topology on function spaces is the topology of uniform convergence on compacta. Using new techniques based on selection theorems we will characterize many cardinal invariants of the topology of uniform convergence on compacta on the space of minimal usco maps.

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Ascoli-type theorems for locally bounded quasicontinuous functions, minimal usco and minimal cusco maps

Cited by 10 publications

References 14 publications

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Ascoli-Type Theorem for Baire 1 Functions

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

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