Quasicontinuous Subcontinuous Functions and Compactness

Holá, Ľubica; Holý, Dušan

doi:10.1007/s00009-016-0759-8

Cited by 10 publications

(7 citation statements)

References 13 publications

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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: 83%

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: 83%

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Holá

Holý

2017

Tatra Mountains Mathematical Publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Fréchet-Urysohn property of quasicontinuous functions

Osipov¹

2023

Preprint

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The aim of this paper is to study the Fréchet-Urysohn property of the space Qp(X,R) of real-valued quasicontinuous functions, defined on a Hausdorff space X, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space X, the space Qp(X,R) is Fréchet-Urysohn if and only if X is countable. In particular, it is true in the class of first-countable regular spaces X. In ZFC, it is proved that for a metrizable space X, the space Qp(X,R) is Fréchet-Urysohn if and only if X is countable. 2010 MSC: 54C35, 54C40

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Section: Introductionmentioning

confidence: 99%

Frechet-Urysohn property of quasicontinuous functions

V.¹

2023

Preprint

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The aim of this paper is to study the Fréchet-Urysohn property of the space Q p (X, R) of real-valued quasicontinuous functions, defined on a Hausdorff space X, endowed with the pointwise convergence topology.It is proved that under Suslin's Hypothesis, for an open Whyburn space X, the space Q p (X, R) is Fréchet-Urysohn if and only if X is countable. In particular, it is true in the class of first-countable regular spaces X.In ZF C, it is proved that for a metrizable space X, the space Q p (X, R) is Fréchet-Urysohn if and only if X is countable.

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Quasicontinuous Subcontinuous Functions and Compactness

Cited by 10 publications

References 13 publications

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Quasicontinuous Functions, Densely Continuous Forms and Compactness

Fréchet-Urysohn property of quasicontinuous functions

Frechet-Urysohn property of quasicontinuous functions

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