Minimal usco maps, densely continuous forms and upper semi-continuous functions
type: Item
instance of: scholarly article
title: Minimal usco maps, densely continuous forms and upper semi-continuous functions
zbMATH Open document ID: 1178.54004
DOI: 10.1216/RMJ-2009-39-2-545
author: Dušan Holý
author: Ľubica Holá
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: Let \(X,Y\) be Hausdorff spaces. The set-valued mapping \(F:X\to Y\) is an USCO map if it is upper semi-continuous with nonempty compact values; a minimal USCO map is one that is not properly contained in any other USCO map from \(X\) to \(Y\). A single-valued map \(f:X\to Y\) is subcontinuous (resp. quasi-continuous) at \(x\in X\) provided for every \(X\)-net \((x_{\sigma})_{\sigma}\) converging to \(x\), \((f(x_{\sigma}))_{\sigma}\) has a converging subnet (resp. for every \(Y\)-open \(V\ni f(x)\) and \(X\)-open \(U\ni x\) there is a nonempty \(X\)-open \(W\) with \(f(W)\subseteq V\)); \(f\) is subcontinuous (resp. quasi-continuous), if it is so at every point of \(X\). The main results is as follows: if \(Y\) is regular, then a set-valued mapping \(F:X\to Y\) is minimally USCO if and only if there is a quasicontinuous, subcontinuous function \(f:X\to Y\) such that the closure of the graph of \(f\) in \(X\times Y\) is equal to the graph of \(F\). The set \(M(X)\) of real-valued minimal USCO maps is shown to be uniformly isomorphic to the space of upper semi-continuous, locally bounded quasi-continuous real-valued functions endowed with the uniformity of uniform convergence. Similar questions are investigated for densely continuous forms and modifications.
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 5566186
zbMATH Keywords: USCO map
zbMATH Keywords: densely continuous form
zbMATH Keywords: quasicontinuous function
zbMATH Keywords: subcontinuous function
zbMATH Keywords: cardinal invariants
reviewed by: Laszlo Zsilinszky
MaRDI profile type: Publication
cites work: Q4277838
cites work: Q5722776
cites work: Theorems of Namioka and R. E. Johnson Type for Upper Semicontinuous and Compact Valued Set-Valued Mappings
cites work: On minimal upper semicontinuous compact-valued maps
cites work: Q5341500
cites work: Q5806961
cites work: Sets of Points of Discontinuity
cites work: Spaces of densely continuous forms
cites work: Spaces of densely continuous forms, USCO and minimal USCO maps
cites work: Cardinal invariants of the topology of uniform convergence on compact sets on the space of minimal USCO maps
cites work: Densely continuous forms, pointwise topology and cardinal functions
cites work: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps
cites work: Sur les fonctions quasicontinues
cites work: Q3421486
cites work: Q4519569
cites work: Quasi-continuity
cites work: On subcontinuity
cites work: Q4955565
description: scientific article; zbMATH DE number 5566186