Topological convexities, selections and fixed points
type: Item
instance of: scholarly article
title: Topological convexities, selections and fixed points
zbMATH Open document ID: 1144.54030
author: Charles D. Horvath
published in: Topology and its Applications
publication date: 2008-05-15T00:00:00Z
review text: The author introduces a convexity on a set \(X\), which is an algebraic closure operator \(A\mapsto [\![A]\!]\) from the power set \(\mathcal P(X)\) to itself such that \([\![\{x\}]\!]=\{x\}\) for all \(x\in X\) or, equivalently, a family \(\mathcal C\) of subsets of \(X\) which contains the whole space and the empty set as well as singletons and which is closed under arbitrary intersections and updirected unions. Uniform convex spaces, that is uniform topological spaces endowed with a convexity for which the convex hull operator is uniformly continuous, are also introduced in this paper. The definition is slightly different from that given by \textit{M. Van de Vel} [Trans. Am. Math. Soc. 336, No. 2, 463--496 (1993; Zbl 0781.52002)]. With these new concepts, the author obtains remarkable generalizations of very important results on selections and fixed points as follows: Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem \textit{à la Michael} holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions. The paper ends with a few examples of uniform convex spaces.
reviewed by: In-Sook Kim
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 5275706
zbMATH Keywords: generalized convexity
zbMATH Keywords: uniform spaces
zbMATH Keywords: continuous selections
zbMATH Keywords: fixed points
MaRDI profile type: Publication
full work available at URL: j.topol.2007.01.024
OpenAlex ID: W1964090313
cites work: The Role of Countable Dimensionality in the Theory of Cell-Like Relations
cites work: Extension of uniformly continuous transformations and hyperconvex metric spaces
cites work: Q3799313
cites work: -convexity
cites work: Topological semilattices on the two-cell
cites work: Approximation of set valued functions and fixed point theorems
cites work: Q4385566
cites work: Complete Regularity as a Separation Axiom
cites work: Q5520521
cites work: Q4421936
cites work: Selections and approximations in finite-dimensional spaces
cites work: Q4104377
cites work: Extension and selection theorems in topological spaces with a generalized convexity structure
cites work: Intersection of sets with 𝑛-connected unions
cites work: Kakutani property of the polytopes implies Kakutani property of the whole space
cites work: KKM and Ky Fan theorems in hyperconvex metric spaces
cites work: Discrete Convex Analysis
cites work: Q3126366
cites work: Abstract convexity and global optimization
cites work: Hyperconvexity and approximate fixed points
cites work: Q4339231
cites work: Q4150325
cites work: A Selection Theorem for Topological Convex Structures
cites work: Q3141898
cites work: Q2957941
DOI: 10.1016/J.TOPOL.2007.01.024
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description: scientific article; zbMATH DE number 5275706