On function extension operators and normal functors
type: Item
instance of: scholarly article
title: On function extension operators and normal functors
zbMATH Open document ID: 0810.54019
published in: Moscow University Mathematics Bulletin
publication date: 1992-09-27T00:00:00Z
review text: All the spaces considered are compact Hausdorff topological spaces. First, a new characterization of spaces from AE(1) is found by proving that a closed subspace \(X\) of \(Y\) is a set-valued retract of \(Y\) iff there exists an extensor \(u: C_ + (X)\to C_ +(Y)\) (here + stands for nonnegative functions) preserving constant functions and the order, and which is semiadditive, semimultiplicative, homogeneous. (Another consequence is a characterization of the exponential functor as that one naturally isomorphic to the functor assigning to \(X\) the set of all functions \(u: C_ +(X)\to R_ +\) that preserve constant functions and the order, and that are semiadditive, semimultiplicative and homogeneous, endowed with the weak*-topology). The next result characterizes \(\kappa\)-metrizable spaces as those \(X\) having an extensor \(C_ +(X)\to C_ +(Y)\) preserving the order, for every embedding \(X\to Y\) (moreover, preserving min and max, if \(X\) is connected). Finally, it is proved that the functor \(P\) assigning to \(X\) the set of probability measures on \(X\) is the initial functor with respect to normal functors assigning to \(X\) convex compact subsets of locally convex spaces (with continuous affine mappings as morphisms).
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 59452
zbMATH Keywords: set-valued retract
zbMATH Keywords: extensor
zbMATH Keywords: functor
zbMATH Keywords: set of probability measures
zbMATH Keywords: locally convex spaces
author: Leonid B. Shapiro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59452