On function extension operators and normal functors
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).
scientific article; zbMATH DE number 59452