Separable subspaces of affine function spaces on convex compact sets
type: Item
instance of: scholarly article
title: Separable subspaces of affine function spaces on convex compact sets
zbMATH Open document ID: 1155.46003
author: Evgenii Reznichenko
author: Warren B. Moors
published in: Topology and its Applications
publication date: 2008-06-24T00:00:00Z
review text: Let \(K\) be a compact convex subset of a separated locally convex space. Let Ext\((K)\) be the set of its extreme points, \(\mathcal A(K)\) the space of all continuous real-valued affine mappings defined on \(K\) and \(\mathcal A_p (K)\) the space \(\mathcal A(K)\) endowed with the topology of pointwise convergence on the extreme points of \(K\). A number of results and examples on topological properties of this space, in the spirit of the following ones, are included in this paper. (Theorem 2.14) If Ext\((K)\) is Lindelöf, then \(\mathcal A_p (K)\) is monolithic. (Theorem 3.1) If Ext\((K)\) contains a dense Lindelöf subset, then every compact subset of \(\mathcal A_p (K)\) is monolithic. (Theorem 2.19) Suppose that every regular Borel probability measure supported on Ext\((K)\) is atomic. Then bounded subsets of \(\mathcal A_p (K)\) are separable if and only if \(K\) is metrizable. The authors also develop several examples and/or use further axioms of set theory to give some insight into compactness and countable tightness in \(\mathcal A_p (K)\). As noted in the Introduction, the study of the space \(\mathcal A_p (K)\) may give useful information on the more frequently considered space \(C_p(X)\) of all continuous real-valued functions on a compact space \(X\) endowed with the topology of pointwise convergence. This space can be identified with \(\mathcal A_p (K)\), where \(K\) is the set of all regular Borel probability measures endowed with the weak* topology.
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 46A50
zbMATH DE Number: 5292207
zbMATH Keywords: compact convex set
zbMATH Keywords: extreme points
zbMATH Keywords: metrizability
zbMATH Keywords: monolithic
zbMATH Keywords: Choquet theory
zbMATH Keywords: Choquet boundary
reviewed by: X. Domínguez
MaRDI profile type: Publication
full work available at URL: j.topol.2008.03.015
OpenAlex ID: W1987924431
cites work: Q4228099
cites work: The representations of linear functionals by measures on sets of extreme points
cites work: Angelicity and the boundary problem
cites work: On the Krein-Šmulian theorem for weaker topologies
cites work: Q3233488
cites work: Q3254057
cites work: Functional analysis and infinite-dimensional geometry
cites work: Q5723419
cites work: Pointwise Compactness on Extreme Points
cites work: A characterisation of weak compactness in Banach spaces
cites work: Radon‐Nikodým compact spaces and fragmentability
cites work: Q2957940
cites work: Lectures on Choquet's theorem
cites work: Convex compact spaces and their maps
cites work: Q5631608
cites work: A convergence theorem with boundary
DOI: 10.1016/J.TOPOL.2008.03.015
description: scientific article; zbMATH DE number 5292207