Separable subspaces of affine function spaces on convex compact sets
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.
scientific article; zbMATH DE number 5292207
scientific article