Semicontinuous functions and convex sets in $C(K)$ spaces

J. Aust. Math. Soc. 82 (2007), 111-121

Semicontinuous functions and convex sets in C (K ) spaces

J. P. Moreno
Dpto. Matemáticas
Facultad de Ciencias
Universidad Autónoma de Madrid
Madrid 28049
Spain
josepedro.moreno@uam.es

Abstract

The stability properties of the family $\mathcal M$ of all intersections of closed balls are investigated in spaces C (K ), where K is an arbitrary Hausdorff compact space. We prove that $\mathcal M$ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that $\mathcal M$ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C [0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

Download the article in PDF format (size 124 Kb)

Australian Mathematical Publishing Association Inc.