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Semicontinuous functions and convex sets in C(K) spaces
Published online by Cambridge University Press: 09 April 2009
Abstract
The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that 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.
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 111 - 121
- Copyright
- Copyright © Australian Mathematical Society 2007
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