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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of the kernel of a closed set


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 51 (1975), 431-433
MSC: Primary 52A05
DOI: https://doi.org/10.1090/S0002-9939-1975-0372749-6
MathSciNet review: 0372749
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Abstract: Let $ S$ be a closed subset of some linear topological space such that int ker $ S \ne \phi $ and ker $ S \ne S$ Let $ \mathcal{C}$ denote the collection of all maximal convex subsets of $ S$ and, for any fixed $ k \geq 1$, let $ \mathfrak{M} = \{ {A_1} \cup \cdots \cup {A_k}:{A_1}, \ldots ,{A_k}$ distinct members of $ \mathcal{C}\} $. Then $ \mathfrak{M} \ne \phi $ and $ \cap \mathfrak{M} = \ker S$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0372749-6
Keywords: Convex kernel, maximal convex subsets, unions of convex sets
Article copyright: © Copyright 1975 American Mathematical Society