Proceedings of the American Mathematical Society

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



A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem

Author: Naoki Shioji
Journal: Proc. Amer. Math. Soc. 111 (1991), 187-195
MSC: Primary 47H10; Secondary 47H19, 54H25, 58C06
MathSciNet review: 1045601
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Abstract: Granas and Dugundji obtained the following generalization of the Knaster-Kuratowski-Mazurkiewicz theorem.

Let $ X$ be a subset of a topological vector space $ E$ and let $ G$ be a set-valued map from $ X$ into $ E$ such that for each finite subset $ \{ {x_1}, \ldots ,{x_n}\} $ of $ X,co\{ {x_1}, \ldots ,{x_n}\} \subset \cup _{i = 1}^nG{x_i}$ and for each $ x \in X,Gx$ is finitely closed, i.e., for any finite-dimensional subspace $ L$ of $ E,Gx \cap L$ is closed in the Euclidean topology of $ L$. Then $ \{ Gx:x \in X\} $ has the finite intersection property.

By relaxing, among others, the condition that $ X$ is a subset of $ E$, we obtain a further generalization of the theorem and show some of its applications.

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Keywords: Fixed point, KKM-map, minimax theorem
Article copyright: © Copyright 1991 American Mathematical Society