A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem

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.