A Further Generalization of the Knaster-kuratowski-mazurkiewicz Theorem

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 setvalued map from X into E such that for each finite subset {xi.xn} of X, co{x,, ... , xn} C (J/Li Gx¡ and for each x e X , Gx is finitely closed, i.e., for any finite-dimensional subspace L of E , Gx D L is closed in the Euclidean topology of L . Then {Gx: x 6 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.