An intersection theorem and related problems (Q1373027)

We give a new intersection theorem for Hausdorff topological spaces. Our arguments for proving this theorem are purely topological and very simple. First we introduce the notion of weakly connected multifunctions relative to a subset of a Hausdorff topological space and consider some classes of these multifunctions (Section 1). This notion is a generalization of the notion of \(\alpha\)-connected function given in [\textit{Hoang Tuy}, Colloq. Math. 33, 145-158 (1975; Zbl 0322.90065)] and of the multifunctions which have been implicitly used in several papers [\textit{G. Allen}, J. Math. Anal. Appl. 58, 1-10 (1977; Zbl 0383.49005); \textit{J. Dugundji} and \textit{A. Granas}, Fixed point theory Vol. I (1982; Zbl 0483.47038); \textit{W. Wu}, Beskonec. Antagonist. Igry 24-30 (1963); translation from Sci. Record, N. Ser. 3, 229-233 (1959; Zbl 0115.38301)]. Using this new notion we obtain an intersection theorem for topological spaces (Section 2). Section 3 is devoted to some related problems including fixed point theorems in Hausdorff topological spaces, a generalization of well-known versions of the minimax theorems in game theory such as Wu Wen-Tsun's [loc. cit., Theorem 1] and a results of Hoang Tuy's minimax theorem [loc. cit.]. We also consider existence theorems in the theory of variational inequalities in Hausdorff topological spaces.