Some topological minimax theorems

Two topological variants of the minimax theorem are proved with no restrictions on one of the spaces except for those related to the function under consideration. The conditions concerning the behavior of the function deal only with the interval between the maximin and minimax. As corollaries, we obtain the well-known theorems of Sion and Hoang-Tui on quasiconvex-quasiconcave semicontinuous functions. The scheme of arguments goes back to the Hahn-Banach theorem and the separating hyperplane theorem. It is shown how this scheme can be explicitly realized in the proof of the Hahn-Banach theorem.