Since then, numerous applications of this interesting theorem have been found. The object of this note is to obtain a generalization of Theorem 1 by relaxing, among the others, the compactness condition. It contains a fixed point theorem for maps with inwardness or outwardness conditions given by Fan [6]. As its direct consequence, we also obtain a new minimax theorem. Theorem 1, as pointed out in [6], is a geometrical formulation of the minimax inequality of Fan, which has been extended in other directions [1, 3, 4], mainly for the purpose of studying variational inequalities. Our main result is the following Theorem 3. Its proof relies on an extension of the well-known fixed point theorem of Kakutani [7] (see also [2], p. 174), which is stated below as Lemma 2 and is proved by a modification of Kakutani's method. Let X, Y be topological spaces. By a set-valued map f defined on X with values in Y,, we mean that to each point x e X , f assigns a unique nonempty subset f (x) of Y. f is called upper semi-continuous if for each open subset G of Y, the set {xeX : f(x) C G} is open in X. It is easy to show (e.g. [2], p. 112) that if Y is compact Hausdorff and if f (x) is closed for each xeX , then f is upper semi-continuous if and only if the graph {(x,y)eX x Y :ye f ( x ) } of f is closed in X x Y.