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In this review, we introduce a new KKM-type theorem for intersectionally closed-valued KKM map on abstract convex spaces and its direct consequences such as a Fan-Browder-type fixed point theorem and maximal element theorems. For these basic theorems of the KKM theory, we review previously obtained particular consequences of them mainly due to the author and their recent applications obtained by other authors. Therefore, those applications might be improved by following our new results. 2010 Mathematics Subject Classification: 47H04; 47H10; 52A99; 54H25; 55M20; 58E35; 90D13.

Applications of some basic theorems in the KKM theory