Let denote the set of all nonempty finite subsets of a set . An abstract convex space consists of a nonempty set , a nonempty set , and a multimap with nonempty values. Let be an abstract convex space and a set. For a multimap with nonempty values, if a multimap satisfies
then is called a KKM map with respect to . A KKM map is a KKM map with respect to the identity map .
In this paper, the author introduces abstract convex spaces which are adequate to establish the KKM theory. In these spaces, he generalizes and simplifies some known results in the theory on convex spaces, -convex spaces, and others. The KKM type maps are used to obtain coincidence theorems and fixed point theorems. A number of examples of abstract convex spaces and generalizations of the KKM principle are then discussed by the author.