Abstract: KKMS Theorem is the generalization of the famous Sperner Lemma. In the study of existence of many kinds of cores in cooperative game theory, and in the equilibrium analysis of mathematical economics, KKMS Theorem plays an important and fundamental role. Based on the importance of KKMS Theorem in the application of game theory and economics, this paper introduces the conception of KKMS points, constructs the space of KKMS mappings, and studies the stability of the KKMS points, and obtains the semi-continuous and continuous results of KKMS mappings. The results show that KKMS mappings have upper semi-continuity. Further, by constructing a specific counterexample in 2-simplices, this shows that KKMS mappings do not have lower semi-continuity generally. This also gives a sufficient and necessary condition for the continuity of KKMS mappings. Furthermore, this obtains that the KKMS points have generic stability and essential stability, which includes the existing stability results on KKM points as special cases.

Key words: KKMS Lemma, balanced sets, generic stability, continuity