Abstract convexity in KKM theorems via set-valued maps (Q828720)

\textit{B. Knaster} et al. [Fundam. Math. 14, 132--137 (1929; JFM 55.0972.01)] proved the well-known KKM lemma which is an equivalent form of the Brouwer fixed point theorem. Some researchers tried to present the KKM results such as generalized L-KKM type theorems in the absence of usual convexity. In this paper, the authors present a counterexample to show that the main results in some generalized KKM type theorems and their consequence are not valid. Then, by using upper semicontinuous set-valued maps, the authors introduce the concept of uniform map convex space and prove a nonempty intersection result in this space.