| The Knaster-Kuratowski-Mazurkiewicz Covering Theorem (KKM), is the basic ingredient in the proofs of many so-called "intersection" theorems and related fixed-point theorems (including the famous Brouwer Fixed Point Theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space. Among them we can cite the H-spaces, introduced by C. Bardaro and R. Cepitelli, and the G-spaces, introduced by S. Park and H. Kim in. Inspired by some earlier work of E. Michael, we have introduced the concept of M-space, that turns out to be closely related to H-spaces and G-spaces. We introduce also the concept of L-space, which is motivated by the MC-spaces of J. V. Llinares. We study the relationship between all the convexities mentioned above. We prove some KKM, minimax, and selection theorems for G-spaces. Then using our results relating the different convex structures, we obtain a collection of similar results for M-spaces and G-spaces. |
