Abstract :
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 the
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 others we could mention H-spaces and G-spaces. We have introduced a new abstract
convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael,
Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556–575] and called a topological space endowed with
this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible
multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173–187], the concepts of G-spaces and metric
spaces with Michael’s convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove
that G-spaces andM-spaces are close related.We also introduce here the concept of an L-space, which is inspired in the MC-spaces
of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization,
J. Math. Econom. 29 (1998) 285–302], and establish relationships between the convexities of these spaces with the spaces
previously mentioned.
© 2007 Elsevier Inc. All rights reserved.
