A review of the KKM theory on φA-spaces or GFC-spaces

Abstract

In the KKM theory, G-convex spaces are extended to KKM spaces or abstract convex spaces in 2006. Various types of φA-spaces (X,D;{φA}A∈hDi) appeared until 2007 can be made into G-convex spaces in several ways. Moreover, various types of generalized KKM maps on φA-spaces are simply KKM maps on G-convex spaces. Therefore, our G-convex space theory can be applied to various types of φA- spaces. However, Khanh et al. in 2009 introduced a disguised form of φA-spaces called GFC-spaces. In the present paper, we review their works on GFC-spaces and clarify that their basic results are consequences of known ones. Finally, further comments on each of seven papers on GFC-spaces are given.