Abstract
In 1994, Llinares introduced mc-spaces and began to study KKM theoretic results on them. Since 1998, he became an L-space theorist and repeated to claim that his mc-spaces generalize G-convex spaces without any justifications. Later he insisted that his mc-spaces are the same as L-spaces. Hence his study on mc-spaces is
 useless now as the L-space case shown by our previous works. The present article is a continuation of our previous works on L-spaces and concerns with the rise and fall of mc-spaces. This paper will be an important record for the history of the KKM theory.
Highlights
Since we introduced generalized convex spaces or G-convex spaces in the KKM theory initiated by ourselves, a large number of their imitations, modications, and fake generalizations were appeared
Since 1998, he became an L-space theorist and repeated to claim that his mc-spaces generalize G-convex spaces without any justications
The present article is a continuation of our previous works on L-spaces and concerns with the rise and fall of mc-spaces
Summary
Since we introduced generalized convex spaces or G-convex spaces in the KKM theory initiated by ourselves, a large number of their imitations, modications, and fake generalizations were appeared. We showed that better forms of results using several coercing families can be deduced from a general KKM theorem on abstract convex spaces in our previous works. The KKM theory becomes the study on abstract convex spaces including partial KKM spaces and we obtained a large number of new results in such frame; see the references at the end of this article. From the introduction of [PK] (without citations): Recently the rst author introduced certain general classes of upper semicontinuous multimaps dened on convex spaces which were shown to be adequate to establish theories on xed points, coincidence points, KKM maps, variational inequalities, best approximations, and many others. In this paper we investigate fundamental properties of many examples of such classes of multimaps and generalized convex spaces In his Ph.D. Thesis [8] in 1994, the fourth L-space theorist Llinares introduced mc-spaces as follows: Denition 1. As we claimed in [34], mc-spaces as well as L-spaces should be destroyed
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