Remarks on some basic concepts in the KKM theory

✍ Scribed by Sehie Park

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 269 KB
• Volume: 74
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.10.035

✦ Synopsis

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with ''transfer'' attached is equivalent to the corresponding one without ''transfer''. Moreover, we can invalidate terms with ''compactly'' attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan-Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.