Function and colorful extensions of the KKM theorem

Abstract

We deal with an aggregate version of the KKM theorem. Given $n$ KKM families of special type on an $( n-1)$-dimensional simplex, we show that it is possible to choose a single element from every KKM family to get a KKM family on that simplex. We also introduce and study function KKM families as surrogates of KKM families on simplexes. We show a function version of the KKM theorem. The Coincidence Theorem is our main result, Brouwer's fixed is a special cases of that theorem.