Generalized KKM theorem on \(H\)-space with applications (Q1191752)

An \(H\)-space is a pair \((X,(\Gamma_ A)_{A\in{\mathcal F}})\) consisting of a topological space \(X\) and a family \((\Gamma_ A)_{A\in{\mathcal F}}\) of nonempty contractible subsets of \(X\) indexed by the set \(\mathcal F\) of finite subsets of \(X\) such that \(\Gamma_ A\subset \Gamma_ B\) whenever \(A\subset B\). Let \((Y,(\Gamma_ A))\) be an \(H\)-space and \(X\) a nonempty set. A map \(F\) which assigns to \(x\in X\) a nonempty subset of \(Y\) is called a \(KKM\)-mapping if for each finite set \(\{x_ 1,\dots,x_ n\}\) in \(X\) there exists a finite set \(\{y_ 1,\dots,y_ n\}\subset Y\) such that \(\Gamma_{\{y_{i_ 1},\dots,y_{i_ k}\}} \subset\bigcup^ k_{j=1} F(x_{i_ j})\) whenever \(\{i_ 1,\dots,i_ k\} \subset \{1,\dots,n\}\). The authors prove the following KKM-type theorem: Let \(X\), \(Y\), \(F\) be as above and assume that either \(F(x)\) is closed relative to each compact subset of \(Y\) for all \(x\in X\) or that \(F(x)\) is open relative to each compact subset of \(Y\) for all \(x\in X\). If at least one \(F(x_ 0)\) is compact then \(\bigcap_{x\in X} F(x) \neq \emptyset\). It is then a routine matter to derive minimax and coincidence theorems from this result.