On Ky Fan's section theorem
The authors generalize Ky Fan's classical section theorem to the context of \(H\)-spaces. A pair \((X,\Gamma)\) is called an \(H\)-space if \(X\) is a topological space and \(\Gamma\) is a mapping which assigns to each non-empty finite subset \(A\subset X\) a non-empty contractible subset \(\Gamma_A\subset X\) such that \(\Gamma\) is monotone with respect to inclusions. A subset \(D\subset X\) is said to be \(H\)-convex if \(\Gamma_A\subset D\) whenever \(A\subset D\) is a finite set. The \(H\)-convex hull \(H\text{-conv}K\) of a set \(K\) is the smallest \(H\)-convex set containing \(K\). If finally, \(X\) and \(Y\) are sets and \(M\subset X\times Y\) then \(M[y]:=\{x\in X| \;(x,y)\in M\}\). The authors' main result can then be stated as follows: Let \(X\) be a Hausdorff space and \((Y,\Gamma)\) an \(H\)-space. Assume that \(M\) and \(N\) are subsets of \( X\times Y\) such that the following conditions hold: there is a compact set \(K\subset X\) and a subset \(P\) of \(M\) which is closed in \(X\times Y\) such that \(H\text{-conv}\{y\in Y| \;(x,y)\notin N\}\subset \{y\in Y| \;(x,y)\notin M\}\) whenever \(x\in K\), \(\bigcap_{y\in Y}\text{cl}N[y]=\bigcap_{y\in Y}N[y]\) and such that \(\{x\in K| \;(x,y)\in P\}\) is a nonempty acyclic set for each \( y\in Y\). The conclusion is that there is an \(x_0\in K\) such that \(\{x_0\}\times Y\subset N\).
scientific article; zbMATH DE number 1236165
scientific article