Un problème de connectivité sur les espaces métriques (Q801084)

If x and y are two points of a compact metric space E, we call connectivity in E between x and y the smallest number of points other than x or y that we must withdraw from E in order to get a metric space which does not admit any continuous path between x and y. We denote this number by \(C_ E(x,y)\) and we prove, under some regularity conditions on E, that if \(C_ E(x,y)<\infty\), then there exists in E a system of \(C_ E(x,y)\) innerly disjoint continuous paths between x and y. This result provides us with an extension of Menger's theorem for finite graphs and we show how it may become incorrect when E presents some kinds of irregularities.