有限个任意小同胚连结区域的弧连通性

The path - connectedness of the domain joined by finite arbitrarily small homeomorphisms

证明了在一定条件下,有限个任意小同胚连结区域Cx是弧连通的.Cx表示一个集合,其中的任意两点可以由任意小的同胚连接.当(X,p)为紧致度量空间,X(?)Rn,x∈X,若有Rn的开球B(x)使其中仅含有Cx中点x的弧连通分支的点,则Cx是弧连通的.这一结论揭示了这种弧连通性与空间复杂性之间存在着内在的联系.利用这一结论判断了在一定条件下,某些集合的弧连通性.

Under some conditions, the path -connectedness of the domain joined by finite arbitrarily small homeomorphisms is proved. Suppose Cx indicates a set whose any 2 points can be joined by finite arbitrarily small homeomorphisms and (X,p) is a compact metric space, X(?)Rn. x∈X. If there is a ball B(x) of Rn which contains only points of a connected branch with x of Cx, then Cx is path -connected. With this result, the connectedness of some sets is proved.