Connectedness im kleinen and local connectedness in 2XandC(X)

Abstract

Let X be a compact connected metric space and 2X(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. In this paper the hyperspaces are investigated with respect to point-wise connectivity properties. Let MeC(X). Then 2X is locally connected (connected im kleinen) at M if and only if for each open set U containing M there is a connected open set V such that M <zV <zU (there is a component of U which contains M in its interior). This theorem is used to prove the following main result. Let Ae2 x. Then 2X is locally connected (connected im kleinen) at A if and only if 2 X is locally connected (connected im kleinen) at each component of A. Several related results about C(X) are also obtained. A continuum X will be a compact connected metric space. 2X(C(X)) denotes the hyperspace of closed subsets (subcontinua) of Xy each with the finite (Vietoris) topology, and since X is a continuum, each of 2X and C(X) is also a continuum (see [5]). One of the earliest results about hyperspaces of continua, due to Wojdyslawski [7], was that each of 2X and C(X) is locally connected if and only if X is locally connected. As a point-wise property, local connectedness is stronger than connectedness im kleinen, which in turn is stronger than aposyndesis. The author [1] has shown that if X is any continuum, then each of 2X and C(X) is aposyndetic. It is the purpose of this paper to investigate the internal structure of 2 X and C(X) with respect to these properties. In particular, we determine necessary and sufficient conditions (in terms of the neighborhood structure in X) that 2X be locally connected at a point and that 2X be connected im kleinen at a point. We also determine that C(X) has, in general, stronger point-wise connectivity properties that either 2X or X. For notational purposes, small letters will denote elements of X, capital letters will denote subsets of X and elements of 2X, and script letters will denote subsets of 2X. If A c X, then A* (int A) (bd A) will denote the closure (interior) (boundary) of A in X. Let x e X. Then X is locally connected (I.e.) at x if for each open set U containing x there is a connected open set V such that xe Vc U. X is connected im kleinen (c.i.k.) at x if for each open set U containing x there is a component of U which contains x in its interior. X is aposyndetic at x if for each y e X — x there is a