Local connectedness and connectedness im kleinen in the hyperspace of a metric space

Abstract

This work is a detailed study of the paper “Connectedness im kleinen and local connect-edness in 2 X and C(X)” by J. T. Goodykoontz, Jr. Our assumption throughout the study is that X is a compact connected metric space. We study the hyperspace 2 X, of all closed subsets of X, and C(X), the hyperspace of all connected elements of 2 X. The hyperspaces 2 X and C(X) are endowed with the Hausdorff metric topology and we show that the Haus-dorff metric topology is equivalent to the Vietoris topology. Our purpose is to study the connectivity properties of local connectedness and connectedness im kleinen on X and its hyperspaces. We show that for M ∈ C(X), 2 X is connected im kleinen at M if and only if for each open set U containing M there is a component of U which contains M in its interior. We also show that 2 X is locally connected at M if and only if for each open set U containing M there exists a connected open set V such that M ⊆ V ⊆ U. These two results are used to show the main results. For A ∈ 2 X, 2 X 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. Finally, we prove the first two results of the paper “More on connectedness im kleinen and local connectedness in C(X)” by J. T. Goodykoontz, Jr. We show that for M ∈ C(X), 2 X connected im kleinen at M implies that C(X) is locally arcwise connected at M. The second results gives a characterisation of connectedness im kleinen in C(X).