More on connectedness im kleinen and local connectedness in 𝐶(𝑋)

Abstract

Let X be a compact connected metric space and 2 X ( C ( X ) ) {2^X}(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. Let M ∈ C ( X ) M \in C(X) . If 2 X {2^X} is connected im kleinen at M, then C ( X ) C(X) is locally arcwise connected at M. A characterization of connectedness im kleinen in C ( X ) C(X) is given. Indecomposability of X is related to an absence of local connectedness in 2 X {2^X} and C ( X ) C(X) . An example is given of a continuum X and a subcontinuum M such that C ( X ) C(X) is connected im kleinen at M but not locally connected at M.