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

Abstract:

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