Journal of the
Korean Mathematical Society JKMS

Let $C(X)$ ($C_{K}(X)$) denote the hyperspace of all nonempty closed connected subsets (subcontinua) of a locally compact Hausdorff space $X$ with the Fell topology. We prove that the following statements are equivalent:\\ (1) $X$ is locally connected. (2) $C(X)$ is locally connected. (3) $C(X)$ is locally connected at each $E\in C_{K}(X)$. (4) $C_{K}(X)$ is locally connected.

Keywords: Fell topology, Vietoris topology, hyperspace, continuum, local compactness, local connectedness, connected im kleinen

MSC numbers: Primary 54B20, 54B15