Hyperspace of closed connected subsets

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Let $(X,\tau)$ be a topological space. The hyperspace of closed connected sets is defined by

$\mathrm{CLC}(X) = \{A \in \mathrm{CL}(X) \colon A$ is connected $\},$

where $\mathrm{CL}(X)$ denotes the hyperspace of closed subsets.