Hyperspace of continua

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

$\mathrm{C}(X)=2^X \bigcap \mathrm{CLC}(X)$,

where $2^X$ denotes the hyperspace of compact subsets and $\mathrm{CLC}(X)$ denotes the hyperspace of closed connected subsets.

Properties

Peano implies contractible hyperspaces