hyperspace

Given a continuum X with a metric d, we let 2^X to denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by

$$H(A,B) = \max \{\sup \{d(a,B): a \in A \},\, \sup \{d(b,A): b \in B\}\} \quad \hbox {for} \; A, B \in 2^X$$

(see e.g. [64, (0.1), p. 1 and (0.12), p. 10]). If H(A, A_n) tends to zero as n tends to infinity, we put $A = \mathrm{Lim}\,A_n$. Further, we denote by F₁(X) the hyperspace of singletons of X, and by C(X) the hyperspace of all subcontinua of X, i.e., of all connected elements of 2^X. Since X is homeomorphic to F₁(X), there is a natural embedding of X into C(X), and so we can write $X \subset C(X) \subset 2^X$. Thus one can consider a retraction from either C(X) or 2^X onto X.