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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

\displaystyle 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. [Nadler 1978, (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$. Further, we denote by F_1(X)$ 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 F_1(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$.
next up previous contents index
Next: indecomposable Up: Definitions Previous: HU-terminal
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30