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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cut points of $ X$ and the hyperspace of subcontinua $ C(X)$


Authors: Togo Nishiura and Choon Jai Rhee
Journal: Proc. Amer. Math. Soc. 82 (1981), 149-154
MSC: Primary 54B20; Secondary 54E40, 54F20
MathSciNet review: 603619
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Abstract: Let $ X$ be a nondegenerate metric continuum and $ {p_0}$ a point with $ X = {X_1} \cup {X_2}$, $ \{ {p_0}\} = {X_1} \cap {X_2}$, $ {X_1}$ and $ {X_2}$ continua. Denote by $ C(X)$, $ C({X_1})$ and $ C({X_2})$ the hyperspaces of nonempty subcontinua of $ X$, $ {X_1}$ and $ {X_2}$ respectively.

Theorem. $ C(X)$ is contractible if and only if $ C({X_1})$ and $ C({X_2})$ are contractible and either $ {X_1}$ or $ {X_2}$ is contractible im kleinen at $ {p_0}$ (a modification of connected im kleinen at $ {p_0}$).

Theorem. Let $ {X_1}$ and $ {X_2}$ satisfy Kelley's condition $ K$. Then $ C(X)$ is contractible when and only when either $ {X_1}$ or $ {X_2}$ is connected im kleinen at $ {p_0}$. Examples are given.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0603619-6
Keywords: Hyperspace of subcontinua $ C(X)$, cut point, contractibility, connected im kleinen
Article copyright: © Copyright 1981 American Mathematical Society