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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

More on connectedness im kleinen and local connectedness in $ C(X)$


Author: Jack T. Goodykoontz
Journal: Proc. Amer. Math. Soc. 65 (1977), 357-364
MSC: Primary 54B20; Secondary 54D05
MathSciNet review: 0451188
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Abstract: Let X be a compact connected metric space and $ {2^X}(C(X))$ denote the hyperspace of closed subsets (subcontinua) of X. Let $ M \in C(X)$. If $ {2^X}$ is connected im kleinen at M, then $ C(X)$ is locally arcwise connected at M. A characterization of connectedness im kleinen in $ C(X)$ is given. Indecomposability of X is related to an absence of local connectedness in $ {2^X}$ and $ C(X)$. An example is given of a continuum X and a subcontinuum M such that $ C(X)$ is connected im kleinen at M but not locally connected at M.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0451188-5
PII: S 0002-9939(1977)0451188-5
Keywords: Continuum, hyperspace, connected im kleinen, locally connected
Article copyright: © Copyright 1977 American Mathematical Society