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

 

A characterization of locally connected continua by hyperspace retractions


Author: Sam B. Nadler
Journal: Proc. Amer. Math. Soc. 67 (1977), 167-176
MSC: Primary 54F15; Secondary 54B20
MathSciNet review: 0458378
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Abstract: Let X be a (metric) continuum. It is shown that X is locally connected if and only if there is special type of retraction from $ {2^X}$ onto $ C(X)$, where $ {2^X}$ [resp., $ C(X)$] is the space of all nonempty compact subsets [resp., subcontinua] of X with the Hausdorff metric. Also, necessary and sufficient conditions are given for the ``continuity of balls".


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0458378-6
PII: S 0002-9939(1977)0458378-6
Keywords: Convex metric, dendrite, Hausdorff metric, hereditarily unicoherent continuum, locally connected continuum, retraction, selection
Article copyright: © Copyright 1977 American Mathematical Society