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Proceedings of the American Mathematical Society

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

 

 

A strong form of the Phragmén-Brouwer theorem


Author: R. F. Dickman
Journal: Proc. Amer. Math. Soc. 90 (1984), 333-337
MSC: Primary 54F55; Secondary 54D05
DOI: https://doi.org/10.1090/S0002-9939-1984-0727261-4
MathSciNet review: 727261
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Abstract: In this paper we prove the following form of the Phragmen-Brouwer Theorem: a locally connected, connected normal $ {T_1}$-space $ X$ is unicoherent if and only if for every pair of disjoint nonseparating continua $ C$ and $ D$ in $ X$, $ C \cup D$ does not separate $ X$. Among the several corollaries is the proposition: $ X$ is multicoherent if and only if $ X$ is the union of a circular chain of continua $ \left\{ {{A_0},{A_1},{A_2},{A_3}} \right\}$ where no three of the $ {A_i}$'s have a point in common.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0727261-4
Keywords: Unicoherence, circular chain of continua
Article copyright: © Copyright 1984 American Mathematical Society