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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 proof of a conjecture of A. H. Stone


Author: R. F. Dickman
Journal: Proc. Amer. Math. Soc. 80 (1980), 177-180
MSC: Primary 54F55
MathSciNet review: 574531
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Abstract: In this paper we use the techniques of analytic topology to establish a conjecture of A. H. Stone: A perfectly normal, locally connected, connected space is multicoherent if and only if there exist four nonempty, closed and connected subsets $ {A_0},{A_1},{A_2},{A_3}$ of X such that $ \bigcup\nolimits_{i = 0}^3 {{A_i} = X} $ and the nerve of $ \{ {A_0},{A_1},{A_2},{A_3}\} $ forms a closed 4-gon, i.e. $ {A_i}$ meets $ {A_{i + 1}}$ and $ {A_{i - 1}}$ and no others (the suffices being taken $ \bmod \; 4$).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0574531-5
PII: S 0002-9939(1980)0574531-5
Article copyright: © Copyright 1980 American Mathematical Society