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Proceedings of the American Mathematical Society

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



A counterexample to a conjecture of A. H. Stone

Authors: Harold Bell and R. F. Dickman
Journal: Proc. Amer. Math. Soc. 71 (1978), 305-308
MSC: Primary 54F55
MathSciNet review: 0482699
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Abstract: A. H. Stone has offered a sequence, $\{ S(n);n > 2\}$, of conjectures characterizing multicoherence for locally connected, connected, normal spaces. The conjecture $S(n)$ is, “X is multicoherent if and only if X can be represented as the union of a circular chain of continua containing exactly n elements". It is known that $S(3)$ always obtains and that $S(6)$ obtains if the space is compact. In this paper, we construct a multicoherent plane Peano continuum C for which $S(7)$ fails. Since $S(n + 1)$ implies $S(n),n > 2,S(n)$ fails for C for all $n > 6$. Furthermore we show that for any integer $n \geqslant 3$ there exists a plane Peano continuum for which $S(2n)$ obtains while $S(2n + 1)$ fails.

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Keywords: Multicoherence, circular chain of continua
Article copyright: © Copyright 1978 American Mathematical Society