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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A counterexample to a conjecture of A. H. Stone
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by Harold Bell and R. F. Dickman PDF
Proc. Amer. Math. Soc. 71 (1978), 305-308 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 71 (1978), 305-308
  • MSC: Primary 54F55
  • DOI: https://doi.org/10.1090/S0002-9939-1978-0482699-5
  • MathSciNet review: 0482699