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.References
- R. F. Dickman Jr., Some mapping characterizations of unicoherence, Fund. Math. 78 (1973), no. 1, 27–35. MR 322834, DOI 10.4064/fm-78-1-27-35
- R. F. Dickman Jr., Multicoherent spaces, Fund. Math. 91 (1976), no. 3, 219–229. MR 420578, DOI 10.4064/fm-91-3-219-229
- A. H. Stone, Incidence relations in unicoherent spaces, Trans. Amer. Math. Soc. 65 (1949), 427–447. MR 30743, DOI 10.1090/S0002-9947-1949-0030743-8 A. D. Wallace, A characterization of unicoherence, Bull. Amer. Math. Soc. 48 (1952), 445.
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