A proof of a conjecture of A. H. Stone

Dickman, R. F.

doi:10.1090/S0002-9939-1980-0574531-5

A proof of a conjecture of A. H. Stone
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by R. F. Dickman

Proc. Amer. Math. Soc. 80 (1980), 177-180

DOI: https://doi.org/10.1090/S0002-9939-1980-0574531-5

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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 $\bmod \; 4$).

References

Harold Bell and R. F. Dickman Jr., A counterexample to a conjecture of A. H. Stone, Proc. Amer. Math. Soc. 71 (1978), no. 2, 305–308. MR 482699, DOI 10.1090/S0002-9939-1978-0482699-5
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
R. F. Dickman Jr., R. A. McCoy, and L. R. Rubin, $C$-separated sets in certain metric spaces, Proc. Amer. Math. Soc. 40 (1973), 285–290. MR 328882, DOI 10.1090/S0002-9939-1973-0328882-6
R. F. Dickman Jr. and L. R. Rubin, $C$-separated sets and unicoherence, Fund. Math. 83 (1973), no. 1, 25–33. MR 334166, DOI 10.4064/fm-83-1-25-33
J. H. V. Hunt, The Phragmén-Brouwer theorem for separated sets, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 1, 26–35. MR 431108
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
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491