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

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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