A counterexample to a conjecture of A. H. Stone

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.