A counterexample to a conjecture of A. H. Stone

Abstract

A. H. Stone has offered a sequence, { S ( n ) ; n > 2 } \{ S(n);n > 2\} , of conjectures characterizing multicoherence for locally connected, connected, normal spaces. The conjecture S ( n ) 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 ) S(3) always obtains and that S ( 6 ) S(6) obtains if the space is compact. In this paper, we construct a multicoherent plane Peano continuum C for which S ( 7 ) S(7) fails. Since S ( n + 1 ) S(n + 1) implies S ( n ) , n > 2 , S ( n ) S(n),n > 2,S(n) fails for C for all n > 6 n > 6 . Furthermore we show that for any integer n ⩾ 3 n \geqslant 3 there exists a plane Peano continuum for which S ( 2 n ) S(2n) obtains while S ( 2 n + 1 ) S(2n + 1) fails.