The existence of widely connected and biconnected semigroups

Abstract

A. D. Wallace in [10] and Wallace and R. J. Koch in Theorem 3 of [3, p. 683] have shown that if a topological semigroup with unit is a metric indecomposable continuium then it is a topological group: it is known that a solenoid is both a compact connected group and an indecomposable continuum. We have shown [5 ] that any compact indecomposable continuum I contains a widely connected subset W such that W = I: below it is shown that if I is also a topological group with unit and other stated simple properties, then W can be taken so that it is a widely connected topological group. Here W is nonbicompact and it nowhere contains an arc in contrast to the well known result by Pontrjagin that a bicompact connected topological group does contain an arc. Koch in [2] has shown recently that a bicompact connected topological semigroup with unit contains an arc; in contrast to this we also give below an example of a biconnected topological semigroup: it is nonbicompact, it has a unit and also a dispersion point at its zero. To show the existence of both of these we make strong use of Zermelo's well ordering theorem and assume the hypothesis of the continuum. The existence proof is a modification of the well known one used by Knaster and Kuratowski in [1, pp. 241, 250] to show the existence of a biconnected set. It is a modification of this by Wilder, which does not make a strong use of Zermelo's well ordering theorem, which we use in [5] to show the existence of a widely connected set: however we are unable to use this below because the construction must give also the desired algebraic properties. We recall the following definitions. The connected set W is widely connected if and only if every nondegenerate connected subset C of W has the same closure that W does, i.e. C = W. The connected set B is biconnected if and only if it is not the union of two disjoint nondegenerate connected subsets; d EB is a dispersion point of B if B -d is totally disconnected. For more basic topological definitions see Moore in [4] or Wilder in [11]; bicompact is as in [11, p. 34]. For theorems on indecomposable continua see Moore in [4, pp. 75-77]. We denote the null set by 0. For basic semigroup definitions see Wallace in [9].