Topological semigroups and continua with cut points

Abstract

In this paper we are concerned with continua that possess cut points and that admit the structure of topological semigroups, herein called mobs. First, we discuss the relations between the existence of cut points in compact connected mobs and the algebraic structure of the mob. Next, we consider D-chains, as defined by Remage [6], in compact connected mobs. D-chains may be considered to be generalizations of the classical A-sets. In the final section, we center our attention on compact metric trees, or dendrites, that are mobs and show that under certain circumstances the algebraic structure of the non-cut points has a marked influence upon the algebraic structure of the entire mob. The author wishes to acknowledge the advice and helpful suggestions of Professor A. D. Wallace in the preparation of this paper.