The Structure of Locally Connected Topological Spaces

0.1. This paper presents an investigation of the following problem. Exhibit a class X of topological spaces which contains all peano spaces and which has the following properties: (1) a cyclic element theory exists in each space of the class, (2) the abstract set consisting of all cyclic element of any space X of the class can be topologized so as to be a member of the class X, and (3) the hyperspace thus obtained is acyclic. Since the class <P of all peano spaces does not satisfy the condition (2), it is clear that any solution of the problem lies in a generalization of peano spaces. (For cyclic element theory, see Whyburn [5](1), and [6], or Kuratowski and Whyburn [4].) One such generalization has been proposed by R. L. Moore [5 ] ; another by one of the authors (Youngs [9]). Moore employed two primitive concepts: region and contiguity (compare this with satelliticity 4.8). Youngs used the notion of arc as primitive. In the following pages a solution is given which is based upon the usual concept of open set. 0.2. The work is divided into four sections. The first of these is devoted to the definition and a brief discussion of the class X of spaces to be used in the remainder of the paper; namely, locally connected topological spaces. No separation or countability properties are assumed. Thus, in particular, a single point need not form a closed point set. The use of such a weak topology is not dictated merely by a desire for generality; indeed, it is shown in later sections that this weakness is fundamental in the consideration of hyperspaces of peano spaces. The development of a theory of cyclic elements for spaces of the class X occupies the second section. In such spaces, the standard definitions of "cut point" and "cyclic element" (Kuratowski and Whyburn [4]) fail to yield certain important properties of these concepts. However, the properties are easily recovered by generalizations of the definitions mentioned. The degree of similarity achieved between the cyclic structure of spaces of the general class X and of peano spaces seems remarkable in view of the weak topology assumed in the former.