Cyclic Element Theory in Connected and Locally Connected Hausdorff Spaces

Abstract

G. T. Whyburn, in 1926, began the development of cyclic element theory for Peano continua. This theory proved fruitful in the study of Peano spaces and a comprehensive development of the theory for metric spaces was presented in [6]. An excellent history of the theory is to be found in [4]. In [7] and [5] the generalization of cyclic element theory to more general spaces was begun. However, in each of these papers only basic definitions were set forth and fundamental results obtained. In this paper, we concern ourselves primarily with connected and locally connected Hausdorff spaces, developing the cyclic element theory initiated in [7] and demonstrating that the theory has many of the applications to connected and locally connected Hausdorff spaces that the classical theory has to Peano spaces.