Isometrically Homogeneous And Topologically Homogeneous Continua

Based on the past study of homogeneous continua, this paper concludes that compact connected metric topological groups and isometrically homogeneous continua fall into the following three mutually disjoint classes:(1) indecomposable;(2) aposyndetic and semi-indecomposable;(3) mutually aposyndetic.Among all continua these are special classes with members having extremal properties. Indecomposable isometrically homogeneous continua are characterized as solenoids, and one-dimensional isometrically homogeneous continua are characterized as solenoids or circles. It is shown that path connected isometrically homogeneous continua are locally connected.