One-Dimensional Continuous Curves and A Homogeneity Theorem

Abstract

Let M be the class of all one-dimensional compact metric locally connected continua up to topological equivalence. Let SU be the subclass of M consisting of those elements of M having no local cut points2 Each of the universal curve and the universal plane curve' is an element of WL. One of the principal results of this paper asserts that if an element M of JJ contains no open subset imbeddable in the plane then M is the universal curve. Thus we have a simple characterization of the universal curve. As a corollary of this result and known theorems, it follows that an element K of M is homogeneous if and only if K is a simple closed curve or a universal curve.4 In other words, there are exactly two homogeneous one-dimensional compact continuous curves. A further characterization of the universal curve is also given, as are a number of corollaries of the characterizations. Generalizations to the locally compact, non-compact cases are included in Section 6.