The cyclic connectivity of homogeneous arcwise connected continua

Abstract:

A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn’s theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua. Theorem. The Hausdorff continuum $X$ is cyclicly connected if and only if for each point $x \in X$, $x$ lies in the relative interior of some arc in $X$ and $X - \{ x\}$ is arcwise connected. We then prove that arcwise connected homogeneous metric continua are cyclicly connected.