Continuously homogeneous continua and their arc components

Abstract:

Let $X$ be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence ${A_1},{A_2}, \ldots$ of its arc components with $X = {\text {c1}}{A_1} \cup {\text {c1}}{A_2} \cup \cdots$, and there is an arc component of $X$ with nonempty interior, then $X$ is arcwise connected. As consequences and applications we get: (1) if $X$ is the countable union of arcwise connected continua, then $X$ is arcwise connected; (2) if $X$ is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.