Planarity of compactifications of $\mathbb{R}$ with arc-like remainder

Abstract

We show that if $X$ is an arc-like continuum, then any continuum which is the union of $X$ and a ray $R$ such that $X \cap R = \emptyset$ and $\overline{R} \setminus R \subseteq X$ can be embedded in the plane $\mathbb{R}^2$. Further, we prove that any compactification of a line with remainder $X$ is also embeddable in $\mathbb{R}^2$ -- answering a question of Sam B. Nadler from 1972.