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 $X\cap R=\varnothing$ 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.

8 pages, 2 figures