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

TL;DR

The paper proves that certain compactifications of the real line with arc-like continua as remainders can be embedded in the plane, resolving a longstanding question about their planarity.

Contribution

It establishes the planarity of compactifications of the real line with arc-like remainders, answering a question posed by Nadler in 1972.

Findings

Any union of an arc-like continuum and a disjoint ray with specific closure properties embeds in the plane.

Compactifications of a line with arc-like remainders are planar.

Addresses a 50-year-old open problem in continuum theory.

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.