Planarity of compactifications of ℝ with arc-like remainder

2024/09/30 by Andrea Ammerlaan, Ammerlaan, Andrea, Logan C. Hoehn +1
Engineering · Mathematics · #54C25 #54D35 #Advanced Banach Space Theory #Advanced Numerical Analysis Techniques #Advanced Topology and Set Theory #Arc (geometry) #Arithmetic #Combinatorics #FOS: Mathematics #General Topology (math.GN) #Geometry #Mathematics #Planarity testing #Primary 54F15 #Remainder #Secondary 54F50

paper · pdf · doi:10.48550/arxiv.2409.20455

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∩ R = ∅ and R ∖ R ⊆ X can be embedded in the plane ℝ². Further, we prove that any compactification of a line with remainder X is also embeddable in ℝ² -- answering a question of Sam B. Nadler from 1972.

Related

• Planar embeddings of chainable continua
• The Nadler-Quinn problem on accessible points of arc-like continua
• Radial departures and plane embeddings of arc-like continua
• A compactification of outer space which is an absolute retract
• The Structure and Singularities of Arc Complexes