Planar embeddings of chainable continua

2018/06/13 by Ana Anušić, Anušić, Ana, Henk Bruin +3
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.1806.05225

Corrected some typos and the proof of Theorem 8.5

arxiv created 2019/11/21 · arxiv updated 2019/11/25

Abstract

We prove that for a chainable continuum X and every non-zigzag x∈ X there exists a planar embedding ϕ:X→ ϕ(X)⊂\mathbb R² such that ϕ(x) is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings ϕ,ψ:X → \mathbb R² are called strongly equivalent if ϕ∘ ψ^-1: ψ(X) → ϕ(X) can be extended to a homeomorphism of \mathbb R². We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.

Related

• Continuum Many Planar Embeddings of the Knaster Accordion
• Planar embeddings of Minc's continuum and generalizations
• Accessibility of countable sets in plane embeddings of arc-like continua
• The Nadler-Quinn problem on accessible points of arc-like continua
• Planarity of compactifications of ℝ with arc-like remainder