An answer to a question of J.W. Cannon and S.G. Wayment

TL;DR

This paper constructs specific compact sets in higher-dimensional Euclidean spaces that can be approximated by disjoint sets but cannot be decomposed into an uncountable family of disjoint, equivalently embedded copies, answering a longstanding question.

Contribution

It provides a novel construction of compacta in \\mathbb{R}^N for N ≥ 4 that exhibit unique embedding and approximation properties, utilizing Krushkal's sticky Cantor sets.

Findings

Constructed compacta with specific approximation properties

Demonstrated the non-existence of uncountable disjoint families of equivalent embeddings

Answered a question posed by Cannon and Wayment in 1970

Abstract

Solving R.J. Daverman's problem, V. Krushkal described sticky Cantor sets in $\mathbb R^N$ for $N\geqslant 4$; these sets cannot be isotoped off of itself by small ambient isotopies. Using Krushkal sets, we answer a question of J.W. Cannon and S.G. Wayment (1970). Namely, for $N\geqslant 4$ we construct compacta $X\subset \mathbb R^N$ with the following two properties: some sequence $\{ X_i \subset \mathbb R^N \setminus X, \ i\in\mathbb N \}$ converges homeomorphically to $X$, but there is no uncountable family of pairwise disjoint sets $Y_\alpha \subset \mathbb R^N$ each of which is embedded equivalently to $X$.