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

Abstract

Solving R.J. Daverman's problem, V. Krushkal described sticky Cantor sets in ${\mathbb{R}}^N$ for $N⩾4$; 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 $N⩾4$ 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 $X$.

arXiv comment13 pages