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 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 ⊂ R N with the following two properties: some sequence {X i ⊂ R N \ X, i ∈ N} converges homeomorphically to X, but there is no uncountable family of pairwise disjoint sets Y α ⊂ R N each of which is embedded equivalently to X.