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

Solving R. J. Daverman’s problem, V. Krushkal described sticky Cantor sets in $\mathbb R^N$ for $N>3$; these sets cannot be isotoped off themselves by small ambient isotopies. Using Krushkal sets, we answer a question of J. W. Cannon and S. G. Wayment (1970). Namely, for $N>3$ we construct compacta $X$ in $\mathbb R^N$ with the following two properties: some sequence of compacta $X_k$ in $\mathbb R^N \setminus X$ converges homeomorphically to $X$, but there is no uncountable family of pairwise disjoint subsets of $\mathbb R^N$ each of which is embedded equivalently to $X$. Such examples were described by Cannon and Wayment for $N=3$ and for $N>4$. Our construction works for any $N>3$ thus giving the answer for the hitherto open case $N=4$. Our proof is based on the impossibility of placing uncountably many pairwise disjoint equivalent Krushkal Cantor sets in $\mathbb R^N$ which holds true for any $N>3$. The proof of Cannon and Wayment used other results. Namely, the theorem of R. H. Bing: any disjoint family of wild closed surfaces in $\mathbb R^3$ is no more than countable; and its analogue for $\mathbb R^N$, $N>4$, which follows from J. Bryant’s reasoning together with the homotopy-theoretic criterion for local flatness by A. V. Chernavsky and R. J. Daverman. The question if the analogue of Bing–Bryant theorem holds for $N=4$ is open.

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