# Cantor set

## Sticky Cantor sets ★★

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\begin{conjecture} Let $C$ be a \Def{Cantor set} embedded in $\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $0$ so that $f$ moves every point by less than $\epsilon$ and $f(C)$ does not intersect $C$? Such an embedded Cantor set for which no such $f$ exists (for some $\epsilon$) is called "sticky". For what dimensions $n$ do sticky Cantor sets exist? \end{conjecture}

Keywords: Cantor set