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Let $n\ge 2$ and let $C$ be a Cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the Cantor ternary set.

Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the straight line segment joining them. Then for any given $\varepsilon>0$ we would like to to find a homeomorphism $\phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that

  1. $\phi(x)=x$ and $\phi(y)=y$.
  2. $\phi$ is the identity outside an $\varepsilon$-neighborhood of $L_{xy}$, and $\phi$ moves points less than a distance of $\varepsilon$.
  3. $\phi(L_{xy})\cap C=\emptyset$.

How do we construct/show existence of such a homeomorphism $\phi?$

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