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Dear Mathoverflow Community,

I am looking for a reference for the following topological fact:

Fact

Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if you want). Then every homeomorphism $f: S^2 \setminus E \to S^2 \setminus F$ extends to a homeomorphism of $S^2$ onto itself.

Thank you, Malik

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  • $\begingroup$ For the perfect case, i.e. for embeddings of the Cantor set in $\mathbb R^2$, there might be some information in Moise's book "Geometric topology in dimensions 2 and 3". $\endgroup$
    – Lee Mosher
    Commented May 3, 2019 at 20:48
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    $\begingroup$ It's false: just take $E=F$ to be $\{0\}$ and define $f(x)=x/\|x\|^2$. However it's true with the plane replaced by the 2-sphere, just because then the embedding into sphere is the end compactification. $\endgroup$
    – YCor
    Commented May 3, 2019 at 22:06
  • $\begingroup$ @YCor Right, sorry, I meant the sphere. $\endgroup$
    – Malik Younsi
    Commented May 3, 2019 at 23:54

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