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A Vitali set is a subset $V$ of the interval $[ 0 , 1 ]$ of real numbers such that, for each real number $r$, there is exactly one number $v ∈ V$ such that $v − r$ is a rational number. It's an example of a set which isn't Lebesgue measurable.

This is a set of representatives of $\Bbb R/\Bbb Q$. But is it homeomorphic to a set of representatives of $\Bbb R/\Bbb Z[\frac12]$? I.e. a subset $W$ of the interval $[ 0 , 1 ]$ of real numbers such that, for each real number $r$, there is exactly one number $v ∈ V$ such that $v − r$ is a dyadic rational.

I have that by a theorem of Sierpinski every countable set with no isolated points is homeomorphic to the others. So we have a quotient of $\Bbb R$ by two homeomorphic sets. Does that make the quotients homeomorphic too?

Eric Wofsey
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    These Vitali spaces are topological group quotients, not just set quotients. And even if they are set quotients, their homotopy types still depend on how these subsets are embedded. – Zerox Sep 14 '23 at 13:10
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    A set of representatives of each equivalence class (as a subspace of $\mathbb{R}$) has a very different topology from the quotient space $\mathbb{R}/\mathbb{Q}$ (or $\mathbb{R}/\mathbb{Z}[1/2]$). Which one are you interested in? – Eric Wofsey Sep 14 '23 at 14:28
  • Ah good point @EricWofsey I had not picked up on the distinction until you said it. Well the question only makes sense if I say the set of representatives so I have edited the title to reflect that. – it's a hire car baby Sep 14 '23 at 14:31
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    Also, Vitali sets are not all homeomorphic to each other (by the argument here). So conceivably the answer depends on which Vitali set you choose. – Eric Wofsey Sep 14 '23 at 14:36
  • Thanks @EricWofsey it all became a lot clearer once the distinction was made between the set of representatives and the quotient itself. – it's a hire car baby Sep 15 '23 at 08:20
  • @EricWofsey although I guess we have not yet stated here whether there is one set which is homeomorphic to some VItali set. – it's a hire car baby Sep 15 '23 at 09:02

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