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Quick question:

Is $\mathbb{R}$ and $\mathbb{R}\backslash \mathbb{Q}$ homeomorphic?

Assuming equipped with the usual topology.

I am guessing no, because a homeomorphism is a cardinality preserving closed map. Singletons are closed in $\mathbb{R}$, but they are neither closed nor open in $\mathbb{R} \backslash \mathbb{Q}$. So $f(\{a\})$ is not closed. Not homeomorphic, bad!

Right?

Olórin
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  • A subspace of a Hausdorff space is always Hausdorff ($\Bbb R$ is Hausdorff, and so $\Bbb R\setminus\Bbb Q$ is) – AdLibitum Aug 01 '16 at 11:11

1 Answers1

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They are not, but cardinality isn’t good enough to show this, because $\Bbb R$ and $\Bbb R\setminus\Bbb Q$ have the same cardinality. However, $\Bbb R$ is connected, while $\Bbb R\setminus\Bbb Q$ is not, so they definitely are not homeomorphic.

Note that singletons are closed in $\Bbb R\setminus\Bbb Q$, as indeed they are in any $T_1$ space.

Brian M. Scott
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  • Yes that is indeed easy, but what does it have to do with the question if singletons are closed in $\mathbb R\setminus\mathbb Q$? – Lukas Betz Aug 01 '16 at 08:44
  • @John: Let $x\in\Bbb R\setminus\Bbb Q$, and let $U=\Bbb R\setminus{x}$. You know that $U$ is open in $\Bbb R$, so $U\cap(\Bbb R\setminus\Bbb Q)$ is open in $\Bbb R\setminus\Bbb Q$ by definition of the subspace topology, and $U\cap(\Bbb R\setminus\Bbb Q)=(\Bbb R\setminus\Bbb Q)\setminus{x}$. Since the complement of ${x}$ is open in $\Bbb R\setminus\Bbb Q$, ${x}$ is closed. – Brian M. Scott Aug 01 '16 at 08:45
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    @LeBtz: It’s relevant: if you have spaces $X$ and $Y$ such that all singletons of $X$ are closed, while $Y$ has a singleton that is not closed, then $X$ and $Y$ are not homeomorphic. The OP was simply mistaken in thinking that this was the case for the two spaces in question here. – Brian M. Scott Aug 01 '16 at 08:47
  • Yes but that is not what OP wrote in his comment (which is now deleted). I unterstood it the way that he thought that having a sequence in $\mathbb R\setminus\mathbb Q$ which does not converge has something to do with singletons not being closed which is wrong of course. No need to talk about that though since the comment was deleted anyway. – Lukas Betz Aug 01 '16 at 09:05