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I'm trying to find an example of a non-separable subspace of a separable space.

What kind of examples are there?

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    $\begingroup$ The standard example is the antidiagonal in the Sorgenfrey plane (if I remember the name correctly). $\endgroup$ Apr 17, 2014 at 22:20
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    $\begingroup$ Wikipedia mentions that subspace of a separable space need not be separable and lists some examples. $\endgroup$ Sep 9, 2015 at 14:11

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If you don't care about separation axioms (e.g. Hausdorff, etc.) then you can take the following example:

$\Bbb R$ with the topology defined as $U$ is open if and only if $0\in U$ or $U=\varnothing$. Then $\{0\}$ is dense in this topology so the space is separable.

But $\Bbb R\setminus\{0\}$ is discrete (since given $x\in\Bbb R\setminus\{0\}$ the set $\{x,0\}$ is open, so $\{x\}$ is relatively open). And uncountable discrete spaces cannot be separable.

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  • $\begingroup$ Can you explain why a discrete topology is not separable? $\endgroup$ Apr 17, 2014 at 22:28
  • $\begingroup$ Because $\Bbb R\setminus\{0\}$ is uncountable? $\endgroup$
    – Asaf Karagila
    Apr 17, 2014 at 22:28
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    $\begingroup$ I think separable here means that has a countable subset which is dense. $\endgroup$ Apr 17, 2014 at 22:29
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    $\begingroup$ @Matt: You're right about that. And uncountable discrete spaces are not separable. $\endgroup$
    – Asaf Karagila
    Apr 17, 2014 at 22:30
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    $\begingroup$ In a discrete space, no set is dense but the space itself--every subset is already closed in the space. $\endgroup$
    – MPW
    Apr 17, 2014 at 22:47
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Antidiagonal (i.e. $(x,-x)$) of Sorgenfrey plane or $(x,0)$ in the Nemytskii plane both work.

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