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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Martin Sleziak
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nomadicmathematician
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3The standard example is the antidiagonal in the Sorgenfrey plane (if I remember the name correctly). – Daniel Fischer Apr 17 '14 at 22:20
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2Wikipedia mentions that subspace of a separable space need not be separable and lists some examples. – Martin Sleziak Sep 09 '15 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.
Asaf Karagila
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Can you explain why a discrete topology is not separable? – nomadicmathematician Apr 17 '14 at 22:28
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1I think separable here means that has a countable subset which is dense. – MattAllegro Apr 17 '14 at 22:29
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1@Matt: You're right about that. And uncountable discrete spaces are not separable. – Asaf Karagila Apr 17 '14 at 22:30
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8In a discrete space, no set is dense but the space itself--every subset is already closed in the space. – MPW Apr 17 '14 at 22:47
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Antidiagonal (i.e. $(x,-x)$) of Sorgenfrey plane or $(x,0)$ in the Nemytskii plane both work.
Batman
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