# Give an example of a non-separable subspace of a separable space

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

What kind of examples are there?

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

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.

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

Antidiagonal (i.e. $(x,-x)$) of Sorgenfrey plane or $(x,0)$ in the Nemytskii plane both work.