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?

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.

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