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I'm trying to show that on $S \times S$ where $S$ is the Sorgenfrey line, the subspace $Y=\{(-x,x)\}$ is discrete so I can show that $Y$ isn't separable. I'm taking $u=[-x,a) \times [x,b)$ such that $-x<a,~ x<b$ and saying $u \cap Y=\{(-x,x)\}~ \forall x \in \mathbb{R} $. $S$ is $T_1$ so $u \cap Y$ is closed. Then $Y$ is discrete and the only dense set of a discrete set is the set of all points from that set, so there is no countable subset of $Y$ dense in $Y$, so $Y$ isn't separable.

Anything I'm missing? How should I state the last part? Can I use a similar method to show $S \times S$ isn't Lindelöf?

Jyrki Lahtonen
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chris
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  • I assume you mean $Y={(-x,x):x\in\Bbb R}$, because otherwise $Y$ is just a singleton and thus not a counterexample to separability. – Mario Carneiro Apr 07 '14 at 22:28
  • Related (not quite a duplicate, but this question is a corrolary of that one): http://math.stackexchange.com/questions/162582/what-topology-does-a-straight-line-in-the-plane-inherit-as-a-subspace-of-mathb?rq=1 – Mario Carneiro Apr 07 '14 at 22:38

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You want to show that singletons are open in order to prove that $Y=\{(-x,x):x\in\Bbb R\}$ is discrete. Then from the uncountability of $Y$ you have that $Y$ is not separable and hence $S\times S$ is not separable (because a subspace of a separable space is separable).

You don't need to know anything about $T_1$; the definitions will suffice. The set $u=[-x,a)\times[x,b)$ (picking any $a>-x,b>x$) is open by definition of the product topology and the Sorgenfrey line, so $u\cap Y=\{(-x,x)\}$ is open by definition of a subspace. Thus singletons are open.

Mario Carneiro
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