Or subspace ${(x,x): x \in \mathbb{R}})$ space $(\mathbb{r},\tau_\rightarrow)\times(\mathbb{R},\tau_\rightarrow)$ is discrete space?

Question

Or subspace ${((x,x): x \in \mathbb{R}})$ space $(\mathbb{R},\tau_\rightarrow)\times(\mathbb{R},\tau_\rightarrow)$ is discrete space?

$\tau_\rightarrow$ is topology of Sorgenfrey arrow

Answer 1

The diagonal of any space $X$ is just homeomorphic to $X$ again, so no in this case.