The usual argument is as follows: Lutzer showed in 1969 (in this paper) that an orderable space $X$ is metrisable iff it has a $G_\delta$ diagonal (i.e. the set $\Delta_X = \{(x,x) : x \in X\}$ is a countable intersection of open sets in $X \times X$ in the product topology).
And the Sorgenfrey ...
I know there are lists of convex subsets of $\mathbb{R}$ up to homeomorphism, and closed convex subsets of $\mathbb{R}^2$ up to homeomorphism, but what about just closed subsets in general of $\mathbb{R}$?
I know there are lists of convex subsets of $\mathbb{R}$ up to homeomorphism, and closed convex subsets of $\mathbb{R}^2$ up to homeomorphism, but what about just closed subsets in general of $\mathbb{R}$?
