$\mathbb{R}_\ell$ is not locally compact

Question

Consider $\mathbb{R}_\ell$ be the the 'Sorgenfrey line':

Real line with the topology constructed from the intervals $\{[a,b):a

Prove that $\mathbb{R}_\ell$ is not locally compact.

Answer 1

I show here that every compact subset of $\mathbb{R}_\ell$ is countable, so has empty interior in particular. So there are no open sets with compact closure, whatever your definition of local compactness, $\mathbb{R}_\ell$ fails it.