The Sorgenfrey Line is $\mathbb R_/ = (\mathbb R, \tau_s)$ where $\tau_s$ is the topology on $\mathbb R$ with base $\{[a, b)\ |\ a, b \in \mathbb R\}$. I know how to show $\mathbb R_/$ is not locally compact. It turns out that the only compact sets in $\mathbb R_/$ are at best countable.
What I want to prove is that it is paracompact. You can't use the usual proof that works on $\mathbb R$ because that uses a refinement of any open cover of $\mathbb R$ that is constructed using the fact that closed balls are compact in $\mathbb R$, which can't be used here.
I'm sure I'll figure this out eventually but any feedback would be good. This is not a homework question, it's a sidetrack from my Honours project, where I have to use paracompactness as a prerequisite in some of the proofs I'm studying. I got sucked into studying the Sorgenfrey line, but I need to get back to my project, I'm too easily distracted.
Glossary:
Locally Compact: $X$ is locally compact if for every $x \in X$ there exists a compact set $K \subset X$ that itself contains an open neighbourhood of $x$.
Paracompact: $X$ is paracompact if every open cover $A$ of $X$ has a refinement $B$ so that every $x \in X$ has an open neighbourhood that intersects with finitely many members of $B$.
Best Answer
HINT: Prove that the Sorgenfrey line is Lindelöf, and use the fact that a regular Lindelöf space is paracompact. To prove that it’s Lindelöf, start with a basic open cover $\mathscr{U}$ (i.e., a cover by sets of the form $[a,b)$). Show that $\{(a,b):[a,b)\in\mathscr{U}\}$ covers all but a countable subset of $\Bbb R$, and use the fact that $\Bbb R$ with the usual topology, being second countable, is hereditarily Lindelöf.