TITLE: Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $
QUESTION [5 upvotes]: I'm trying to find a sufficient and necessary condition for a subset to be compact in  $\mathbb{R} $  when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, a\in\mathbb{R}\right\} $. I thought it might be something similar to the standard characterization for compact sets in $\mathbb{R}$ with the standard topology like perhaps "closed and upper bounded" but that hasn't really led anywhere.
Help would be appreciated.

REPLY [5 votes]: HINT: Let $A\subseteq\Bbb R$. Suppose that $A$ has a largest element, say $a$; what happens when you’ve covered $a$ with an open set? Now suppose that $A$ does not have a largest element, and consider the cover $\{(\leftarrow,a):a\in A\}$.