How $ [- \infty,x], x \in \mathbb{R} $ is always compact in extended real line with order topology?

Question

How $ [- \infty,x], x \in \mathbb{R} $ is compact in extended real line with order topology?

Answer 1

A set $X$ is compact in the order topology iff every subset has a supremum, or equivalently: both $\min(X), \max(X)$ exist (as they do here, $-\infty$ and $x$) and every non-empty subset has a supremum (which is clear from the order proprties of $\Bbb R$), so yes it is.

Alternatively you could show it is homeomorphic to $[0,1]$.