Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / E / E4 / a / iv / Solution

By the Heine–Borel theorem a subset of $\mathbb{R}$ is compact exactly when it is closed and bounded. Consequently the nonempty compact intervals are exactly $\boxed{[a,b](-\infty <a\le b<\infty )}$. This includes the degenerate interval $[a,a]$. The empty interval is compact as well; nonclosed bounded intervals and all unbounded intervals fail the criterion.