Compact complement topology

In mathematics, the compact complement topology is a topology defined on the set $\scriptstyle\mathbb{R}$ of real numbers, defined by declaring a subset $\scriptstyle X\subseteq \mathbb {R}$ open if and only if it is either empty or its complement $\scriptstyle \mathbb {R} \setminus X$ is compact in the standard Euclidean topology on $\scriptstyle\mathbb{R}$ .