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 $$\scriptstyle\mathbb{R}$$.