Compact complement topology

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