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