Compact complement topology

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