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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \scriptstyle\mathbb{R} \setminus X} is compact in the standard Euclidean topology on ${\displaystyle \scriptstyle \mathbb {R} }$.