Compact complement topology

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

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3