Definition:Finite Complement Topology

Definition
Let $S$ be a set whose cardinality is usually specified as being infinite.

Let $\tau$ be the set of subsets of $S$ defined as:
 * $H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$

where $\relcomp S H$ denotes the complement of $H$ relative to $S$.

Then $\tau$ is the finite complement topology on $S$, and the topological space $T = \struct {S, \tau}$ is a finite complement space.

Also see

 * Finite Complement Topology is Topology