Finite complement topology

Definition: Finite complement topology

The finite complement topology of a set $X$ is the collection of all subsets $U\subseteq X$ such that the complement $X\backslash U$ is either finite or all of $X$. We write $X_f$ to denote the set $X$ equipped with the finite complement topology.

A set $X$ equipped with the finite complement topology ${\mathcal{T}}_f$ has the following properties:

• ${\mathcal{T}}_f$ is minimal T1 topology—the condition where finite subsets are closed—meaning it is the coarsest topology on $X$ such that singletons are closed.
• If $A\subseteq X$ is infinite, then every point $x\in X$ is a limit point of $A$, since any open set of $X$ is infinite and must contain a point of $A$.
• $X$ and all its subsets are compact.

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