Compact Complement Topology is T1

Theorem
Let $T = \left({S, \tau}\right)$ be a compact complement space.

Then $T$ is a $T_1$ (Fréchet) space.

Proof
We have that a Finite Topological Space is Compact.

So any finite subspace of $T$ is compact.

Let $\tau^*$ be the set defined as:


 * $\tau^* = \left\{{S \subseteq \R: S = \varnothing \text { or } \complement_\R \left({S}\right)}\right.$ is finite $\left.{}\right\}$

where $\complement_\R \left({S}\right)$ denotes the complement of $S$ in $\R$.

Then $\tau^*$ is a subset of $\tau$ by definition of the compact complement topology.

But $\tau^*$ is the finite complement topology.

The result follows from Finite Complement Topology is Minimal $T_1$ Topology.