Compact Complement Topology is T1
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Theorem
Let $T = \struct {S, \tau}$ 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:
- $\tau^* = \leftset {S \subseteq \R: S = \O \text { or } \relcomp \R S}$ is finite $\rightset {}$
where $\relcomp \R S$ 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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology: $1$