Countable Complement Space is T1
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Theorem
Let $T = \struct {S, \tau}$ be a countable complement topology.
Then $T$ is a $T_1$ (Fréchet) space.
Proof
We have that the Countable Complement Topology is Expansion of Finite Complement Topology.
We also have that a Finite Complement Space is $T_1$.
Then from Separation Properties Preserved by Expansion, we have that $T$ is a $T_1$ (Fréchet) space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $1$