Odd-Even Topology is Separable
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Theorem
Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is separable.
Proof
From Odd-Even Topology is Second-Countable, $T$ is second-countable.
The result follows from Second-Countable Space is Separable.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $6$. Odd-Even Topology: $3$