Odd-Even Topology is First-Countable

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 first-countable.

Proof

From Odd-Even Topology is Second-Countable, $T$ is second-countable.

The result follows from Second-Countable Space is First-Countable.

$\blacksquare$

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: $3$