Odd-Even Topology is Weakly Countably Compact
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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 weakly countably compact.
Proof
Let $H \subseteq \Z_{>0}$ such that $H$ is infinite.
Let $x \in H$.
By definition, the odd-even topology is a partition topology.
So $U$ is a union of sets of the form $\set {2 k - 1, 2 k}$.
Now if $x \in U$, it will be of the form $2 k - 1$ or $2 k$.
So there will exist $y \in U$ of the form $2 k$ or $2 k - 1$.
So, by definition, $x$ is a limit point of $H$.
So, by definition, $T$ is weakly countably compact.
$\blacksquare$
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: $3$