Odd-Even Topology is not Countably Compact

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Theorem

Let $T = \left({\Z_{>0}, \tau}\right)$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.


Then $T$ is not countably compact.


Proof

By definition, the odd-even topology is a partition topology.

Let $\mathcal P$ be the partition which is the basis for $T$:

$\mathcal P = \left\{{\left\{{2 k - 1, 2 k}\right\}: k \in \Z_{>0} }\right\}$

Then $\mathcal P$ is a countable open cover of $S$ which has no finite subcover.

Hence the result.

$\blacksquare$


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