Odd-Even Topology is not Countably Compact

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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 not countably compact.


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

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

$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$

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

Hence the result.