Deleted Integer Topology is not Countably Compact

Theorem
Let $S = \R_{\ge 0} \setminus \Z$, and let $\vartheta$ be the deleted integer topology on $S$.

Then the topological space $T = \left({S, \vartheta}\right)$ is not countably compact.

Proof 1
By definition, the deleted integer topology is a partition topology.

Let $\mathcal P$ be the partition which is the basis for $T$, i.e.:


 * $\mathcal P = \left\{{\left({n - 1 . . n}\right): n \in \Z_{> 0}}\right\}$

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

Hence the result.

Proof 2
We have that a Countably Compact Space is Weakly Countably Compact.

However, we also have that the Deleted Integer Topology is Not Weakly Countably Compact.

Hence the result by the Rule of Transposition.