Deleted Integer Topology is not Countably Compact

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the deleted integer topology on the set $S = \R_+ \setminus \Z$.

Then $T$ is not countably compact.

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

Let $\mathcal P$ be the partition which is the basis for $T$:
 * $\mathcal P = \left\{{\left({n - 1 . . n}\right): n \in \Z, n > 0}\right\}$

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

Hence the result.