Deleted Integer Topology is not Countably Compact

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

Then the topological space $T = \struct {S, \tau}$ is not countably compact.

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

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


 * $\PP = \set {\openint {n - 1} n: n \in \Z_{> 0} }$

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

Hence the result.