Deleted Integer Topology is Weakly 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 weakly countably compact.

Proof
Let $A \subseteq S$ such that $A$ is infinite.

Let $x \in A$.

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

Let $U$ be the union of open real intervals of the form $\left({n-1 .. n}\right)$, and hence open in $T$.

Now if $x \in U$, it will be an element in some $\left({n-1 .. n}\right)$.

So there will exist $y \in U$ which will also be an element in that $\left({n-1 .. n}\right)$.

So, by definition, $x$ is a limit point of $A$.

Hence $T$ is weakly countably compact.