Weak Countable Compactness is not Preserved under Continuous Maps

Theorem
Let $T_A = \left({S_A, \tau_A}\right)$ be a topological space which is weakly countably compact.

Let $T_B = \left({S_B, \tau_B}\right)$ be another topological space.

Let $\phi: T_A \to T_B$ be a continuous mapping.

Then $T_B$ is not necessarily weakly countably compact.

Proof
Let $\Z_{>0}$ be the strictly positive integers:
 * $\Z_{>0} = \left\{{1, 2, 3, \ldots}\right\}$

Let $T_A = \left({\Z_{>0}, \tau_A}\right)$ be the odd-even topology.

Let $T_B = \left({\Z_{>0}, \tau_B}\right)$ be the discrete topology on $\Z_{>0}$.

Let $\phi: T_A \to T_B$ be the mapping:
 * $\phi \left({2k}\right) = k, \phi \left({2k-1}\right) = k$

Then:
 * $\phi^{-1} \left({k}\right) = \left\{{2k, 2k-1}\right\} \in \tau_A$

demonstrating that $\phi$ is continuous.

Now we have that the Odd-Even Topology is Weakly Countably Compact.

But we also have that a Countable Discrete Space is not Weakly Countably Compact.

Hence the result.