Weak Countable Compactness is not Preserved under Continuous Maps

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

Let $T_B = \struct {S_B, \tau_B}$ 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:
 * $\Z_{>0} = \set {1, 2, 3, \ldots}$

Let $T_A = \struct {\Z_{>0}, \tau_A}$ be the odd-even topology.

Let $T_B = \struct {\Z_{>0}, \tau_B}$ be the discrete topology on $\Z_{>0}$.

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

Then:
 * $\map {\phi^{-1} } k = \set {2 k, 2 k - 1} \in \tau_A$

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.