Double Pointed Real Number Space is Weakly Countably Compact

Theorem

Let $T_R = \left({\R, \tau_R}\right)$ be the (uncountable) discrete space on the set of real numbers.

Let $T_D = \left({D, \tau_D}\right)$ be the indiscrete topology on the doubleton $D = \left\{{a, b}\right\}$.

Let $T = T_R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_R$ and $T_D$.

Then $T$ is weakly countably compact.

Proof

We have that $T$ is a partition topology, whose basis $\mathcal P$ is defined as:

$\mathcal P = \left\{{\left\{{\left({s, a}\right), \left({s, b}\right)}\right\}: s \in \R}\right\}$

Let $A \subseteq \R \times D$ such that $A$ is infinite.

Let $x \in A$.

Let $U$ be the union of sets of the form $\left\{{\left({s, a}\right), \left({s, b}\right)}\right\}$, and hence open in $T$.

Now if $x \in U$, it will be an element in some $\left\{{\left({s, a}\right), \left({s, b}\right)}\right\}$.

So there will exist $y \in U$ which will also be an element in that $\left\{{\left({s, a}\right), \left({s, b}\right)}\right\}$.

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

So, by definition, $T$ is weakly countably compact.

$\blacksquare$