Double Pointed Discrete Real Number Space is not Lindelöf

Theorem
Let $T_R = \left({\R, \vartheta_R}\right)$ be the (uncountable) discrete space.

Let $T_D = \left({D, \vartheta_D}\right)$ 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 not a Lindelöf space.

Proof
We have that $T$ is a partition topology, whose basis $\mathcal P$ is defined as:
 * $\mathcal P = \left\{{\left\{{\left({s, a}\right), \left({s, b}\right)}\right\}: s \in \R}\right\}$

We have that $\mathcal P$ is an open cover of $T$.

But as $\mathcal P$ has no countable subcover.

Hence the result, by definition of Lindelöf space.