Double Pointed Discrete Real Number Space is not Lindelöf

Theorem

Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers.

Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$.

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 $\PP$ is defined as:

$\PP = \set {\set {\paren {s, a}, \paren {s, b} }: s \in \R}$

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

But $\PP$ has no countable subcover.

Hence the result, by definition of Lindelöf space.

$\blacksquare$