Either-Or Topology is Lindelöf

Theorem
Let $T = \left({S, \tau}\right)$ be the either-or space.

Then $T$ is a Lindelöf space.

Proof
Any open cover $\mathcal C$ of $T$ must contain an open set of $T$ which contains $0$.

So $\left({-1 \,.\,.\, 1}\right)$ will always be covered by one set in $\mathcal C$, leaving just $-1$ and $1$ possibly needing to be included in at most two other sets.

So $\mathcal C$ has a subcover containing at most three sets.

Hence $T$ is a Lindelöf space by definition.