Countable Complement Space is Lindelöf
Then $T$ is a Lindelöf space.
Let $\mathcal C$ be an open cover of $T$.
Let $U \in \mathcal C$ be any set in $C$.
So for each of those points we pick an element of $\mathcal C$ which covers each of those points.
Hence we have a countable subcover of $T$.
So by definition $T$ is a Lindelöf space.