The proof is very similar to Example 3 of §12. The empty set and $X$ are in the collection because their complements are $X$ and the empty set, the complement of any union of open sets is the intersection of the countable complements of these sets, so it is countable as well, finally, the complement of the finite intersection of open sets is the union of the countable complements, so it is countable.

Now, $\mathcal{T}_{\infty}$ is the trivial topology if $X$ is finite, but if $X$ is infinite then it is not a topology, as, for example, we can partition $X$ into three disjoint sets $X=X_{1}\cup X_{2}\cup\{x\}$ such that the first two sets are infinite, and then $X_{1}$ and $X_{2}$ are open but their union is not.