Past exam of the topology course of the University of Oxford / 2014 / 1 / b / iii / Solution

Any proper closed subset $F$ of $Y$ must be finite: if it were infinite, part (ii) would give $F=\overline{F}=Y$. Therefore every nonempty open subset of $Y$, being the complement of a proper closed set, is cofinite.The empty set belongs to both topologies. Inclusion need not be equality; the target may have fewer open sets.