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

Any two nonempty open sets of $Y$ are cofinite by part (iii). Their complements have finite union, so the two sets must intersect because $Y$ is infinite. Thus $Y$ is a hyperconnected space.

A Hausdorff space with at least two points has disjoint nonempty open neighbourhoods of two distinct points. This is impossible here. Therefore $Y$ is not Hausdorff.