Let ,

First let be Hausdorff. and , so . Since is Hausdorff, there exist open neighbourhoods and of and respectively such that , and thus is an open neighbourhood of contained in . Hence every point in has an open neighbourhood contained in , therefore is open, whence is closed.

Now let be closed, i.e. be open. Let with . Then there exists an open neighbourhood of . Canonical projections are open, hence and are open neighbourhoods of and respectively, which do not intersect. Hence is Hausdorff.