Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 12A / i / Solution

Assume $X$ is Hausdorff. For any $(x_1,x_2)$ outside the diagonal, the points differ, so choose disjoint open neighborhoods $U_1,U_2$. Their rectangle contains $(x_1,x_2)$ and misses the diagonal. Thus the complement of the diagonal is open in the product topology, proving (i) implies (ii). This is one direction of the closed diagonal theorem.