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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.