 Solution

Assume (iii). Take $Y=X\times X$ and let $f,g$ be the two coordinate projections, continuous by the definition of the product topology. Their equality set is the diagonal, so it is closed. For distinct $x_1,x_2$, the point $(x_1,x_2)$ lies in its open complement. Choose a basic rectangle $U_1\times U_2$ containing it and contained in that complement. If $U_1\cap U_2$ were nonempty, a point $u$ in their intersection would give $(u,u)$ in the rectangle, impossible. Thus $U_1,U_2$ are disjoint open neighborhoods of $x_1,x_2$, proving that $X$ is Hausdorff. Hence (iii) implies (i), completing all three equivalences.