= 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 \b[(iii) implies (i)], completing all three equivalences.
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