Solution
= Solution
Assume the diagonal $\Delta\subset X\times X$ is closed. For <continuous> $f,g:Y\to X$, the map $h:Y\to X\times X$, $h(y)=(f(y),g(y))$, is <continuous>: the inverse image of a basic rectangle $U\times V$ is $f^{-1}(U)\cap g^{-1}(V)$. Therefore
$$
\{y:f(y)=g(y)\}=h^{-1}(\Delta)
$$
is closed as the inverse image of a <closed set> under a <continuous map>. This proves \b[(ii) implies (iii)].