Proper maps in terms of projection from pullback

Question

I've read that a continuous $f:X\rightarrow S$ is proper (inverse images of compacts are compacts) iff for all other continuous maps $g:Y\rightarrow S$ the projection $X\times _SY\rightarrow Y$ in the pullback square below (in $\mathsf{Top}$) is a closed map. $$\require{AMScd} \begin{CD} X\times _S Y @>>> Y\\ @VVV @VV{g}V\\ X @>>{f}> S \end{CD}$$

1. How does one prove this?
2. Does it fail if we drop the assmptions $f,g$ are continuous? I.e we let the pullback be in $\mathsf{Set}$, with the pullback still topologized as a subspace of the product.