If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map

$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \ldots, x_n) \mapsto (f_1(x_1), \ldots, f_n(x_n))$$ doesn't need to be closed. However the question is: are there some nice conditions on $X_i$, $Y_i$ (like compactness, connection, Hausdorff...) such that $f$ will be closed? I have looked in "Bourbaki - General Topology Part 1,2" and I have found nothing about it. By the way I'm more interested to the case $X_1 = \cdots = X_n$, $Y_1 = \cdots = Y_n$.

Thank you for your help!