When is the product of two quotient maps a quotient map? It is not true in general that the product of two quotient maps is a quotient maps (I don't know any examples though).
Are any weaker statements true? For example, if $X, Y, Z$ are spaces and $f : X \to Y$ is a quotient map, is it true that $ f \times {\rm id} : X \times Z \to Y \times Z$ is a quotient map?

If the quotient maps $p : W \rightarrow X, q:Y \rightarrow Z$ are open, then the product map is also a quotient map:
$U\times V$ open in $X \times Y \implies p^{-1}U\times q^{-1}{V} = (p\times q)^{-1}(U\times V)$ open.
Conversely, $U\times V = (p\times q)(\dots) = p\circ p^{-1}U\times q\circ q^{-1}V$ is open since $p,q$ are open maps.
