Suppose $V$ is a product space of connected and separable spaces $V_1,...,V_n$. Let $W$ be a subspace of $V$ with the subspace topology. Is $W$ then a product space of subsets of $V_1,...,V_n$?
My attempt: It is not a product space of subsets of $V_i$. Is $V=(\{a,b\}\times\{c,d\},\tau)$ with $\tau$ the product topology and $W=(\{(a,c),(b,d)\},\tau_W)$ with $\tau_W$ the subspace topology a suitable counterexample?
Well, you could literally just write down every possible product of two subspaces, and see if any of them turn out to be $W$.