Suppose $X$ is a countable set. Consider $(X, \tau_x)$ with the discrete topology. Denote another Polish topological space as $(Y, \tau_y)$, which is not neccesarily the discrete topology.
Consider a Polish topology $\tau$ on $X \times Y$, such that $\tau_x \times \tau_y \subseteq \tau$ and their Borel sets are the same $\mathcal{B}(\tau_x \times \tau_y) = \mathcal{B}(\tau)$. My question is, is it always possible to find a Polish topology $\tau'_y$ on the factor space $Y$, such that $\tau_y \subseteq \tau'_y$, $\mathcal{B}(\tau_y) = \mathcal{B}(\tau'_y)$ and importantly, we can write $\tau = \tau_x \times \tau'_y$?
It is also relevant to this question.
Many thanks in advance!
