Let $\tau$ be a topology on a topological space $X \times Y$ which is not a product topology. Consider the subspace topology $\tau_X$ and $\tau_Y$ induced by the topology $\tau$. My question is would the product topology $\tau_X \times \tau_Y$ on $X \times Y$ coincide with the topology $\tau$.
I tried proving this by showing open sets in one topology is contained in other. I think I can show that $\tau \subset \tau_X \times \tau_Y$ but I am not able to show the other way round.