Doubt about open sets in the Product topology So we know that in the product topology a basis element is gonna be of the form $\prod U_\alpha$ where  $ U_\alpha \neq X_\alpha $ for a finite number of $\alpha$. So my thing is that we know that a product of closed sets in the product topology is gonna be closed so why is not true that if i do $ \prod X_\alpha - \prod F_\alpha = \prod X_\alpha-F_\alpha$ im gonna get a contradiction. I know this isnt true i just cant seem to grasp it, Thanks in advance.

Because $\prod_\alpha X_\alpha\setminus\prod_\alpha F_\alpha\neq\prod_\alpha(X_\alpha\setminus F_\alpha)$. In fact, $\prod_\alpha X_\alpha\setminus\prod_\alpha F_\alpha$ is the union of all the products $\prod_\alpha Y_\alpha$ where $Y_\alpha=X_\alpha\setminus F_\alpha$ for one specific $\alpha$, whereas for all other $\alpha$'s you have $Y_\alpha=X_\alpha$.
