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.

If $x \notin \prod_{\alpha \in J} F_\alpha$ this means that $x(\beta) \notin F_\beta$ for some $\beta \in J$, and then the basic open set $\prod_{\alpha \in J} O_\alpha$ with $O_\beta=X-F_\beta$ and all other $O_\alpha = X_\alpha$ is a neighbourhood of $x$ that misses $\prod_{\alpha \in J} F_\alpha$ entirely.
