I know that the projections are not closed maps in presence of the product topology. However, later in a chapter, they show that $\prod_\alpha F_\alpha$ is closed in $\prod_\alpha X_\alpha$ if and only if $\forall \alpha: F_\alpha$ is closed in $X_\alpha$.
But does this not contradict the fact that projections are not closed maps, since $P_\beta(\prod_\alpha F_\alpha) = F_\beta$.
Could anyone point out the flaw in my reasoning?