# Talk:Filter on Product Space Converges to Point iff Projections Converge to Projections of Point

By the definition of the natural basis, there is a finite set $J \subseteq I$ such that $\displaystyle U = \bigcap_{j \in J}U_j$ where $U_j \subseteq X_j$ is an open set for all $j \in J$.
By the definition of the natural basis, there is a finite set $J \subseteq I$ such that $\displaystyle U = \bigcap_{j \in J} \pi^{-1}(U_j)$ where $U_j \subseteq X_j$ is an open set for all $j \in J$.