I know that the product topology on a collection of topologies $X_i$ is defined by the basis $$ B = \left \{ \Pi \, U_i : \text{Finitely many } U_i \text{ are open in } X_i, \text{ all other } U_i = X_i \right \}.$$
Why is this said to be the basis of $X = \Pi X_i$ and not simply the topology for $X$? The union of any two elements in $B$ is simply going to be another element of $B$. What am I missing?
The collection $B$ is closed under finite intersections, and covers $X$. So it is a base for some topology on $X$ which we call the product topology by definition.
Even finite unions of such rectangles need not be open at all, consider the plane. And an open circle in that plane is a union of (infinitely many) open rectangles, but certainly not in $B$. So $B$ is not a topology, merely a base for one. But to describe the topology it's enough to know a base for it (this already determines continuity, convergence, compactness etc.)