Natural Basis of Product Topology/Finite Product

Theorem
Let $n \in \N$.

For all $k \in \set {1, \ldots, n}$, let $\struct {X_k, \tau_k}$ be topological spaces.

Let $\displaystyle X = \prod_{k \mathop = 1}^n X_k$ be the cartesian product of $X_1, \ldots, X_n$.

Then the natural basis on $X$ is:
 * $\BB = \set{\displaystyle \prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$