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:
 * $\BB = \set{\displaystyle \prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$

Proof
From Natural Basis of Tychonoff Topology, the natural basis on $X$ is the set $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where:
 * for all $k = 1, \dots, n: U_k \in \tau_i$
 * for all but finitely many indices $k : U_k = X_k$

Let $\BB' = \set{\displaystyle \prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$

By definition of $\BB$:
 * $\BB \subseteq \BB'$

Let $B \in \BB'$, that is:
 * $B = \displaystyle \prod_{k \mathop = 1}^n U_k$

where $\forall k : U_k \in \tau_k$.

Let $J = \set{k : U_k \neq X_k}$.

Since $J \subseteq \set{ 1, \dots, n}$ is finite then:
 * for all but finitely many indices $k : U_k = X_k$

Thus $B \in \BB$.

Since $B$ was arbitrary, then $\BB' \subseteq \BB$.

It follows that $\BB = \BB'$ from the definition of set equality.