Basis for Box Topology

Theorem
Let $\left \langle {\left({X_i, \tau_i}\right)} \right \rangle_{i \in I}$ be an $I$-indexed family of topological spaces.

Let $X$ be the cartesian product of $\left \langle {X_i} \right \rangle_{i \in I}$, i.e.:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

Define:
 * $\displaystyle \mathcal B := \left\{{\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}\right\}$

Then $\mathcal B$ is a synthetic basis on $X$.

Proof
Let us check the two conditions for $\mathcal B$ to be a synthetic basis in turn.

$(B1)$: Covering
From open set axiom $(O3)$, we have $X_i \in \tau_i$ for all $i \in I$.

Thus $X = \displaystyle \prod_{i \mathop \in I} X_i \in \mathcal B$.

Hence by Subset of Union: General Result, we have:


 * $X \subseteq \displaystyle \bigcup \mathcal B$

so $\mathcal B$ is a cover for $X$.

$(B2)$: Intersections are Unions
Let $A = \displaystyle \prod_{i \mathop \in I} U_i$ and $B = \displaystyle \prod_{i \mathop \in I} V_i$ be in $\mathcal B$.

Then by Cartesian Product of Intersections, we have:


 * $A \cap B = \displaystyle \prod_{i \mathop \in I} \left({U_i \cap V_i}\right)$

By open set axiom $(O2)$, $U_i \cap V_i \in \tau_i$ for all $i \in I$.

Hence $A \cap B \in \mathcal B$, and in particular, by Union of Singleton:


 * $A \cap B = \displaystyle \bigcup \left\{{A \cap B}\right\}$

Having verified both conditions, we conclude $\mathcal B$ is a synthetic basis on $X$.

Also see

 * Box Topology
 * Synthetic Basis and Analytic Basis are Compatible