Definition:Product Topology/Natural Basis

Definition
Let $\left\langle {\left({X_i, \tau_i}\right)} \right\rangle_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

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

For each $i \in I$, let $\operatorname {pr}_i: X \to X_i$ denote the $i$th projection on $X$:
 * $\forall \left\langle{x_j}\right\rangle_{j \mathop \in I} \in X: \operatorname {pr}_i \left({\left\langle{x_j}\right\rangle_{j \mathop \in I}}\right) = x_i$

Let $\mathcal S$ be the natural sub-basis on $X$:
 * $\mathcal S = \left\{{\operatorname {pr}_i^{-1} \left({U}\right): i \in I, \, U \in \tau_i}\right\}$

The natural basis on $X$ is defined as the basis generated by $\mathcal S$.