Definition:Product Topology/Natural Basis

Definition
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$:
 * $\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

Let $\mathcal S$ be the natural sub-basis on $X$:
 * $\mathcal S = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

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