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 $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:

$\ds \XX := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:

$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

Let $\tau$ be the product topology on $S$.

Let $\SS$ be the natural sub-basis on $X$:

$\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

The natural basis on $\SS$ is defined as the basis generated by $\SS$.