Definition:Product Space (Topology)/Two Factor Spaces

Definition
Let $\mathbb X = \left \langle {\left({X_i, \vartheta_i}\right)}\right \rangle_{i \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

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

For each $i \in I$, let $\operatorname {pr}_i : X \to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$:


 * $\forall \left({x_i}\right)_{i \in I} \in X: \operatorname {pr}_i \left({\left({x_i}\right)_{i \in I}}\right) = x_i$

The initial topology $\mathcal T$ on $X$ with respect to the family $\left \langle {\operatorname {pr}_i}\right \rangle_{i \in I}$ is called the product topology on $X$.

The topological space $\left({X, \mathcal T}\right)$ is called the direct product of the $\left \langle \left({X_i, \vartheta_i}\right)\right \rangle_{i \in I}$.

$\mathcal T$ is generated by $\mathcal S = \left\{{ \operatorname {pr}_i^{-1} \left({U}\right) : i \in I, U \in \vartheta_i}\right\}$.

This topology is called the Tychonoff topology, and is the coarsest topology on $X$ such that all the $\operatorname {pr}_i$ are continuous.

The basis $\displaystyle \mathcal S^* = \left\{{\bigcap S : S \subseteq \mathcal S \text{ finite}}\right\}$ (which is generated by $\mathcal S$) is called the natural basis of $X$.