Definition:Product Topology

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 $\mathbb X$:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

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 $X_i$:


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

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

Then $\mathcal T$ is called the Tychonoff topology on $X$.

Alternatively, $\mathcal T$ is the initial topology on $X$ with respect to $\left\{{\operatorname {pr}_i: i \in I}\right\}$.

Natural Subbasis
$\mathcal S$ is called the natural subbasis of $X$.

Natural Basis
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 $X$.

Also see

 * Tychonoff Topology is Coarsest Topology such that Projections are Continuous