Definition:Product Topology

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

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

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


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

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

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

Natural Sub-Basis
The natural sub-basis on $X$ is defined as:
 * $\mathcal S = \left\{{\operatorname {pr}_i^{-1} \left({U}\right): i \in I, \, U \in \tau_i}\right\}$

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

Also known as
The Tychonoff product is also known as the direct product or topological product, but these terms are less precise and there exists the danger of confusion with other similar uses of these terms in different contexts.

Also see

 * Tychonoff Topology is Coarsest Topology such that Projections are Continuous
 * Projection from Product Topology is Open