Definition:Product Topology

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

The Tychonoff topology on $X$ is defined as the initial topology $\tau$ on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

By definition of the initial topology on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$, $\tau$ is generated by the sub-basis:
 * $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

Also known as
The Tychonoff topology is also known as the product topology and both of these terms are commonly used on.

Also see

 * Natural Basis of Tychonoff Topology
 * Natural Basis of Tychonoff Topology of Finite Product
 * Tychonoff Topology is Coarsest Topology such that Projections are Continuous
 * Product Space is Product in Category of Topological Spaces
 * Definition:Product Space (Topology)

Relation between Tychonoff and Box Topology

 * Results about the relation between the Tychonoff topology and the box topology can be found here.