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

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

That is, $\tau$ is the topology generated by:
 * $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.

Natural Sub-Basis
By definition of the initial topology on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$, $\tau$ is generated by the natural sub-basis.

Also see

 * Definition:Product Space (Topology): the topological space $\struct {X, \tau}$


 * Natural Basis of Product Topology: $\tau$ is generated from the basis $\BB$ of cartesian products of the form $\ds \prod_{i \mathop \in I} U_i$ where:
 * for all $i \in I : U_i \in \tau_i$
 * for all but finitely many indices $i$: $U_i = X_i$


 * Natural Basis of Product Topology of Finite Product
 * Product Topology is Coarsest Topology such that Projections are Continuous
 * Product Space is Product in Category of Topological Spaces

Relation between Product and Box Topology

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