Definition:Box Topology

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

Let $S$ be the cartesian product of $\mathbb S$:
 * $\displaystyle S := \prod_{i \mathop \in I} S_i$

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


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

Let $\mathcal S$ be the set defined as:
 * $\mathcal S = \left\{{\operatorname {pr}_i^{-1} \left({U}\right) : i \in I, U \in \tau_i}\right\}$

That is, let $\mathcal S$ be the set whose elements are of the form:
 * $S_1 \times S_2 \times \cdots \times S_{i-1} \times U_i \times S_{i+1} \times \cdots$

where $U_i$ is an open set of $\left({S_i, \tau_i}\right)$.

Define:
 * $\displaystyle \tau := \left\{{\bigcap \mathcal T: \mathcal T \subseteq \mathcal S}\right\}$

That is, $\tau$ is the set of intersections of all subsets of $\mathcal S$.

Then $\tau$ is known as the box topology on $S$.

The topological space $\left({S, \tau}\right)$ is called the box product of $\mathbb S$.

Also see

 * Box Topology is a Topology