box topology

Let ${\{(X_{\alpha },{𝒯}_{\alpha })\}}_{\alpha \in A}$be a family of topological spaces.Let $Y$ denote the generalized Cartesian product of the sets $X_{\alpha }$,that is

$$Y=\prod _{\alpha \in A}{X}_{\alpha }.$$

Let $ℬ$ denote the set of all products of open sets of the correspondingspaces, that is

$$ℬ=\left\{\prod _{\alpha \in A}{U}_{\alpha }\right|U_{\alpha }\in {𝒯}_{\alpha }\text{ for all }\alpha \in A\}.$$

Now we can construct the box product $(Y,𝒮)$, where $𝒮$,referred to as the box topology,is the topology the base $ℬ$.

When $A$ is a finite (http://planetmath.org/Finite) set,the box topology coincides with the product topology.

Example

As an example,the box product of two topological spaces $(X_0,{𝒯}_0)$ and $(X_1,{𝒯}_1)$is $(X_0\times X_1,𝒮)$,where the box topology $𝒮$ (which is the same as the product topology)consists of all sets of the form${\bigcup }_{i\in I}(U_i\times V_i)$,where $I$ is some index setand for each $i\in I$ we have $U_i\in {𝒯}_0$ and $V_i\in {𝒯}_1$.