# box topology

Let ${\{({X}_{\alpha},{\mathcal{T}}_{\alpha})\}}_{\alpha \in A}$
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 $\mathcal{B}$ denote the set of all products^{} of open sets of the corresponding
spaces, that is

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

Now we can construct the *box product* $(Y,\mathcal{S})$, where $\mathcal{S}$,
referred to as the box topology,
is the topology^{} the base $\mathcal{B}$.

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 $({X}_{0},{\mathcal{T}}_{0})$ and $({X}_{1},{\mathcal{T}}_{1})$
is $({X}_{0}\times {X}_{1},\mathcal{S})$,
where the box topology $\mathcal{S}$ (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 set^{}
and for each $i\in I$ we have ${U}_{i}\in {\mathcal{T}}_{0}$ and ${V}_{i}\in {\mathcal{T}}_{1}$.

Title | box topology |
---|---|

Canonical name | BoxTopology |

Date of creation | 2013-03-22 12:46:55 |

Last modified on | 2013-03-22 12:46:55 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54A99 |

Synonym | box product topology |

Related topic | ProductTopology |

Defines | box product |