# Product topology

This article is about a basic definition in topology.VIEW: Definitions built on this | Facts about this | Survey articles about this

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## Definition for two spaces

Let be topological spaces. Then, we can consider the set : the Cartesian product of and , which is the set of ordered pairs where and . The **product topology** on is defined in the following equivalent ways:

- It is a topology with basis given by
*open rectangles*in , viz., sets of the form where is open in and is open in . - Suppose we choose a basis for and a basis for . Then, it is a topology with basis given by
*basis rectangles*in , viz., sets of the form , where is a*basis element*of and is a*basis element*of - It is a topology with subbasis given by
*open cylinders*in , viz., sets of the form where is open in , or of the form where is open in . - It is the coarsest topology on for which the projection maps to and are both continuous.

## Definition for an arbitrary family of spaces

Let be an indexing set and be a family of topological spaces. Consider the set:

viz., the Cartesian product of all the s. The **product topology** on is given in the following equivalent ways:

- It is a topology with basis given by subsets of the form , where are open subsets of , and for all but finitely many , .
- Suppose we choose a basis for each . Then, it is a topology with basis given by subsets of the form , where are
*basis elements*of , and for all but finitely many , . - It is a topology with subbasis given by
*open cylinders*: subsets of the form , where all the s are open subsets of , and at most one is a proper subset of the corresponding - It is the coarsest topology on so that the projection maps to each of the is continuous.

## Related notions

Box topology is another topology on the Cartesian product of topological spaces, where the basis is all *open boxes* or *open rectangles* (i.e., we don't have the *all but finitely many* condition). For products of finitely many topological spaces, the box topology coincides with the product topology; in general, it is a finer topology.

Properties of topological spaces that are closed under taking products with the product topology, are listed in Category:Properties of topological spaces closed under products. Those properties that are closed under taking products of finitely many spaces, are listed in Category:Properties of topological spaces closed under finite products.