Product topology

From Topospaces

This article is about a basic definition in topology.
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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:

  1. It is a topology with basis given by open rectangles in , viz., sets of the form where is open in and is open in .
  2. 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
  3. 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 .
  4. 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:

  1. It is a topology with basis given by subsets of the form , where are open subsets of , and for all but finitely many , .
  2. 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 , .
  3. 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
  4. 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.