Product topology

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This article is about a basic definition in topology.
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Definition for two spaces

Let A,B be topological spaces. Then, we can consider the set A \times B: the Cartesian product of A and B, which is the set of ordered pairs (a,b) where a \in A and b \in B. The product topology on A \times B is defined in the following equivalent ways:

  1. It is a topology with basis given by open rectangles in A \times B, viz., sets of the form U \times V where U is open in A and V is open in B.
  2. Suppose we choose a basis for A and a basis for B. Then, it is a topology with basis given by basis rectangles in A \times B, viz., sets of the form U \times V, where U is a basis element of A and V is a basis element of B
  3. It is a topology with subbasis given by open cylinders in A \times B, viz., sets of the form A \times V where V is open in B, or of the form U \times B where U is open in A.
  4. It is the coarsest topology on A \times B for which the projection maps to A and B are both continuous.

Definition for an arbitrary family of spaces

Let I be an indexing set and \{ A_i \}_{i \in I} be a family of topological spaces. Consider the set:

P := \times_{i \in I} A_i

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

  1. It is a topology with basis given by subsets of the form \times_{i \in I} U_i, where U_i are open subsets of A_i, and for all but finitely many i, U_i = A_i.
  2. Suppose we choose a basis for each A_i. Then, it is a topology with basis given by subsets of the form \times_{i \in I} U_i, where U_i are basis elements of A_i, and for all but finitely many i, U_i = A_i.
  3. It is a topology with subbasis given by open cylinders: subsets of the form \times_{i \in I} U_i, where all the U_is are open subsets of A_i, and at most one U_i is a proper subset of the corresponding A_i
  4. It is the coarsest topology on P so that the projection maps to each of the A_i 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.