Let be a family of topological spaces, and let be the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) of the sets , that is
Recall that an element is a function such that for each , and that for each the projection map is defined by for each .
The (Tychonoff) product topology for is defined to be the initial topology with respect to the projection maps; that is, is the smallest topology such that each is continuous (http://planetmath.org/Continuous).
If is open, then is an open set in . Note that this is the set of all elements of in which the component is restricted to and all other components are unrestricted. The open sets of are the unions of finite intersections of such sets. That is,
is a subbase for the topology on .
The following theorems assume the product topology on . Notation is as above.
Let be a topological space and let be a function. Then is continuous if and only if is continuous for each .
The product topology on is the topology induced by the subbase
The product topology on is the topology induced by the base
A net in converges to if and only if each coordinate converges to in .
Each projection map is continuous and open (http://planetmath.org/OpenMapping).
For each , let . Then
In particular, any product of closed sets is closed.
(Tychonoff’s Theorem) If each is compact, then is compact.
Comparison with box topology
There is another well-known way to topologize , namely the box topology. The product topology is a subset of the box topology; if is finite, then the two topologies are the same.
The product topology is generally more useful than the box topology. The main reason for this can be expressed in terms of category theory: the product topology is the topology of the direct categorical product (http://planetmath.org/CategoricalDirectProduct) in the category Top (see Theorem 1 above).
- 1 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
|Date of creation||2013-03-22 12:47:09|
|Last modified on||2013-03-22 12:47:09|
|Last modified by||CWoo (3771)|
|Synonym||Tychonoff product topology|