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).
(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|