Products of Sets
If and are two non-empty sets, then the Cartesian product is defined as .
Let and be non-empty sets, then they can be defined by the following two functions:
(1) defined as for all
(2) defined as for all .
The above maps are called the projection maps on and respectively.
Note: Let be non-empty sets, then the projection maps can be defined similarly.
Let be the product of topological spaces and . The coarsest topology on with respect to which the projection maps and are continuous, is said to be a product topology and thus the space is said to be the product space.
• It may be observed that if and are distinct topological spaces then the collection form a subbase for product topology on .
• It may be noted that if and are any open interval, then will be open rectangle strips. A collection of open rectangles form a basis for the usual topology on . So, generalizing this fact to the product of a finite number of spaces and are topological spaces then form a basis for product topology.