Products of Sets:
If and are two non-empty sets, then Cartesian product is defined as .
Let and be non-empty sets, then it can be defined 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 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. Collection of open rectangle form a bases for usual topology on . So, generalizing this fact to the product of finite number of spaces and are topological spaces then form a bases for product topology.