Product Topology

Products of Sets

If {X_1} and {X_2} are two non-empty sets, then the Cartesian product {X_1} \times {X_2} is defined as {X_1} \times {X_2} = \left\{ {\left( {{x_i},{x_j}:{x_i} \in {X_1},{x_j} \in {X_2}} \right)} \right\}.


Projection Maps

Let A and B be non-empty sets, then they can be defined by the following two functions:
(1) {p_1}:A \times B \to A defined as {p_1}\left( {a,b} \right) = a for all \left( {a,b} \right) \in A \times B
(2) {p_2}:A \times B \to B defined as {p_2}\left( {a,b} \right) = b for all \left( {a,b} \right) \in A \times B.

The above maps are called the projection maps on A and B respectively.

Note: Let {X_1},{X_2},{X_3}, \ldots {X_n} be non-empty sets, then the projection maps {p_1},{p_2},{p_3}, \ldots {p_n} can be defined similarly.


Product Topology

Let {X_1} \times {X_2} be the product of topological spaces {X_1} and {X_2}. The coarsest topology \tau on {X_1} \times {X_2} with respect to which the projection maps {p_1}:{X_1} \times {X_2} \to {X_1} and {p_2}:{X_1} \times {X_2} \to {X_2} are continuous, is said to be a product topology and thus the space \left( {{X_1} \times {X_2},\tau } \right) is said to be the product space.


• It may be observed that if {X_1} and {X_2} are distinct topological spaces then the collection S = \left\{ {p_1^{ - 1}\left( {{G_1}} \right)p_2^{ - 1}\left( {{G_2}} \right):{G_1} \in {\tau _1},{G_2} \in {\tau _2}} \right\} form a subbase for product topology on {X_1} \times {X_2}.
• It may be noted that if A and B are any open interval, then A \times B will be open rectangle strips. A collection of open rectangles form a basis for the usual topology on {\mathbb{R}^2}. So, generalizing this fact to the product of a finite number of spaces \left( {{X_1},{\tau _1}} \right) and \left( {{X_2},{\tau _2}} \right) are topological spaces then {\rm B} = \left\{ {{G_1} \times {G_2}:{G_1} \in {\tau _1},{G_2} \in {\tau _2}} \right\} form a basis for product topology.