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.