# 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.

Remarks
• 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.