Definition:Product Space (Topology)/Two Factor Spaces

Definition
Let $T_1 = \left({X_1, \tau_1}\right)$ and $T_2 = \left({X_2, \tau_2}\right)$ be topological spaces.

Let $X_1 \times X_2$ be the cartesian product of $X_1$ and $X_2$.

The product topology $\tau$ for $X_1 \times X_2$ is the topology with basis $\mathcal B = \left\{{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}\right\}$.

Factor Space
Each of the topological spaces $\left({X_i, \tau_i}\right)$ are called the factors of $\left({X, \mathcal T}\right)$, and can be referred to as factor spaces.

Also see

 * Product Topology is Topology