Definition:Product Space (Topology)/Two Factor Spaces

Definition
Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces.

Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

Let $\tau$ be the product topology on $S_1 \times S_2$.

From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the natural basis:
 * $\BB = \set {U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$

The topological space $\struct {S_1 \times S_2, \tau}$ is called the product space of $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$.

Also see

 * Natural Basis of Tychonoff Topology
 * Natural Basis of Tychonoff Topology of Finite Product
 * Tychonoff Topology is Coarsest Topology such that Projections are Continuous
 * Product Space is Product in Category of Topological Spaces