Product topology

Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. The product topology on the Cartesian product $X \times Y$ is the topology with basis $$\{ U \times V : U \in \tau_X, \ V \in \tau_Y \}.$$

Remarks

This family is a basis of a topology, because $(U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2)$. A set $W \subseteq X \times Y$ is open if and only if for each $(x, y) \in W$ there are open $U \ni x$ and $V \ni y$ with $U \times V \subseteq W$. For metric spaces, it is the metric topology of the product metric. The product topology on $\mathbb{R} \times \mathbb{R}$ is the usual topology of $\mathbb{R}^2$. The projections are continuous. For products of any family of spaces, see the product topology on an arbitrary product.

Explanations

L1 human

The open sets of $X \times Y$ are built from open rectangles $U \times V$. In the plane, open rectangles and open discs give the same topology.