---
tag: 0Z57EC6YX3
kind: definition
name: Product topology
msc: [54B10]
defines: 0GBG8B49AA
uses: [0CNQXVBHCX, 06EX0FHBTF, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Product_topology
    locator: Definition
---

## Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be [[topological spaces|0CNQXVBHCX]].
The product topology on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the topology with [[basis|06EX0FHBTF]]
$$\{ U \times V : U \in \tau_X, \ V \in \tau_Y \}.$$

## 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|044D4AQV8V]]. The product topology on $\mathbb{R} \times \mathbb{R}$ is the usual topology of $\mathbb{R}^2$.
The [[projections are continuous|0PM88Z9W4M]]. For products of any family of spaces, see the [[product topology on an arbitrary product|04S9HX3GAP]].

## 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.