---
tag: 04S9HX3GAP
kind: definition
name: Product topology on an arbitrary product
aliases: [product-of-topological-spaces]
msc: [54B10]
defines: 0GBG8B49AA
uses: [0CNQXVBHCX, 0CHR35YNJQ, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Product_topology
    locator: Definition
---

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]], indexed by any set $I$, and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]], with projections $\pi_i \colon X \to X_i$.
The product topology on $X$ is the topology generated by the [[subbasis|0CHR35YNJQ]]
$$\{ \pi_i^{-1}(U) : i \in I, \ U \text{ open in } X_i \}.$$

## Remarks

A basis consists of the sets $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$.
For two factors, this is the [[product topology|0Z57EC6YX3]] of $X_1 \times X_2$.
It is the coarsest topology on $X$ for which each projection $\pi_i$ is continuous.
A sequence converges in $X$ if and only if each of its coordinates converges. The [[box topology|0CTM1EHG9P]] is finer and lacks this property.

## Explanations

### L1 human

In an infinite product, a basic open set can restrict only finitely many coordinates.
This choice makes the [[universal property|0M6W4XNST6]] and [[Tychonoff's theorem|0JTRSJKCFB]] work.