---
tag: 0CTM1EHG9P
kind: definition
name: Box topology
msc: [54B10]
defines: 0BWJTRE3X3
uses: [0CNQXVBHCX, 06EX0FHBTF, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Box_topology
    locator: Definition
---

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]] and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]].
The box topology on $X$ is the topology with [[basis|06EX0FHBTF]] the set of all products $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$.

## Remarks

For finitely many factors, the box topology is the [[product topology|04S9HX3GAP]].
For infinite products, the box topology is finer than the product topology, and often strictly finer.
On $\mathbb{R}^{\mathbb{N}}$ with the box topology, the map $t \mapsto (t, t, t, \ldots)$ from $\mathbb{R}$ is not continuous, although each coordinate is continuous. The preimage of $\prod_{n \ge 1} (-1/n, 1/n)$ is $\{0\}$, which is not open.

## Explanations

### L1 human

The box topology lets you restrict all coordinates at the same time.
This looks natural, but it gives too many open sets, and simple maps into the product stop being continuous.