---
tag: W5S52WZZ9T
kind: definition
name: Zero-dimensional space
msc: [54F45, 54G05]
defines: JJ7D0CYECZ
uses: [0CQHFPGJ45, DQ4E55T3BW]
sources:
  - url: https://en.wikipedia.org/wiki/Zero-dimensional_space
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *zero-dimensional* if it has a base consisting of [[clopen sets|DQ4E55T3BW]], that is, of sets that are both open and closed.

## Remarks

Example: $\Q$ with the subspace topology from $\R$ is zero-dimensional.
For irrational numbers $a < b$, the set $\Q \cap (a, b) = \Q \cap [a, b]$ is clopen in $\Q$.
These sets form a base, since irrational numbers are dense in $\R$.

Non-example: $\R$ is not zero-dimensional.
Since $\R$ is [[connected|0V032VWTHD]], its only clopen sets are $\emptyset$ and $\R$, and they do not form a base.

Some authors define zero-dimensional by [[Lebesgue covering dimension|A3W7RGXXK9]] zero instead.
The two definitions agree for separable metric spaces, but not for all spaces.