---
tag: JJ7D0CYECZ
kind: concept
name: Zero-dimensional space
msc: [54F45, 54G05]
sources:
  - url: https://en.wikipedia.org/wiki/Zero-dimensional_space
same_as:
  wikidata: Q1801244
  wikipedia: Zero-dimensional space
---

## Summary

A zero-dimensional space is a [[topological space|0CQHFPGJ45]] that has a base of [[clopen sets|DQ4E55T3BW]].
In terms of dimension theory, a nonempty space with this property has small [[inductive dimension|MKS9H497AM]] zero.
Examples include discrete spaces, the rational numbers and the [[Cantor set|0BXNDZFQME]].
Zero-dimensional $T_0$ spaces are [[totally disconnected|0RJSBR9GW1]], and for compact Hausdorff spaces the two notions agree.

## Explanations

### L1 human

In a zero-dimensional space, every small neighbourhood can be chosen with no boundary at all.
Such a set is both open and closed, so the space falls apart into tiny separate pieces.
The rational numbers are like this: the set of rationals between two irrational numbers is both open and closed in $\Q$.
The real line is not, because the only sets that are both open and closed in $\R$ are $\emptyset$ and $\R$.