---
tag: 0W41XG4W60
kind: statement
role: proposition
name: Complete subspaces are closed
msc: ["54E50"]
uses: [08EWT7J0SC, 003QP7A8HG, 0T2WYDTRP2]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_metric_space
    locator: Some theorems
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A \subseteq X$.
If $A$ with the [[subspace metric|0T2WYDTRP2]] is [[complete|08EWT7J0SC]], then $A$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: let $x \in \overline{A}$. By [[closure is the set of limits of sequences|003QP7A8HG]], there is a sequence in $A$ with limit $x$. It is a Cauchy sequence in $A$, so it has a limit in $A$. Limits are unique in $X$, so $x \in A$.
The space $X$ need not be complete here.
For example, $\mathbb{Q}$ is not closed in $\mathbb{R}$, so $\mathbb{Q}$ is not complete.
In a complete space, a subset is complete if and only if it is closed, by [[closed subsets of complete spaces are complete|0FBXJZX3MK]].

## Explanations

### L1 human

A complete subset contains the limits of its Cauchy sequences, and thus of its convergent sequences.
A set that keeps all its limits is closed.