# Complete metric space

A **complete metric space** is a metric space \((X, d)\) in which every [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) of points of \(X\) converges to a point of \(X\). A sequence \((x_n)\) is Cauchy when for every \(\epsilon > 0\) there is an \(N\) such that \(d(x_n, x_m) < \epsilon\) for all \(n, m \geq N\); its terms eventually lie arbitrarily close to each other, whether or not they approach a limit. Completeness asks that this kind of mutual convergence always be witnessed by an actual limit inside the space.<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup><sup> • </sup><sup>[2](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \((X, d)\) is complete when every Cauchy sequence in \(X\) converges to an element of \(X\)<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup> |
| Basic examples | \(\mathbb{R}\) and \(\mathbb{R}^d\) are complete; \(\mathbb{Q}\) is not<sup>[2](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)</sup> |
| Subspaces | A subspace of a complete metric space is complete if and only if it is closed<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup> |
| Compactness | A metric space is compact if and only if it is complete and totally bounded<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup> |
| Completion | Every metric space has a completion, a complete space containing a dense isometric copy of it, unique up to isometry<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup> |
| Fixed points | A contraction \(T: X \to X\) with \(d(Tx, Ty) \leq \lambda\, d(x, y)\), \(\lambda \in (0,1)\), on a complete \(X\) has a unique fixed point<sup>[4](https://www.ma.imperial.ac.uk/~svanstri/Files/de-3rd.pdf)</sup> |
| Baire property | In a complete metric space, a countable family of open dense sets has a dense intersection<sup>[2](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)</sup> |

## Cauchy sequences and the definition

In any metric space, every convergent sequence is Cauchy: if \(x_n \to x\), then for \(\epsilon > 0\) choose \(N\) with \(d(x_n, x) < \epsilon/2\) for \(n > N\), and the triangle inequality gives \(d(x_m, x_n) \leq d(x_m, x) + d(x_n, x) < \epsilon\) for all \(m, n > N\).<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup> The converse fails in general, and completeness is exactly the statement that the converse holds: a space is complete when Cauchy and convergent mean the same thing for its sequences.<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup><sup> • </sup><sup>[6](https://www.studocu.com/en-us/document/university-of-california-los-angeles/analysis/131b-f23-outline-lecture-notes/79344072)</sup>

The definition depends on the metric, not merely on the underlying set or its topology. The interval \((0, 1)\) with the usual metric is not complete, because a sequence tending to 0 is Cauchy but has no limit in \((0,1)\), while the closed interval \([0, 1]\) is complete.<sup>[6](https://www.studocu.com/en-us/document/university-of-california-los-angeles/analysis/131b-f23-outline-lecture-notes/79344072)</sup> Yet \((0,1)\) and \(\mathbb{R}\) are homeomorphic as topological spaces, one complete and the other not, so completeness is a metric property rather than a topological one.<sup>[7](https://momsbasementuniversity.com/topology/compactness-in-metric-spaces/)</sup> Changing the metric can also create completeness: on \((0, \infty)\), the metric \(d(x, y) = |x - y| + |x^{-1} - y^{-1}|\) is complete even though the usual metric is not, because a Cauchy sequence for this metric forces both \(x_n\) and \(x_n^{-1}\) to converge, ruling out escape toward 0.<sup>[8](https://www2.math.upenn.edu/~gressman/analysis/08a-completions.html)</sup>

## Standard examples

The real line \(\mathbb{R}\) with its usual metric is complete; the standard proof uses the least upper bound axiom.<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup> From it, \(\mathbb{R}^d\) is complete by showing each coordinate sequence is Cauchy and converging coordinate by coordinate.<sup>[2](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)</sup> The rationals \(\mathbb{Q}\) are not complete: a sequence of rationals converging in \(\mathbb{R}\) to an irrational such as \(\sqrt{2}\) is Cauchy but has no rational limit.<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup>

**Subspaces inherit completeness through closedness.** A subspace \(A\) of a complete space \((X, d)\) is complete if and only if \(A\) is closed in \(X\). If \(A\) is closed, a Cauchy sequence in \(A\) converges in \(X\), and closedness places the limit in \(A\); conversely, if \(A\) is not closed, some sequence in \(A\) converges to a point outside \(A\), giving a non-convergent Cauchy sequence in \(A\).<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup><sup> • </sup><sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup>  This is why closed intervals in \(\mathbb{R}\) are complete.<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup>

Any set with the discrete metric, \(d(m, n) = 0\) if \(m = n\) and 1 otherwise, is complete, since every Cauchy sequence is eventually constant.<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup> Incomplete spaces also arise naturally in analysis: the polynomials on a closed interval \([a, b]\) with the metric of uniform convergence, \(d(x, y) = \max_{t \in [a,b]} |x(t) - y(t)|\), are not complete, because a sequence of polynomials converging uniformly to a continuous non-polynomial function is Cauchy without a limit in the space; likewise, continuous functions on \([0,1]\) with the metric \(d(x,y) = \int_0^1 |x(t) - y(t)|\,dt\) form an incomplete space.<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup>

The most important complete metric spaces in analysis are the complete normed vector spaces, called Banach spaces; \(\mathbb{R}^n\) with any \(\ell^p\)-norm, \(1 \leq p \leq \infty\), is a [Banach space](https://www.edgepedia.org/banach-space).<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup>

## Relation to compactness

Completeness and total boundedness together characterize compactness in metric spaces. A space is totally bounded when for every \(\epsilon > 0\) it can be covered by finitely many balls of radius \(\epsilon\).<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup> The theorem states that a metric space is compact if and only if it is both complete and totally bounded.<sup>[1](https://math.hws.edu/eck/metric-spaces/completeness.html)</sup><sup> • </sup><sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup>  This is the correct generalization of the Heine–Borel picture: in a general metric space, "closed" must be replaced by "complete" and "bounded" by "totally bounded".<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup>

Both halves of the condition are needed. The set \([0, 2] \cap \mathbb{Q}\) is closed and bounded in \(\mathbb{Q}\) but not compact, since a rational sequence converging to \(\sqrt{2}\) has no convergent subsequence in \(\mathbb{Q}\). Conversely, \(\mathbb{N}\) with the discrete metric is complete and bounded but not compact, because the sequence \(x_n = n\) has no convergent subsequence; it is bounded but not totally bounded.<sup>[5](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)</sup> In a complete space the criterion simplifies: a subset is compact if and only if it is closed and totally bounded.<sup>[9](https://notes.amittai.studio/real-analysis/metric-spaces/compactness)</sup>

## Completion

Every metric space can be completed. The completion theorem says that for a metric space \(X\) there exists a complete metric space \(\hat{X}\) containing a subspace \(W\) isometric with \(X\) and dense in \(\hat{X}\), and that \(\hat{X}\) is unique up to isometry: any other complete space with a dense subspace isometric to \(X\) is isometric to \(\hat{X}\).<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup> An isometry is a distance-preserving map, and isometric spaces are indistinguishable from the metric viewpoint.<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup>

The construction assigns limits to Cauchy sequences in \(X\) that do not converge, identifying two Cauchy sequences \((x_n)\) and \((x'_n)\) when \(\lim_{n \to \infty} d(x_n, x'_n) = 0\), and taking equivalence classes with the metric \(d_{\hat{X}}(\hat{x}, \hat{y}) = \lim_{n \to \infty} d(x_n, y_n)\).<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup> The process is suggested by the completion of the rational line: the real numbers are the completion of \(\mathbb{Q}\).<sup>[3](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)</sup> If \(X\) is already complete, every Cauchy sequence converges, so every class contains a constant sequence and \(\hat{X}\) is \(X\) itself.<sup>[10](https://calc-path.com/real-analysis/modules/metric-spaces)</sup>

## Why completeness matters

**Fixed point theorems.** The Banach fixed point theorem states that if \(X\) is a complete metric space and \(F: X \to X\) satisfies \(d(F(x), F(y)) \leq \lambda\, d(x, y)\) for all \(x, y\), with \(\lambda \in (0,1)\), then \(F\) has a unique fixed point \(p\) with \(F(p) = p\). The iteration \(x_{n+1} = F(x_n)\) produces a Cauchy sequence, and completeness supplies its limit, which the contraction property forces to be a fixed point; uniqueness follows from \(d(p, q) \leq \lambda\, d(p, q)\).<sup>[4](https://www.ma.imperial.ac.uk/~svanstri/Files/de-3rd.pdf)</sup> This theorem underlies the Picard–Lindelöf existence and uniqueness theorem for ordinary differential equations, where the contraction is defined on the complete metric space of continuous functions on a compact interval.<sup>[11](https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/resources/ocw_18100b-lec23-2025may08_mp4/)</sup>

**Baire category.** The Baire category theorem says that in a complete metric space, the intersection of a countable family of open dense sets is again dense.<sup>[2](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)</sup> Completeness is the hypothesis that makes such countable-intersection arguments work, and they in turn yield existence results throughout analysis.<sup>[12](https://arxiv.org/abs/2402.04565)</sup>

Completeness also behaves well under products: the completeness of a product of complete spaces follows from the completeness of each component.<sup>[13](https://arxiv.org/abs/2111.02904)</sup>

## History and further directions

Maurice Fréchet, a French mathematician who founded the theory of abstract spaces, introduced the concept of a metric space in his 1906 doctoral dissertation *Sur quelques points du calcul fonctionnel*; the name "metric space" itself is due to Hausdorff.<sup>[14](https://mathshistory.st-andrews.ac.uk/Biographies/Frechet/)</sup> During the 1920s, metric spaces were central in analysis through Banach spaces, complete normed vector spaces in which the norm provides the metric.<sup>[15](https://doi.org/10.1016%2Fj.hm.2008.01.001)</sup>

Completeness remains an active organizing notion. A 2024 result by Yoshito Ishiki shows that for every metrizable space, the space \(\mathrm{Met}(X)\) of all metrics generating the same topology, equipped with the supremum metric, is Baire, and that when \(X\) is completely metrizable the set of all complete metrics is comeager in \(\mathrm{Met}(X)\); in this sense, generic metrics on such a space are complete.<sup>[12](https://arxiv.org/abs/2402.04565)</sup>

## References

1. [Metric Spaces: Completeness, David Eck, Hobart and William Smith Colleges](https://math.hws.edu/eck/metric-spaces/completeness.html)
2. [Sequences and Series, Gautam Iyer, Carnegie Mellon University](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)
3. [Metric Spaces — Completion of Metric Spaces, George Voutsadakis, Lake Superior State University](https://www.voutsadakis.com/TEACH/LECTURES/FUNCTIONAL/Chapter1.pdf)
4. [MA2AA1 (ODE's) Lecture Notes, Stefan van Vliet? (Imperial College London), Banach Fixed Point Theorem](https://www.ma.imperial.ac.uk/~svanstri/Files/de-3rd.pdf)
5. [Completeness, Compactness, and Continuity, John Hunter, UC Davis](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch13.pdf)
6. [Math 131B F23 Lecture Notes, UCLA](https://www.studocu.com/en-us/document/university-of-california-los-angeles/analysis/131b-f23-outline-lecture-notes/79344072)
7. [Compactness in Metric Spaces, Mom's Basement University](https://momsbasementuniversity.com/topology/compactness-in-metric-spaces/)
8. [Advanced Analysis: Complete Metric Spaces, Götz Gressman, University of Pennsylvania](https://www2.math.upenn.edu/~gressman/analysis/08a-completions.html)
9. [Compactness, Amittai Axelrod study notes](https://notes.amittai.studio/real-analysis/metric-spaces/compactness)
10. [Real Analysis — Metric Spaces & Topology, calc-path.com](https://calc-path.com/real-analysis/modules/metric-spaces)
11. [Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem, MIT OpenCourseWare 18.100B, Spring 2025](https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/resources/ocw_18100b-lec23-2025may08_mp4/)
12. [Yoshito Ishiki (2024). Spaces of metrics are Baire. arXiv:2402.04565](https://arxiv.org/abs/2402.04565)
13. [Compactness of any Countable Product of Compact Metric Spaces without Tychonoff's Theorem. arXiv:2111.02904](https://arxiv.org/abs/2111.02904)
14. [Maurice Fréchet (1878–1973), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Frechet/)
15. [The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica](https://doi.org/10.1016%2Fj.hm.2008.01.001)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*

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