# Complete metric space

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A metric space is complete when every Cauchy sequence converges to a point still belonging to that space.

mko-complete-metric-space · definition · v0.10.0



## Mathematical statement

A metric space (X,d) is complete if every Cauchy sequence in X converges to some point of X. For a finite space with distinct points, the least positive distance forces every Cauchy tail eventually to be constant; the singleton case is immediate. Testing a few sequences does not establish general completeness.





## Representative formula (exact TeX)

C(X)





## Assumptions


- The finite certificate uses a complete finite metric table on a carrier with at least one point; the minimum-positive-distance exercise protocol uses at least two.
- Sequence exercises accept only an explicit ultimately constant certificate or an explicit non-Cauchy counterexample; a finite prefix is not a universal completeness proof.





## Symbol scope


- X — metric space
- d — metric
- x — sequence within the space





## Intuition, definition, and misconceptions

Completeness is a property of the space, not of one selected sequence.

In a finite metric space, minimum positive separation gives a universal certificate, with the one-point case trivial.

A Cauchy sequence whose limit leaves a subspace can refute its completeness; this obligation is not replaced by a finite simulation here.

A few tested convergent sequences, or convergence in an ambient space, do not establish that the subspace is complete.





## Adopted derivation guide

[Complete metric space: proving every finite metric space complete by the least positive distance, contrasted with (0,1]](https://ocme.evemisslab.com/en/derivations/derivation-mko-complete-metric-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Cauchy sequence](https://ocme.evemisslab.com/en/mko/mko-cauchy-sequence/) — Completeness quantifies over all Cauchy sequences.
- [Convergence in a metric space](https://ocme.evemisslab.com/en/mko/mko-metric-convergence/) — The conclusion requires convergence inside the same space.
- [Metric space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) — The complete metric supplies the distance obligations.





## Verification boundary

A complete finite-metric certificate covers that finite space only; sampled sequences do not establish general-space completeness.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Together with norm-induced metrics, add real Banach spaces.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/complete-metric-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/index.md)