# 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 has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality.

mko-metric-space · definition · v0.10.0



## Mathematical statement

A metric d:X×X→R on a set X satisfies nonnegativity and separation, symmetry and the triangle inequality. A complete finite distance table permits cell-by-cell and triple-by-triple auditing of a specified candidate.





## Representative formula (exact TeX)

d(x,y)





## Assumptions


- Exercises use a finite carrier labeled 0 to n−1 and a complete integer distance table.
- Finite-table acceptance proves only the specified model; exhaustive finite checks do not extend to arbitrary infinite metric spaces.





## Symbol scope


- d — distance-function candidate
- x — finite-carrier point
- y — finite-carrier point





## Intuition, definition, and misconceptions

Diagonal entries must be zero and off-diagonal entries must not be zero; together these express separation.

Symmetry requires d(i,j)=d(j,i).

The triangle inequality must be checked for every i,j,k as d(i,k)≤d(i,j)+d(j,k).

Do not call a table a metric after checking only nonnegativity and symmetry or omit off-diagonal zeros and triangle inequalities.





## Adopted derivation guide

[Metric space: auditing a finite distance table cell by cell and triple by triple](https://ocme.evemisslab.com/en/derivations/derivation-mko-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


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — A metric has a fixed carrier.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Distance assigns a unique real value to every ordered point pair.
- [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) — Pairs of points index the distance table.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Distances are nonnegative real values.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Nonnegativity and triangle inequalities use order.





## Verification boundary

Only complete finite tables are checked; no arbitrary function space or infinite carrier is covered.

not_formalized

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





## Next obligation

Add open balls, metric convergence, Cauchy sequences and completeness.





## 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/metric-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-metric-space/index.md)