Metric space
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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 candidatex— finite-carrier pointy— 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 tripleThe original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.
Prerequisites
- Set — A metric has a fixed carrier.
- Function mapping — Distance assigns a unique real value to every ordered point pair.
- Binary relations and set projections — Pairs of points index the distance table.
- Real number — Distances are nonnegative real values.
- Usual order on the real numbers — 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 · Plain Markdown