ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The Kuratowski embedding of a finite metric space

Example

Let $M = {o, a, b}$ with metric determined by

$d (o, a) = 1, d (a, b) = 1, d (o, b) = 2.$

Using $o$ as basepoint, the based Kuratowski map $K_{o} : M \to C_{b} (M)$ is given by

$K_{o} (o) = (0, 0, 0), K_{o} (a) = (1, - 1, - 1), K_{o} (b) = (2, 0, - 2),$

where each function is recorded by its values on $(o, a, b)$ . These three points in $R^{3}$ are at the same pairwise sup distances as the original metric space.

Facts & Assumptions

Given: The three-point metric space $M = {o, a, b}$ above.

[L1]

The based Kuratowski map preserves all distances (The based Kuratowski distance map is an isometric embedding).

Verification

technique · direct

1.1givenalgebra

By definition, $K_{o} (x) (z) = d (x, z) - d (o, z)$ . Evaluating this on the three points gives $K_{o} (o) = (0, 0, 0)$ , $K_{o} (a) = (1, - 1, - 1)$ , and $K_{o} (b) = (2, 0, - 2)$ .

2.1L1step 1.1algebra∎

The sup distances are $∥ K_{o} (a) - K_{o} (o) ∥_{\infty} = 1$ , $∥ K_{o} (b) - K_{o} (a) ∥_{\infty} = 1$ , and $∥ K_{o} (b) - K_{o} (o) ∥_{\infty} = 2$ , exactly matching $d (o, a)$ , $d (a, b)$ , and $d (o, b)$ . This is the concrete instance promised by [L1].

Depends on

The based Kuratowski distance map is an isometric embedding

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

James Dugundji, Topology (standard reference, not scraped)