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

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The based Kuratowski distance map is an isometric embedding

Statement

Let $(M, d)$ be a nonempty metric space, fix $o \in M$ , and let $K_{o} : M \to C_{b} (M)$ be the based Kuratowski distance map of The based Kuratowski distance map into bounded continuous functions. Then $K_{o}$ is an isometric embedding in the sense of Isometry, isometric embedding, and the subspace metric on a subset:

$∥ K_{o} (x) - K_{o} (y) ∥_{\infty} = d (x, y) (x, y \in M) .$

Facts & Assumptions

Given: A nonempty metric space $(M, d)$ , a basepoint $o \in M$ , and the map $K_{o} : M \to C_{b} (M)$ .

[L1]

For each $x \in M$ , $K_{o} (x) (z) = d (x, z) - d (o, z)$ defines a bounded continuous function on $M$ (The based Kuratowski distance map into bounded continuous functions).

[L2]

An isometric embedding preserves all distances (Isometry, isometric embedding, and the subspace metric on a subset).

Proof

technique · direct

1.1

By [L1], the map $K_{o}$ is well defined as a map into $C_{b} (M)$ . For any $x, y, z \in M$ , the triangle inequality gives $d (x, z) \leq d (x, y) + d (y, z) and d (y, z) \leq d (y, x) + d (x, z),$ so $∣ K_{o} (x) (z) - K_{o} (y) (z) ∣ = ∣ d (x, z) - d (y, z) ∣ \leq d (x, y) .$ Taking the supremum over $z$ yields $∥ K_{o} (x) - K_{o} (y) ∥_{\infty} \leq d (x, y)$ .

L1algebra

2.1

Evaluating at $z = y$ gives $∣ K_{o} (x) (y) - K_{o} (y) (y) ∣ = ∣ d (x, y) - 0 ∣ = d (x, y),$ and evaluating at $z = x$ gives the same value. Therefore the supremum norm is at least $d (x, y)$ .

step 1.1algebra

3.1

Steps 1.1 and 2.1 give $∥ K_{o} (x) - K_{o} (y) ∥_{\infty} = d (x, y)$ for all $x, y \in M$ , which is exactly [L2]. Hence $K_{o}$ is an isometric embedding.

L2step 1.1step 2.1∎

Remarks

The proof is two lines long once the target is chosen correctly. The real work is the based definition, which keeps the functions bounded on unbounded metric spaces.

Depends on

Used by

The Kuratowski embedding of a finite metric space Example
Kuratowski-Wojdyslawski embedding theorem Theorem

Dependency tree · two levels

7 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)