PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The nonempty metric spaces quasi-isometric to a one-point space are exactly those of finite diameter

Statement

The nonempty metric spaces quasi-isometric to a one-point space are exactly those of finite diameter.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L1]

A map is $(L, C)$ -coarse Lipschitz when $d (f (x), f (x^{'})) \leq L d (x, x^{'}) + C$ , and an $(L, C)$ -quasi-isometric embedding when in addition $L^{- 1} d (x, x^{'}) - C \leq d (f (x), f (x^{'}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

[L2]

Bounded subset. $A$ is bounded if $A = \emptyset$ or there are $x_{0} \in X$ and a real $r > 0$ with $A \subseteq B (x_{0}, r)$ . (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Proof

technique · direct

1.1

Let $X$ be nonempty and bounded, so $X \subseteq B (x_{0}, r)$ for some $x_{0} \in X$ and $r > 0$ . The constant map $c : X \to {*}$ and the map $s : {*} \to X$ with $s (*) = x_{0}$ are coarse Lipschitz; one has $c \circ s = {id}_{{*}}$ , and for every $x \in X$ the distance between $s (c (x)) = x_{0}$ and $x$ is less than $r$ . Hence $c$ is a quasi-isometry.

F1L1L2choose

2.1

Conversely, if $c : X \to {*}$ is a quasi-isometry and $s : {*} \to X$ is a quasi-inverse, then for some $r > 0$ every $x \in X$ satisfies $d_{X} (s (c (x)), x) = d_{X} (s (*), x) < r$ . Thus $X \subseteq B (s (*), r)$ and is bounded, hence has finite diameter.

F1L2step 1.1∎

Depends on

Used by

A finitely generated group is finite if and only if it is quasi-isometric to a point Proposition

Dependency tree · two levels

16 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)