LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The unit-interval circle is a nonempty compact metric space

Statement

The circle $T = [0, 1)$ with

$d (x, y) = \min (∣ x - y ∣, 1 - ∣ x - y ∣)$

is a nonempty compact metric space.

Facts & Assumptions

Given: The interval model and circle metric of The circle, rotations and the doubling map.

[F1]

The ordinary interval $[0, 1]$ is compact by Heine–Borel (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F2]

A continuous image of a compact space is compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F3]

Topological and metric compactness agree for a metric topology (For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide).

Proof

technique · direct continuous-image argument

1.1

Define $q : [0, 1] \to T$ by $q (t) = t$ for $t < 1$ and $q (1) = 0$ . Equivalently $q (t) = {t}$ . For $s, t \in [0, 1]$ , the circle-distance formula gives $d (q (s), q (t)) \leq ∣ s - t ∣,$ including when one endpoint is $1$ . Thus $q$ is continuous. It is surjective because every $x \in [0, 1)$ equals $q (x)$ .

givenalgebra

2.1

By [F1], $[0, 1]$ is compact. Its continuous image under $q$ is all of $T$ , so [F2] makes the circle compact; [F3] reads this in the metric sense. Finally $0 \in T$ , so it is nonempty.

F1F2F3step 1.1∎

Depends on

Used by

A free irrational torus action that is not proper Counterexample
Irrational circle rotations are uniquely ergodic Theorem

Dependency tree · two levels

55 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

Stacks Project, Section 5.12 (Tag 0059): Quasi-compact spaces and maps (standard reference, not scraped)