Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
For , every Euclidean closed ball and every Euclidean sphere of positive radius is compact
Statement
For , , and , the Euclidean closed ball and Euclidean sphere are compact.
Facts & Assumptions
Given: , , and .
The sets and are respectively the points satisfying and (Euclidean spheres and closed balls as subspaces of ).
The Euclidean norm is continuous and satisfies (The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for ).
The Euclidean compactness theorem identifies compactness with closedness and boundedness (For a nonempty subset of with , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).
Euclidean open sets are the metric-open sets, and metric boundedness means containment in a ball (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).
Proof
Both sets are bounded: lies in the ball of radius about , and .
The complement of is open: if , then the ball about of radius stays in the complement by [L2].
The complement of is open: if , then a ball about of radius avoids the sphere by [L2].
Thus both sets are closed and bounded, hence compact by [L3].
Depends on
- Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$
- For a nonempty subset of $\mathbb{R}^n$ with $n\ge1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
Used by
- ℝⁿ is locally compact and σ-compact Corollary
- The Euclidean closed ball and sphere worked through the compactness equivalence chart Example
- Coordinate balls form a basis of a topological manifold Lemma
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point Theorem
- Every boundary point belonging to a nonempty Euclidean convex set has a supporting hyperplane Theorem
- Stereographic projection identifies the Riemann sphere with the unit two-sphere Theorem
Dependency tree · two levels
48 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
- Heine-Borel theorem (standard reference, not scraped)