Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$

Depends on

• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
• Open ball, closed ball and sphere in a metric space
• $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it

Used by

• A closed three-dimensional ball of radius r≥0 has volume 4π r³/3 Corollary
• For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact Corollary
• For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected Corollary
• One member of every three-set closed cover of S² contains an antipodal pair Corollary
• There is no continuous injection from S² into ℝ² Corollary
• A connected plane domain that is not homologically simply connected Counterexample
• Four closed sets can cover S² without any one containing an antipodal pair Counterexample
• Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
• Solids of revolution about a coordinate axis Definition
• Orthogonal projection S²→ℝ² has exactly one antipodal pair with equal image Example
• Radial normalization retracts the punctured disk, but it cannot extend to the disk Example
• The Euclidean closed ball and sphere worked through the compactness equivalence chart Example
• π₁(ℝ²∖{0})≅ℤ Example
• A fixed-point-free self-map of the disk produces a continuous retraction onto the unit circle Lemma
• Antipodal complements cover Sⁿ by simply connected sets with path-connected overlap for n≥2 Lemma
• Based sphere maps have finite affine bubble normal forms Lemma
• Radial normalisation x↦ x/‖ x‖₂ is continuous on ℝⁿ∖{0} Lemma
• The exterior of a closed disc in the plane is path-connected Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The Hawaiian earring is compact and path-connected Proposition
• Borsuk–Ulam theorem in dimension two Theorem
• Closed Euclidean balls are Jordan measurable and their volumes satisfy the slicing recursion Theorem
• For n≥1, radial normalisation is a deformation retraction of ℝⁿ∖{0} onto Sⁿ⁻¹ Theorem
• For n≥1, the map H(x,t)=((1-t)+t/‖ x‖₂)x is continuous on (ℝⁿ∖{0})×[0,1], starts at x, ends at radial normalisation, fixes the unit sphere, and never reaches 0 Theorem
• Lower-dimensional sphere maps are based nullhomotopic Theorem
• The volume of a three-ball by Cavalieri's cylinder-minus-cones proof Theorem
• There is no retraction of the closed disk onto the unit circle Theorem

Dependency tree · two levels

35 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

• Euclidean space (standard reference, not scraped)
• Sphere (standard reference, not scraped)