For $n\ge 2$, the sphere $S^{n-1}$ is path-connected and connected

Depends on

• For $n\ge2$, the punctured space $\mathbb{R}^n\setminus\{0\}$ is polygonally connected
• Radial normalisation $x\mapsto x/\lVert x\rVert_2$ is continuous on $\mathbb{R}^n\setminus\{0\}$
• A finite concatenation of straight segments in $\mathbb{R}^n$ is a continuous path
• Every path-connected space is connected, and every path component lies inside a component
• Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$

Used by

• Hartogs extension by a compact-support dbar correction Corollary
• Sphere self-maps are homotopic exactly when their degrees agree Corollary
• A connected plane domain that is not homologically simply connected Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• Quaternionic clutching bundles ξ_h,j over S⁴ Definition
• Cut locus of a point on a round sphere Example
• Cutoff extension across a puncture in complex dimension two Example
• The formal frame homotopy behind sphere eversion Example
• A section curvature lower bound makes triangles thinner than the model False statement
• FALSE: every connected subset of ℝⁿ is polygonally connected False statement
• The cut locus of a point is always a smooth hypersurface False statement
• The laplace beltrami definition licenses the use of all euclidean harmonic function theory on manifolds False statement
• A simply connected surface without a Green kernel is plane or sphere Lemma
• Degree two of ℘ and its four branch points Lemma
• Every integer is realized by a map to the sphere Lemma
• General linear frame spaces have the Stiefel cohomology gap Lemma
• Milnor sphere bundles are simply connected Lemma
• Puncturing a connected open subset of ℝⁿ preserves path-connectedness for n≥2 Lemma
• Stiefel manifolds are connected in positive codimension and simply connected in codimension at least two 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
• Compactly supported dbar solutions on complex Euclidean space Theorem

Dependency tree · two levels

24 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

• Sphere (standard reference, not scraped)
• Path-connected space (standard reference, not scraped)