Topology MOC

Sphere space

The 𝑛-sphere 𝕊𝑛 is usually defined as the boundary of the unit ball in (𝑛 +1)-dimensional Real coördinate space ℝ𝑛+1, topology i.e.

𝕊𝑛={𝑣∈ℝ𝑛+1:‖𝑛‖=1}⊆ℝ𝑛+1

along with the subspace metric and subspace topology. It is thereby an 𝑛-dimensional Topological manifold, and forms the boundary of the (𝑛 +1)-Ball space.

Properties

• The 𝑛-sphere is a Compact space.
• Retraction theorem: No retraction from ball to boundary sphere
• Borsuk-Ulam theorem: 𝑓 :𝕊𝑛 →ℝ𝑛 has a set of equal-valued antipodes

1-sphere

• Degree of a circle endomorphism

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