Sphere space

The $𝑛$ -sphere $𝕊 𝑛$ is usually defined as the boundary of the unit ball in $(𝑛 + 1)$ -dimensional Real coördinate space $ℝ 𝑛 + 1$ , #m/def/topology i.e.

𝕊 𝑛 = {𝑣 \in ℝ 𝑛 + 1 : ‖ 𝑛 ‖ = 1} \subseteq ℝ 𝑛 + 1

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

Properties

The $𝑛$ -sphere is a Compact space.
Retraction theorem: No retraction from ball to boundary sphere
Borsuk-Ulam theorem: $𝑓 : 𝕊 𝑛 \to ℝ 𝑛$ has a set of equal-valued antipodes

1-sphere

Degree of a circle endomorphism

#state/tidy | #lang/en | #SemBr