350: n-Sphere Is Path-Connected

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A description/proof of that n-sphere is path-connected

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of n-sphere.
• The reader knows a definition of path-connected topological space.
• The reader admits the proposition that any topological spaces map is continuous at any point if there are super $C^{\mathrm{\infty }}$ manifolds of the topological spaces and a map between them which (the map) restricts to the original map on a chart open set of the domain manifold around the point whose (the manifolds map's) coordinates function is continuous.

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Target Context

• The reader will have a description and a proof of the proposition that the n-sphere, $S^n$, is path-connected.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

The n-sphere, $S^n$, is path-connected.

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2: Proof

$S^n$ is a subspace of the ${\mathbb{R}}^{n+1}$ Euclidean topological space. For any points, $p_1,p_2\in S^n$, a global chart for ${\mathbb{R}}^{n+1}$ can be chosen such that the origin is at the center of $S^n$ and $p_1=(0,0,...,1)$ and $p_2=(0,0,...,sin{\theta }_2,cos{\theta }_2)$ where $0\le {\theta }_2<\pi$. Choose the path, $\lambda :[0,1]\to S^n,t\mapsto (0,0,...,sin({\theta }_{2}t),cos({\theta }_{2}t))$, which is continuous as the coordinates function. By the proposition that any topological spaces map is continuous at any point if there are super $C^{\mathrm{\infty }}$ manifolds of the topological spaces and a map between them which (the map) restricts to the original map on a chart open set of the domain manifold around the point whose (the manifolds map's) coordinates function is continuous, $\lambda$ is continuous as the $[0,1]\to S^n$ map.

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References

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