A description/proof of that n-sphere is path-connected
Topics
About: topological space
The table of contents of this article
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 \(C^\infty\) 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.
Target Context
- The reader will have a description and a proof of the proposition that the n-sphere, \(S^n\), is path-connected.
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.
Main Body
1: Description
The n-sphere, \(S^n\), is path-connected.
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 \(0 \leq \theta_2 \lt \pi\). Choose the path, \(\lambda: [0, 1] \rightarrow S^n, t \mapsto (0, 0, . . ., sin (\theta_2 t), cos (\theta_2 t))\), which is continuous as the coordinates function. By the proposition that any topological spaces map is continuous at any point if there are super \(C^\infty\) 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 \([0, 1] \rightarrow S^n\) map.
