Is the sphere $S^n$ always arcwise connected?

Question

I have a small question about the connectedness of the sphere;

Is the sphere $S^n$ always arcwise connected ?

Thank you.

Answer 1

The answer is yes. For $n>0$, we can use the sterographic projection. This is the jist of the result. Given two points, choose some third point, and remove it, then apply the sterographic projection to flatten our space into $\mathbb{R}^n$, and our two points can be connected by a straight line, and then unapply it to get our line. For $n=0$, $S^0$ is composed of $\{-1, 1\}$, and so there is clearly no arc between them.