# Thread: Show that I^n and S^n are path-connected?

1. ## Show that I^n and S^n are path-connected?

Show that I^n (unit cube) and S^n (n > 0) are both path-connected by constructing a path.

2. ## Re: Show that I^n and S^n are path-connected?

Originally Posted by Borkborkmath
Show that I^n (unit cube) and S^n (n > 0) are both path-connected by constructing a path.
The cube you can literally just do a piecewise linear map (go down the appropriate amount and over the appropriate amount), for $\displaystyle \mathbb{S}^n$ just paramaterize it with two charts and go from there. Try this and report back if you have any issues.

3. ## Re: Show that I^n and S^n are path-connected?

What do you mean by paramaterize it with two charts?
Sorry, if my question is silly

4. ## Re: Show that I^n and S^n are path-connected?

Originally Posted by Borkborkmath
What do you mean by paramaterize it with two charts?
Sorry, if my question is silly
No, I'm sorry, it's not a silly question, and you really need more than two charts theoretically. But, even that's just stupid. If it's on a sphere, just think about a great circle which contains the two points and take either of the paths induced from that.