My Math Forum Path Connected Set?

 Real Analysis Real Analysis Math Forum

 October 4th, 2011, 02:45 PM #1 Member     Joined: Jun 2011 From: California Posts: 82 Thanks: 3 Math Focus: Topology Path Connected Set? Hi all! This is an old comp problem: how can I show that, in $\mathbb{R}^n$, the set $S_n := \left\{ x \in \mathbb{R}^n : ||x|| = 1\right\}$ is path connected?
 October 4th, 2011, 03:11 PM #2 Member   Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0 Re: Path Connected Set? You can show that the set is convex. So you have to show: $t\vec{y} + (1-t)\vec{x} \in \mathbb{R}^n$, whenever $0 \leq t \leq 1$ and $\vec{x},\vec{y} \in \mathbb{R}^n.$
October 4th, 2011, 05:24 PM   #3
Global Moderator

Joined: May 2007

Posts: 6,755
Thanks: 695

Re: Path Connected Set?

Quote:
 Originally Posted by guynamedluis You can show that the set is convex. So you have to show: $t\vec{y} + (1-t)\vec{x} \in \mathbb{R}^n$, whenever $0 \leq t \leq 1$ and $\vec{x},\vec{y} \in \mathbb{R}^n.$
Since the set is points on the surface of the unit sphere in n dimensions, it is not convex.

October 4th, 2011, 05:56 PM   #4
Member

Joined: Dec 2010
From: Miami, FL

Posts: 96
Thanks: 0

Re: Path Connected Set?

Quote:
 Originally Posted by mathman Since the set is points on the surface of the unit sphere in n dimensions, it is not convex.
Oh right.
We are looking at $S_n$, not $\mathbb{R}^n$ .

Then what can you do?

 October 5th, 2011, 01:20 PM #5 Global Moderator   Joined: May 2007 Posts: 6,755 Thanks: 695 Re: Path Connected Set? I presume each point can be expressed in the n dimensional equivalent of polar coordinates, i.e. radius (r) and n-1 angles. Take any two points (r=1) and for each angle (one at a time) get the curve by changing the angle for the first point to the angle for the second point. You will a set of "arcs" on the surface of the hypersphere connecting the first point to the second.

 Tags connected, path, set

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post aise5668 Real Analysis 3 March 5th, 2012 06:36 PM lu5t Real Analysis 2 November 29th, 2011 07:24 PM johnmath Real Analysis 5 May 18th, 2011 04:02 AM TTB3 Real Analysis 1 December 11th, 2009 02:55 PM cknapp Real Analysis 3 October 16th, 2009 01:41 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top