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 , the set 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: , whenever and 
October 4th, 2011, 05:24 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Re: Path Connected Set? Quote:
 
October 4th, 2011, 05:56 PM  #4  
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Path Connected Set? Quote:
We are looking at , not . Then what can you do?  
October 5th, 2011, 01:20 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Re: Path Connected Set?
I presume each point can be expressed in the n dimensional equivalent of polar coordinates, i.e. radius (r) and n1 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.


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