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January 26th, 2018, 04:45 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 Thanks: 2  Show the Annulus is polygonally path connected.
Hello, I am asked to show that the annulus A(0,r,R) is polygonally path connected. For this question could i just give one specific example where i find points $w,z$in$U$ there are points $z=z_{0},z_{1},...z_{n}$=w where these points are contained inside the annulus? For example i could give a vector rotated by 2pi/n for n lots of z(points) For the second part of the question i am asked to give a sketch of my points to show intuitively that it is polygonally path connected. Thanks very much 
January 26th, 2018, 07:33 AM  #2 
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 246 
Showing that an annulus is path connected is trivial. Now try to show in general for an open subset of $\mathbb{R}^n$ that path connected is the same as polygonally connected. Do this by creating balls inside the open subset. A ball is convex. 

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annulus, connected, path, polygonally, show 
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