March 23rd, 2018, 11:08 AM  #1 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3  Topology
Is this picture show if the "inside & outside" is defined in topology? Or this example isn't good? 
March 23rd, 2018, 11:11 AM  #2 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3  topology picture
Is this picture better than the above? And the picture above is wrong? 
March 23rd, 2018, 12:36 PM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out. 
I’m not sure what you’re asking. But you might like to look at the Jordan curve theorem. It turns out to be quite a subtle thing to define exactly what ‘inside’ and ‘outside’ mean.

March 23rd, 2018, 12:54 PM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out. 
A nice little explanation from a time when people actually bothered. 
March 23rd, 2018, 01:16 PM  #5 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
I have now read the definition of Jordan curve theorem. Jordan curve theorem is curve in a plain that not intersect itself and close (the ending point is also the starting point). So the first isn't a Jordan curve and the second it is a Jordan curve. Last edited by policer; March 23rd, 2018 at 01:19 PM. 
March 23rd, 2018, 01:31 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out.  That's right. Topologically, any figure equivalent to the circle is a Jordan curve. If you want, you can try prove the Jordan curve theorem using the intermediate value theorem for say, $x + y = 1$.

March 23rd, 2018, 01:35 PM  #7 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
Can you give more directions and hints: How to prove the intermediate value theorem for: x+y=1. I need a some directions... 
March 23rd, 2018, 04:35 PM  #8  
Senior Member Joined: Sep 2016 From: USA Posts: 398 Thanks: 212 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
This does not seem to have anything to do with what you are asking about in this thread.  
March 23rd, 2018, 05:01 PM  #9 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out.  

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