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March 23rd, 2018, 12:08 PM   #1
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Topology

Is this picture show if the "inside & outside" is defined in topology?
Or this example isn't good?
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March 23rd, 2018, 12:11 PM   #2
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topology picture

Is this picture better than the above?
And the picture above is wrong?
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 March 23rd, 2018, 01:36 PM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,739 Thanks: 609 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. Thanks from policer
 March 23rd, 2018, 01:54 PM #4 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,739 Thanks: 609 Math Focus: Yet to find out. A nice little explanation from a time when people actually bothered. Thanks from topsquark
 March 23rd, 2018, 02:16 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 02:19 PM.
March 23rd, 2018, 02:31 PM   #6
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Quote:
 Originally Posted by policer 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.
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, 02:35 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, 05:35 PM   #8
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Quote:
 Originally Posted by policer Can you give more directions and hints: How to prove the intermediate value theorem for: |x|+|y|=1. I need a some directions...
As stated this does not make sense. The intermediate value theorem says that if $X$ is a connected topological space and $f: X \to \mathbb{R}$ is continuous, then $f(X)$ is an interval.

This does not seem to have anything to do with what you are asking about in this thread.

March 23rd, 2018, 06:01 PM   #9
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Quote:
 Originally Posted by SDK This does not seem to have anything to do with what you are asking about in this thread.
I was trying to provide an exercise with which the OP could prove the JCT for a special case using the IVT.

As for what exactly is being asked in the thread, I've no idea.

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