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immanuel78
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 Question about Homotopy   « on: Sep 20th, 2006, 6:18am » Quote Modify

Let G be a region.
Let L : [0,1] -> G be a closed rectifiable path.
Then there exists a closed, piecewise-smooth path in G which is homotopic to L in G.

This question seems to be true to me, but it is not easy for me to prove it.
 « Last Edit: Sep 20th, 2006, 9:51am by immanuel78 » IP Logged
Icarus
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 Re: Question about Homotopy   « Reply #1 on: Sep 20th, 2006, 6:18pm » Quote Modify

The image of L is compact and lies in the interior of G. Cover the image with balls that lie within G, then choose a finite number of them that still cover it. You can order the balls B1, B2, ..., Bn in the order in which they first occur along L. Choose points xi on L such that xi is in the intersection of Bi and Bi+1 (since L is closed, let xn be  in the intersection of Bn and B1). The piecewise linear path determined by these points satisfies the conditions.
 « Last Edit: Dec 23rd, 2006, 7:18am by Icarus » IP Logged

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I wonder: Which is larger
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immanuel78
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 Re: Question about Homotopy   « Reply #2 on: Sep 21st, 2006, 6:14am » Quote Modify

Thank you for solving it.
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Icarus
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 Re: Question about Homotopy   « Reply #3 on: Sep 21st, 2006, 4:23pm » Quote Modify

Compactness and connectedness are the hammer and screwdriver of the analyst's toolbox. They are often the first things you should look to in solving problems.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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