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wu :: forums    general    complex analysis (Moderators: SMQ, towr, Grimbal, william wu, ThudnBlunder, Eigenray, Icarus)    Question about Homotopy « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Question about Homotopy  (Read 2092 times) immanuel78 Newbie     Gender: Posts: 23 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 wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 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 "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous immanuel78 Newbie     Gender: Posts: 23 Re: Question about Homotopy   « Reply #2 on: Sep 21st, 2006, 6:14am » Quote Modify Thank you for solving it. IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 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. IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth => complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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