Locally star-shaped space and piecewise linear path
For each $x \in [0,1]$ let $N_{\gamma(x)}$ be a starshaped nbhd of $\gamma(x)$. Because $\gamma([0,1])$ is compact and the $N_{\gamma(x)}$ cover it, you can thin them to a finite sub-cover. The rest should be easy.
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Rudy the Reindeer
Updated on August 01, 2022Comments
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Rudy the Reindeer over 1 year
I'm trying to do exercise 4 on page 38 in Hatcher. Can you tell me if this is right?
claim: $X$ locally star-shaped, $\gamma$ a path in $X$ then there is a path consisting of a finite number of line segments that is homotopic to $\gamma$
proof:
By assumption $\gamma(0)$ has a star-shaped neighbourhood, let's call it $U_0$. Consider $\gamma(t_1) \in U_0$. Because $U_0$ is star-shaped there exists a straight line between $\gamma(0)$ and $\gamma(t_1)$. Repeat the same for $\gamma(t_1)$ and so on to get a path consisting of straight lines: $$\gamma(0) \rightarrow \gamma(t_1) \dots \rightarrow \gamma(1)$$
This is homotopic to $\gamma$ because every point of it can be connected to $\gamma$ via a straight line.
Thanks for your help!
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t.b. about 12 yearsI think you need to be a bit more careful: what if $\gamma([0,t_1]) \not\subset U_0$?
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Rudy the Reindeer about 12 yearsCould I not choose a smaller $t_1$ in that case?
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t.b. about 12 yearsSure, but you should mention that. Also, you should probably spell out what exactly the homotopy is and why it is continuous. The idea is of course right but there are a few more details to be checked.
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