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 $\gamma$
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!