I'm trying to proof the $\epsilon$-Neighborhood Theorem from Guillemin and Pollack's book. I'm not good at topology, and I'm having some difficulties to completely understand the theorem. For the proof is necessary do some exercises, and the first one is this:
Show that any neighborhood $\tilde{U}$ of $Y$ in $\mathbb{R}^M$ cointains some $Y^{\epsilon}$; moreover, if $Y$ is compact, $\epsilon$ may be taken constant. [HINT: Find covering open sets $U_{\alpha}^{\epsilon} \subset Y$ and $\epsilon_{\alpha} > 0$, such that $U_{\alpha}^{\epsilon} \propto \subset \tilde{U}$. Let {$\theta_i$} be a subordinate partition of unity, and show that $\epsilon = \sum \theta_i \epsilon_i$ works.]
Remember that the set $Y^{\epsilon}$ is defined as:
$$Y^{\epsilon} = \{w \in \mathbb{R}^M: |w-y|<\epsilon(y) \text{ for some } y \in Y\}$$
I'm stuck on this right now, and is supposed to be an easy exercise according the proof of the theorem on the book.