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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.

Keith Nale
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1 Answers1

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This may be a bit late. Let's take the general case first.

The key is to be as wasteful as possible with how you choose the open sets $\{U_\alpha\}$ and how you pick the indexing set of $\alpha$'s. For each $\alpha\in Y$ there is some open ball $B(\alpha,2\epsilon_\alpha)\subseteq\tilde U$ around $\alpha$. Let $U_\alpha$ be $B(\alpha,\epsilon_\alpha)\cap Y$. Then $U_\alpha^{\epsilon_\alpha}$ fits inside $\tilde U$ by the triangle inequality.

Because the open cover is indexed by points of $Y$, it's not terribly hard to show that the $\epsilon$ function described in the hint satisfies $Y^\epsilon\subseteq \tilde U$. It follows more or less from the definition of $Y^\epsilon$.

In the compact case you can use the usual trick. Throw away all but finitely many of the open sets in the cover. Then $\epsilon$ will be bounded by the largest remaining $\epsilon_\alpha$.

Lichko
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