# Munkres, Lemma 16.2: Existence of $C_x$ s.t it is disjoint from $D_{i-2}$

In the book of Analysis on Manifolds by Munkres, at page 137, it is given that

I have doubts about the existence of such a $C_x$, so just to make sure, I wanted the prove the existence of such a set $C_x$.

Proof:

Let assume that for some $x_0 \in B_i$, every neighbourhood $C_{x_0}$ of $x_0$ has a non-empty intersection with $D_{i-2}$. Then since $B_i \subseteq Ext(D_{i-2})$, and $C_{x_0} \cap B_i \not = \emptyset$, then it must be true that $$x_0 \in \partial D_{i-2} \subseteq D_{i-2},$$ but $B_i$ and $D_{i-2}$ are disjoint, a contradiction.

Is there any flaw in the proof ? I'm particularly asking this because it took me 1 day to figure out this contradiction out clearly.

• This is fine as an indirect proof. Directly $B_i = D_i - \text{Int }D_{i-1}$ and $D_{i-2} \subset \text{Int } D_{i-1}$ implies $B_i \subset \text{Ext } D_{i-2}$ as you observed. There is an open ball and, hence, a cube containing $x_0$ that does not meet $D_{i-2}$ by definition of exterior. – RRL Jul 1 '18 at 6:57
• @RRL Oh, you are right. Thanks for pointing out. – onurcanbektas Jul 1 '18 at 7:06
• @onurcanbektas I pointed it out also – mathworker21 Jul 1 '18 at 7:08
• @mathworker21 Well, actually I did not understand what you mean by "there is a distance between disjoint compact sets", and I was distracted by some other thing before writing this as a comment. Thank you for your answer also; it was my mistake that I couldn't ask for further clarification about your answer. – onurcanbektas Jul 1 '18 at 7:21

I don't feel like reading your proof, but here's the proof that comes to my mind. $B_i$ and $D_{i-2}$ are disjoint compact sets, so they are some positive distance apart from each other. So given any $x \in B_i$, we can take a small enough cube centered at $x$ that is disjoint from $D_{i-2}$ (we can insist the cube is contained in $A$, since $A$ is open).