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

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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.

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    $\begingroup$ 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. $\endgroup$ – RRL Jul 1 '18 at 6:57
  • $\begingroup$ @RRL Oh, you are right. Thanks for pointing out. $\endgroup$ – onurcanbektas Jul 1 '18 at 7:06
  • $\begingroup$ @onurcanbektas I pointed it out also $\endgroup$ – mathworker21 Jul 1 '18 at 7:08
  • $\begingroup$ @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. $\endgroup$ – onurcanbektas Jul 1 '18 at 7:21
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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).

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    $\begingroup$ If the question posed is whether a proposed proof is valid, answers starting with "I didn't feel like reading your proof" are at best vacuous. $\endgroup$ – Mark Fischler Jul 1 '18 at 6:48
  • $\begingroup$ Because onurcanbektas said "I'm particularly asking this because it took me 1 day to figure out this contradiction out clearly.", I thought my answer would be of use to him. There are many times on stack exchange when people don't directly answer a question but instead provide something useful to the questioner. I would like if you removed the downvote or explained it further. Thanks $\endgroup$ – mathworker21 Jul 1 '18 at 7:07

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