(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that every nonempty proper subset of R^{n}has a nonempty boundry.

3. The attempt at a solution

First of all, I let S be an nonempty subset of R^{n}and S does not equal R^{n}.

I tried to go about this in 2 different ways:

1) let x be in S and show that B(r,x) ∩ S ≠ ø and B(r,x) ∩ S^{c}≠ ø. I figured this wouldn't work with just one x in S. Or perhaps, I thought I should use induction on the number of elements in S?

2) Assume that bdS is empty and find a contradiction. However, I wasn't able to figure out a contradiction here. Unless, this implies that S equals R^{n}, then that's a contradiction. But I'm not quite sure it implies that. I think that this is the proof you use toshowthat R^{n}and the empty set are the only 2 that are both open and closed.

Thanks for your help! =)

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Intro Topology: boundry Q

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