# Homework Help: Intro Topology: boundry Q

1. May 24, 2007

### redyelloworange

1. The problem statement, all variables and given/known data
Prove that every nonempty proper subset of Rn has a nonempty boundry.

3. The attempt at a solution

First of all, I let S be an nonempty subset of Rn and S does not equal Rn.

1) let x be in S and show that B(r,x) ∩ S ≠ ø and B(r,x) ∩ Sc≠ 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 Rn, then that's a contradiction. But I'm not quite sure it implies that. I think that this is the proof you use to show that Rn and the empty set are the only 2 that are both open and closed.

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

2. Relevant equations

3. The attempt at a solution

Last edited: May 24, 2007
2. May 24, 2007

### NateTG

What's the definition of boundry?

3. May 24, 2007

### redyelloworange

bd(S) = {x in Rn s.t. B(r,x)∩ S ≠ ø and B(r,x) ∩ Sc≠ ø for every r>0}

4. May 24, 2007

### Dick

Take a point x in S and a point y in S^C and consider the line t*x+(1-t)*y for t in [0,1].

5. May 24, 2007

### river_rat

Well you could assume the contrary and first prove that S must be closed and similarly that S must be open. Now which are the only sets in a connected space with that property?

6. May 25, 2007

### NateTG

Redyelloworange is aware of that, he mentions that he thinks the goal is to prove that the n-dimensional reals are connected under the usual topolgy: