Understanding the Boundary Theorem in Analysis: A Proof Guide | PhysicsForums

[RPierre]

Hey PhysicsForums,

This is my first time here but I have seen many knowledgeable responses to tough questions and I truly am stumped. My question is as follows (This a third year Analysis course in my Mathematics undergrad degree):

1. Let A and B be Subsets of R^N with Boundaries B(A) and B(B) respectively. Which, if any, of the following is true? If you think is a result is true, give a proof for your assertion. Otherwise contradict it.

i) B(A$$\cap$$B) = B(A)$$\cap$$B(B)
ii) B(A$$\cap$$B) $$\subseteq$$ B(A)$$\cap$$B(B)
iii) B(A$$\cap$$B) $$\supseteq$$ B(A)$$\cap$$B(B)

2. My Attempt - I graphically represented all three and determined i) was not true, where ii) and iii) are true. My idea for i) is to draw two circles intersecting, and show that the intersection of the boundaries is not the same as the intersection of the two sets' boundaries. My Idea for ii) and iii) is a big Blank

3. Unfortunately, I have no Idea how to scribe my thoughts mathematically. I was hoping someone could help me out in helping me out a lot, or even showing me how to do these proofs so I could learn and do the rest of my assignment

Thank you VERY much in advance. Also Let me know if this is the wrong forum for this post, and if this should be in Calc & Analysis.

 

[aPhilosopher]

for ii) draw one circle inside of another circle.

If ii) and iii) were both true, then the two sets would be equal.

For three, note that a point of intersection of the boundaries will have points from both sets in all of its neighborhoods.