Let A, C [itex]\subseteq[/itex] ℝn with boundaries B(A) and B(C) respectively. Prove or disprove :
B(AUC) O B(A)UB(C)
B(A[itex]\cap[/itex]C) O B(A)[itex]\cap[/itex]B(C)
Where O represents each of these symbols : [itex]\subseteq, \supseteq, =[/itex]
I know that double inclusion is going to cut the work required by 33% :)?
The Attempt at a Solution
I guess I'll try to start with the case B(AUC) [itex]\subseteq[/itex] B(A)UB(C) ( Since intuitively I know the boundary simply can't get bigger when I union two sets, so I have a feeling that testing for [itex]\supseteq[/itex] is going to flop ).
So suppose [itex]x \in B(A \cup C)[/itex] then we know x is a boundary point of AUC, that is : [itex]\forall δ>0, \exists P \in (A \cup C) \wedge Q \in (ℝ^n - A \cup C) | P, Q \in N_δ(x)[/itex]
Now how to proceed from here I'm not sure, any pointers would be great!