Union of boundaries

1. Sep 19, 2007

teleport

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

I'm trying to prove that the boundary of the union of two sets is a subset of the union of the boundaries of the two sets. i.e. Show that B(X u Y) c= B(X) u B(Y).

2. Relevant equations

3. The attempt at a solution

Since I didn't know what to do here, I used the identity A c= B <==> cl(B) c= cl(A), and then a lot of set algebra to prove it. Given how simple the statement looks, I'm wondering if there is an easier way. Thanks.

2. Sep 19, 2007

HallsofIvy

Staff Emeritus
I don't see any need to deal with the closure. If x is a member of B(X \cup Y), what does that tell you about x? Use that to prove it must be a boundary point of X or a boundary point of Y.

3. Sep 19, 2007

teleport

Actually, didn't know what closure is, so confused the cl symbol I've seen, with complement. So, yea, when I wrote cl I meant complement.

O let me try this. If x in B(X u Y) then for all r>0 the r-neighbourhood of x contains at least some point p in X u Y and another point q in C(X u Y) , (C stands for complement). But now I'm having difficulty with the case where p in X but q in C(Y). This gives the possibility that x is in none of the boundaries of the sets. But if you say that x must be in the boundaries of the sets, then there is no need to solve the problem.

4. Sep 20, 2007

teleport

Im good, got it, just had to use C(X u Y) = C(X) n C(Y), so now it is clear what hapens to q. thank you.