Proving Boundary of Union is Subset of Union of Boundaries

[teleport]

Homework Statement

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).

Homework Equations

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.

 

[HallsofIvy]

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.

 

[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.

 

[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.