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Union of boundaries

  1. Sep 19, 2007 #1
    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. jcsd
  3. Sep 19, 2007 #2


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    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.
  4. Sep 19, 2007 #3
    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.
  5. Sep 20, 2007 #4
    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.
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