Boundary of the union of two sets

  • Thread starter painfive
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Main Question or Discussion Point

is it true that:

[tex]\partial(A\cup B) = (\partial(A)\cap \mbox{int}(X-B))\cup (\partial(B)\cap \mbox{int}(X-A))[/tex]

? (where [itex]\partial(A)[/itex] is the boundary of A, int(A) is the interior, and A and B are two subsets of the topological space X)

I can prove that:

[tex]\partial(A\cup B) \subset (\partial(A)\cap (X-\mbox{int}(B)))\cup (\partial(B)\cap (X-\mbox{int}(A)))[/tex]

But I have an example where the equality doesn't hold. (I can show all this if anyone wants). But in the example, the first equalitiy does hold, and it seems like it would always hold, but I can't prove it.

Edit: Actually, now I think I have an example where the first equality doesn't hold. Now I have no idea what [itex]\partial(A\cup B)[/itex] is. (It also isn't [itex](\partial(A)\cap (X-B))\cup (\partial(B)\cap (X-A))[/itex], because I have a counterexample of that too)
 
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Answers and Replies

24
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I might as well show the counterexample I found. For these two sets, none of the above equalities hold (here X is [itex]\Re^2[/itex]):

[tex]A= {0\le x < 1,0\le y \le 1}[/tex]
[tex]B= {1\le x \le 2,0\le y \le 1}[/tex]

Here, the first equality should actually be a "contains" and the second two should be "is a subset of".
 

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