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A and A complement have the same boundary

  1. Nov 23, 2011 #1
    How would you show that if [itex]X[/itex] is a non-empty set and [itex]A\subseteq X[/itex] then [itex]A[/itex] and [itex]A^c[/itex] have the same boundary?

    The definition is [itex]x\in \partial A \iff [/itex] there exists [itex]r>0[/itex] such that the open ball [itex]B(x,r)[/itex] intersects both [itex]A[/itex] and [itex]A^c[/itex]

    but this is precisely the statement that [itex]x\in \partial A^c[/itex]!
     
  2. jcsd
  3. Nov 23, 2011 #2

    micromass

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    Indeed, so the proof is trivial.
     
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