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I Definition of boundary

  1. Mar 2, 2017 #1
    Hello! This is more of a set theory question I guess, but I have that the definition of the boundary of a subset A of a topological space X is ##\partial A = \bar A \cap \bar B##, with ##B = X - A## (I didn't manage to put the bar over X-A, this is why I used B). I think I have a wrong understanding of the complement of a set because if I take (a,b) on the real axis, the boundary should be {a,b}, but ##\bar A = (- \infty, a] \cup [b, \infty)## while ##B=R-A = (- \infty, a] \cup [b, \infty)## so ##\bar B = (a,b)## and ##\bar A \cap \bar B = \emptyset##. So where exactly I got it wrong? Thank you
     
  2. jcsd
  3. Mar 2, 2017 #2

    fresh_42

    Staff: Mentor

    What was ##\overline{A} = \overline{(a,b)}## again? It's not the complement though!
     
  4. Mar 2, 2017 #3
    Doesn't ##\bar A## means all elements not in A? Which in this case is ##(-\infty,a] \cup [b,\infty)##?
     
  5. Mar 2, 2017 #4

    fresh_42

    Staff: Mentor

    No, here it means the closure of ##A##. That is the reason, why the bar isn't a good choice for complements in topology. Some write ##\mathbb{R}-A= A^C## which I find ugly. I prefer to write complements as ##\mathbb{R}-A= \mathbb{R}\backslash A##. In any case, it's a matter of taste, but ##\overline{A}## as the topological closure of ##A## is pretty usual.
     
  6. Mar 2, 2017 #5
    Oh ok makes sense now. Thank you!
     
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