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Prove or disprove the following statement using sets frontier points

  1. Oct 25, 2013 #1

    ppy

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    if A is a subset of B and the frontier of B is a subset of A then A=B.

    I am pretty sure that this is true as I drew I diagram and I think this helped.

    A frontier point has a sequence in the set and a sequence in the compliment that both converge to the same limit. However i'm not really sure how to use this definition to help me

    Thanks
     
  2. jcsd
  3. Oct 25, 2013 #2
    What if A is the frontier of B?
     
  4. Oct 25, 2013 #3
    So you have [itex]\partial B \subseteq A\subseteq B[/itex]. In particular, [itex]\partial B \subseteq B[/itex] exactly says that [itex]B[/itex] is closed. So [itex]B[/itex] is a closed set, and [itex]A[/itex] is a subset of [itex]B[/itex] which includes every non-interior point of [itex]B[/itex].

    The case R136a1 mentioned is, in some sense, the most extreme possible case of [itex]A\neq B[/itex] (a counterexample to your conjecture, as long as [itex]B[/itex] has nonempty interior).
     
  5. Oct 26, 2013 #4
    As a side question, what the heck is a "frontier" of a set? This looks equivalent to the definition of boundary. Is this just another word for boundary? If so, why?

    That is, why have a new word?
     
  6. Oct 26, 2013 #5

    HallsofIvy

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    As both R136a1 and economicsnerd said, you can't prove it. Without some qualification, it is not true:
    Counterexample: Let A= {0, 1}, B= [0, 1].
    (I am assuming that "frontier" is the same as "boundary".)
     
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