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Prove or disprove the following statement using sets frontier points

  1. Oct 25, 2013 #1


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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

  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


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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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