Prove or disprove the following statement using sets frontier points

  • Thread starter ppy
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  • #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
 

Answers and Replies

  • #2
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What if A is the frontier of B?
 
  • #3
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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).
 
  • #4
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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?
 
  • #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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