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Math Help - Boundary of a Set = Empty

  1. #1
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    Boundary of a Set = Empty

    Let A ⊂ R . Can any of the following hold?
    (a) ∂A = (b) ∂A = A (c) A ⊂ ∂A and A not equal to ∂A

    I do not know how to get the answers to the above. I know that the boundary of a set is empty if the boundary is clopen but I was always taught that boundary is closed. And as I am not given a value for A I get really confused Please help.
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  2. #2
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    Re: Boundary of a Set = Empty

    Quote Originally Posted by alexa2012 View Post
    Let A ⊂ R . Can any of the following hold?
    (a) ∂A = (b) ∂A = A (c) A ⊂ ∂A and A not equal to ∂A
    I was always taught that boundary is closed.
    As I read this question, it is about subsets \mathbb{R}.
    As that is a connected space the only clopen sets are \mathbb{R}~\&~\emptyset. Therefore, all other sets would have a nonempty boundary.
    Think about the set of rationals, \mathbb{Q}.
    This may surprise you \partial (\mathbb{Q})=\mathbb{R}, every real number is a boundary point of rationals.

    Think of the integers: \partial (\mathbb{Z})=\mathbb{Z}.

    See what can be done there.
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  3. #3
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    Re: Boundary of a Set = Empty

    A can be any subset of \mathbb{R}. The question is asking you to choose an A such that the frontier of A meets a particular property, or else show that no choice of A will meet that property.

    I'll hint that all three of these can be done. (Plato gave you two of them and the third is trivial, but try to see why they work and which is which.)
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