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Math Help - Verifying the Boundaries/Interiors etc of these two functions

  1. #1
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    Verifying the Boundaries/Interiors etc of these two functions

    Hi, I believe I have the right information for the following two functions but I just need a brief statement as to why it is true.

    Consider the set \mathbb{Q} of rational numbers.

    1)  \partial(Interior(\mathbb{Q})), which I said is \varnothing

    2)  \partial((Interior(\mathbb{Q})^C)), which I said is \varnothing

    3)  Closure(Interior(\mathbb{Q})), which I said is \varnothing


    And for the second function, S={(x,y): y = sin(1/x), x>0} I said the following:

    1) Interior(S), which I said is \varnothing

    2) \partial(S), which I said is  \{(x,y): x=0, -1 \le \right y \le \right 1\}

    3) Closure(S), which I said is  S \cup \{(x,y): x=0, -1 \le \right y \le \right 1\}

    Again, I just need brief statements verifying the validity of these statements using point/ball methods used in elementary topology. Thanks a lot, I appreciate it.
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  2. #2
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    Yes, those are all true. But why do you think they are true? That's what you want to say!

    In R, a "ball" is an interval. And any interval contains both rational and irrational numbers.

    In R^2, a "ball" is a disk. All you need to say is that a disk on any point of a curve must contain some points not on the curve.
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  3. #3
    MHF Contributor

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    Actually \partial(S)=\text{closure}(S).
    Every point in S is boundary point of S.
    Any open disk containing a point of S must contain a point not in S.
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