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Math Help - Real Analysis: Open and Closed Sets

  1. #1
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    Red face Real Analysis: Open and Closed Sets

    I have 2 questions hopefully someone can help with.

    1) Is R the only open set containing Q? Prove or give an example of a set containing Q.

    I do believe R is the only open set containing Q but I have no idea how to prove this.

    2) Is the set of integers a closed set? I'm pretty sure it is, since the boundary points are the integers. But is is also open?


    Thanks so much for the help!
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  2. #2
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    Quote Originally Posted by ziggychick View Post
    I have 2 questions hopefully someone can help with.

    1) Is R the only open set containing Q? Prove or give an example of a set containing Q.

    I do believe R is the only open set containing Q but I have no idea how to prove this.

    2) Is the set of integers a closed set? I'm pretty sure it is, since the boundary points are the integers. But is is also open?


    Thanks so much for the help!
    First number 2: The integers are not open. Any epsilon neighborhood of an integer will contain points not in the set.

    For number 1: I am not entirely sure, but you could try taking M such that Q is contained in M is contained in R. Suppose M is open. Doesn't this imply that M=R? (if M is open then for any point m in M there is an epsilon neighborhood around m contained in M. Use the density of the irrationals to show that M=R)
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  3. #3
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    Quote Originally Posted by ziggychick View Post
    Is R the only open set containing Q? Prove or give an example of a set containing Q.
    What can you say about the set consisting of all the real numbers except \sqrt2 ?
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  4. #4
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    Talking riight!

    The set (-infty, rt(2)) union (rt(2), +infty) is open and contains all of Q.
    Thanks guys!
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  5. #5
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    Quote Originally Posted by ziggychick View Post

    2) Is the set of integers a closed set? I'm pretty sure it is, since the boundary points are the integers. But is is also open?
    As far as I remember, in any topology, sets including a single point are closed.
    Therefore, union of these sets are also closed.
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  6. #6
    Moo
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    Quote Originally Posted by bkarpuz View Post
    As far as I remember, in any topology, sets including a single point are closed.
    Why ?
    Therefore, union of these sets are also closed.
    So it should be a finite union of these sets, isn't it ? (if we consider the complement of the union..)
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  7. #7
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by Moo View Post
    So it should be a finite union of these sets, isn't it ?
    yap must be finite otherwise \mathrm{Q}is a contrary example.

    Also for the first one, I might be wrong too.
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    see here
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  9. #9
    Moo
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    Quote Originally Posted by Jhevon View Post
    see here
    In the discrete topology, of course it's true !
    Since \tau=\{ \{a\} ~:~ a \in X \}

    But if you consider the topology over \mathbb{R} : \tau=\{ \emptyset~,~\mathbb{R}~,~ [0,1] \}, it's clear that the singletons of \mathbb{R} are nor open, nor closed sets (it's the example I gave to bkarpuz >.<)
    So it cannot be true for any topology
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Moo View Post
    In the discrete topology, of course it's true !
    Since \tau=\{ \{a\} ~:~ a \in X \}

    But if you consider the topology over \mathbb{R}=\{ \emptyset~,~\mathbb{R}~,~ [0,1] \}, it's clear that the singletons of \mathbb{R} are nor open, nor closed sets (it's the example I gave to bkarpuz >.<)
    yes, i was responding to what bkarpuz said about "any" topology. his claim was not true for discrete topologies.

    anyway, i know nothing about topology, so i shouldn't even be responding to anything here
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  11. #11
    Moo
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    Quote Originally Posted by Jhevon View Post
    yes, i was responding to what bkarpuz said about "any" topology. his claim was not true for discrete topologies.

    anyway, i know nothing about topology, so i shouldn't even be responding to anything here
    CaptainBlack would have warned you to quote the post you're replying to !

    Anyway, it's true for discrete topologies, not "not true"
    And why do you say you know nothing ? o.O "open" is a topological concept !
    Learn a little about it, it's interesting !



    Note (mainly to bkarpuz) : in the usual topology over R, a singleton is an open set.
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  12. #12
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Moo View Post
    CaptainBlack would have warned you to quote the post you're replying to !
    haha, yeah

    Anyway, it's true for discrete topologies, not "not true"
    And why do you say you know nothing ? o.O "open" is a topological concept !
    i still say i know nothing about topology. we only did a few stuff on it, and i was just about to finish wrapping my head around the concepts in chapter 1 when we stopped.

    Learn a little about it, it's interesting !
    too interesting for me at the moment
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  13. #13
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    Quote Originally Posted by bkarpuz View Post
    As far as I remember, in any topology, sets including a single point are closed.
    This is true in Hausdorff spaces - or "separated spaces" -, i.e. if you assume that two distinct points always possess disjoint neighbourhoods.

    This is not necessary and sufficient, but nevertheless fairly general.
    In fact, the Wikipedia says that the original definition of a topology by Hausdorff included this separation condition. It also gives a pun to remember the definition of Hausdorff space that I let you read by yourselves... (first paragraph)
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  14. #14
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by Jhevon View Post
    yes, i was responding to what bkarpuz said about "any" topology. his claim was not true for discrete topologies.

    anyway, i know nothing about topology, so i shouldn't even be responding to anything here
    I dont think so, it is helpful to point out a mistake.
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