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Math Help - Quick Cluster Points Question

  1. #1
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    Quick Cluster Points Question

    I'm beginning to study Analysis but I'm having the wording of this particular paragraph is confusing to me:

    "The definition of boundary point (when this is not an isolated point) implies that any ball about such a point actually contains infinitely many points from S. Thus, a cluster point can be either a boundary point or an interior point of a set S but never an exterior point of S. If, for instance S is the set of rational numbers contained in the unit interval, S = Q \bigcap [0, 1], then no point in S is an interior point of S and \overline{S} = [0, 1]"

    It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of \overline{S}. Or are they saying that no point in S is an interior point of S... AND... \overline{S} = [0, 1]?
    Last edited by Plato; July 4th 2011 at 11:01 AM. Reason: LaTeX fix
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  2. #2
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    Re: Quick Cluster Points Question

    I'm getting LaTeX errors on most of what I put in there... can others see it? I don't know what the deal is.
    Last edited by jameselmore91; July 4th 2011 at 11:03 AM. Reason: Thanks Plato
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  3. #3
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    Re: Quick Cluster Points Question

    Quote Originally Posted by jameselmore91 View Post
    I'm getting LaTeX errors on most of what I put in there... can others see it? I don't know what the deal is.
    Use [TEX]...[/TEX] not [Math]...[/Math]
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  4. #4
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    Re: Quick Cluster Points Question

    Quote Originally Posted by jameselmore91 View Post
    . If, for instance S is the set of rational numbers contained in the unit interval, S = Q \cap [0, 1], then no point in S is an interior point of S and \overline{S} = [0, 1]"
    It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of \overline{S}
    Any open set that contains a point of S also contains an irrational number. Therefore every point of S is a boundary point if S and not an interior point.
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  5. #5
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    Re: Quick Cluster Points Question

    Quote Originally Posted by jameselmore91 View Post
    I'm beginning to study Analysis but I'm having the wording of this particular paragraph is confusing to me:

    "The definition of boundary point (when this is not an isolated point) implies that any ball about such a point actually contains infinitely many points from S. Thus, a cluster point can be either a boundary point or an interior point of a set S but never an exterior point of S. If, for instance S is the set of rational numbers contained in the unit interval, S = Q \bigcap [0, 1], then no point in S is an interior point of S and \overline{S} = [0, 1]"

    It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of \overline{S}. Or are they saying that no point in S is an interior point of S... AND... \overline{S} = [0, 1]?
    It is the second.
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