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Math Help - Closed Set Requirements

  1. #1
    Senior Member bugatti79's Avatar
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    Closed Set Requirements

    Folks,

    Generally speaking, what must I use to show that a certain set of elements is a closed subset of some other larger set for a certain norm?

    Thanks
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  2. #2
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    Re: Closed Set Requirements

    Quote Originally Posted by bugatti79 View Post
    Generally speaking, what must I use to show that a certain set of elements is a closed subset of some other larger set for a certain norm?
    Is this a question about a general topological space?
    You used the word norm. Does that imply that this is a metric space?

    Or is the some other setting altogether for closed subsets of elements?
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  3. #3
    Senior Member bugatti79's Avatar
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    Re: Closed Set Requirements

    Quote Originally Posted by Plato View Post
    Is this a question about a general topological space?
    You used the word norm. Does that imply that this is a metric space?

    Or is the some other setting altogether for closed subsets of elements?
    Yes, I am referring to a metric space....... (I suspect the question is too vague for a reasonable answer.)

    Thanks Plato.
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    Re: Closed Set Requirements

    Quote Originally Posted by bugatti79 View Post
    Yes, I am referring to a metric space....... (I suspect the question is too vague for a reasonable answer
    Two general approaches come to mind at once.
    First, show that the set contains all of its limit points.

    Second, equivalently show that its complement is open.
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  5. #5
    Senior Member bugatti79's Avatar
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    Re: Closed Set Requirements

    Quote Originally Posted by Plato View Post
    Two general approaches come to mind at once.
    First, show that the set contains all of its limit points.

    Second, equivalently show that its complement is open.
    For this example...is the set

    \{z \in C[a,b]: u\le z(s)\le v \forall s \in [a,b]\}

    a closed subset in C[a,b] with the integral norm?

    How would i tackle this?

    thanks
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