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Math Help - open covers

  1. #1
    Junior Member mremwo's Avatar
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    open covers

    I don't know where to begin on this problem, because I don't understand open covers that great. I feel like if someone were to help get me started and guide me through it, it would be good practice for me.

    I am being asked to exhibit an open cover of N (natural numbers) that has no finite subcover.

    Any help and hints would be greatly appreciated. Thanks!
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    Quote Originally Posted by mremwo View Post
    I
    I am being asked to exhibit an open cover of N (natural numbers) that has no finite subcover.
    If \displaystyle n\in\mathbb{N} let \mathscr{O}_n=(n-0.1,n+0.1).
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    Junior Member mremwo's Avatar
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    Why does that work/ how did you come up with this?
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    Quote Originally Posted by mremwo View Post
    Why does that work/ how did you come up with this?
    What does it mean to cover a set?
    What does it mean to say open cover?
    What does it mean to have a sub-cover?
    What all do you know about this topic?
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    I was looking at this as well. Thank you!
    Last edited by seamstress; February 23rd 2011 at 07:19 PM.
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    Quote Originally Posted by mremwo View Post
    I am being asked to exhibit an open cover of N (natural numbers) that has no finite subcover.
    Any help and hints would be greatly appreciated. Thanks!
    Another interesting example of an open cover for N={1,2,3...} is the collection of open intervals (sets) C = (0,n), n \epsilon N.
    Proof: Given n \epsilon N, n \epsilon (0,n+1).

    If N includes 0, C = (-.5,N), or C = (-a,n+b), n \epsilon N, any positive a and b, cover N.

    A finite cover exists only for a closed and bounded set, and N is not bounded (basic theorem).

    ref: Taylor, Sec 2-5, "Covering Theorems"

    note: (0,n) = {x:0<x<n}
    Last edited by Hartlw; March 3rd 2011 at 11:25 AM. Reason: add "b"
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