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Math Help - Infinite Collection, compact sets

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    Infinite Collection, compact sets

    Find an infinite collection {Sn : n in N} of compact sets in R such that infinite union of Sn is not compact.


    Is it correct to say that the union of infinite sets is a countable infinity?
    So that if the union of infinite sets is going to be bounded by N, but you could find a set N + 1 that is not in the set?
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  2. #2
    Junior Member platinumpimp68plus1's Avatar
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    Well, if you set Sn=[-n,n], then take the union of each Sn for n=1,2,3,4...

    Each [-n,n] is clearly compact. But the union (which is R itself), is not.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by p00ndawg View Post


    Is it correct to say that the union of infinite sets is a countable infinity?
    Be careful here. If you have a set \mathcal{N}=\left\{S_n:n\in\mathbb{N}\right\} then clearly \mathcal{N} is countable for the "worst case scenario" is that all the sets are distinct in which case the mappking f:\mathbb{N}\mapsto\mathcal{N} given by f(n)=S_n is clearly a bijection. Saying that given a countable class of sets the union is automatically countable is just plain wrong. What about \mathcal{N}=\left\{\mathbb{R}^n:n\in\mathbb{N}\rig  ht\}?
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