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    Characterizing the metric space {N}

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    For the metric space { N }, the set of all natural numbers, characterize whether or not it has the following properties: compact, totally bounded, has the Heine-Borel property, complete.

    For compact, we are to show that every sequence converges. For totally bounded, we are to show that it can be covered by finitely many sets of diameter less than epsilon. For Heine-Borel, we are to show there is a finite subcover. And for completeness, we are to show that each Cauchy sequence converges.

    We are not allowed to use the fact that compactness implies completeness, etc. We can use definitions only.

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    Solution Preview

    N, the set of all natural numbers, to be considered a metric space, then we define the Eucliean distance d(x,y)=|x-y| for every x,y in N.
    We denote "<>" as "not equal to".

    (a) N is not compact.
    Compactness means that every sequence in N has a convergent subsequence. But consider N itself as a sequence, N={1,2,3,...}, then N does not have a convergent subsequence. Since ...

    Solution Summary

    This shows how to characterize metric spaces. The expert uses the fact that compactness implies completeness.