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Math Help - covering

  1. #1
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    covering

    Let  S = (0,1) and  \mathcal{O} = \{(1/n,1): n \in \bold{N} \} . Show that  \mathcal{O} is an open cover of  S .

    So choose  x \in (0,1) . Pick  n_0 \in \bold{N} such that  1/n_0 < x . This implies that  x \in (1/n_0, 1) \subset \bigcup_{n \in \bold{N}} (1/n,1) = \bigcup_{G \in \mathcal{O}} G . Is this correct? Could I pick  n_0 in a different way?

    What if we had a finite subset  \mathcal{O'} \subset \mathcal{O} ?
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  2. #2
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    If you had a finite subset of the collection, then the smallest left-hand endpoint would a number 0 < t < 1.
    But \frac{t}{2} \in \left( {0,1} \right) and \frac{t}{2} < t.
    Therefore, the subcollection does not cover \left( {0,1} \right).
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  3. #3
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    The statement that set is compact means every open covering has a finite subcover.
    Having one open cover which has a finite subcover does not imply that the set is compact.

    Consider: \left( { - 1,.000009} \right) \cup \left[ {\bigcup\limits_{n \in N} {\left( {  n^{ - 1} ,1} \right)} } \right]
    That is clearly an open covering of \left( {0,1} \right) and the collection contains a finite subcover.
    But \left( {0,1} \right) is still not compact.
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