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Math Help - open cover with no finite subcover

  1. #1
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    open cover with no finite subcover

    What would be an example of an open cover on (1,2) with no finite subcover?
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  2. #2
    Member Black's Avatar
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    Let \mathcal{O}_n=\left(1,\, 2-\frac{1}{n+1}\right). Since

    \bigcup_{n=1}^{\infty}\mathcal{O}_n=(1,2),

    \{\mathcal{O}_n\} is an open cover for (1,2).

    Now let A=\{(a_1,b_1),(a_2,b_2), \dots , (a_m,b_m) \} be a finite subclass of \{\mathcal{O}_n\}. If we let b=\max\{b_1,b_2, \dots, b_m\}, then b<2 and

    \bigcup_{j=1}^m(a_j,b_j) \subseteq (1,b),

    but (1,b) and [b,2) are disjoint. Therefore, A is not a cover for (1,2).
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