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Math Help - open sets question

  1. #1
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    open sets question

    (a) Let F be a collection of open sets in R^n. Prove that U F (i.e. the union of all of the sets in F) is always an open set in R^n.

    (b) Let F be the collection of closed intervals A_n=[1/n, 1-1/n] for n=1,2,3,.... What do you notice about U F? Is it closed, open, both or nither?
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  2. #2
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    Union of open sets set is an open set. Period.

    \bigcup\limits_{n = 1}^\infty  {\left[ {\frac{1}{n},1 - \frac{1}<br />
{n}} \right]}  = \left( {0,2} \right]
    Last edited by Plato; September 15th 2010 at 04:14 AM.
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  3. #3
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    Let p be a point in the union of all sets in F. Then p is in at least one of the open sets, A, and so is an interior point of that set- there exist an open neighborhood, U, of p that is a subset of A. But since U is a subset of A, it is a subset of any union that includes A- U is a subset of the union of all sets in F.

    The counter example shows that this is NOT true of intersections.

    By the way, please show what you have tried so we can at least know what concepts and methods you have available. Although you did not say it, I assumed that you were working in a metric space. In more general topology, the fact that infinite unions of open sets are open is part of the definition of "open set".
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