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Math Help - Union of open sets proof

  1. #1
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    Union of open sets proof

    Problem: Prove that the union of any collection of open sets is open.

    Is the following proof acceptable?

    Proof: Let U' denote the collection of open sets and let U' = \bigcup U_n , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point s_0 that belongs to U'. Since s_0 belongs to U', it is implied that s_0 belongs to at least one U_n. So for s_0 that belongs to U_1 \in U' , there exists a neighborhood in U_1; For s_0 that belongs to U_2 \in U' , there exists a neighborhood in U_2,...If we take the union of all U_n , there exists a neighborhood in U' for all s_0. Therefore, U' is open.
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  2. #2
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    Re: Union of open sets proof

    Quote Originally Posted by MissMousey View Post
    Problem: Prove that the union of any collection of open sets is open. Is the following proof acceptable?
    Proof: Let U' denote the collection of open sets and let U' = \bigcup U_n , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point s_0 that belongs to U'. Since s_0 belongs to U', it is implied that s_0 belongs to at least one U_n. So for s_0 that belongs to U_1 \in U' , there exists a neighborhood in U_1; For s_0 that belongs to U_2 \in U' , there exists a neighborhood in U_2,...If we take the union of all U_n , there exists a neighborhood in U' for all s_0. Therefore, U' is open.
    How do we know what is acceptable or not?
    I am puzzled by this problem. Topologies are collections of open sets that are closed under arbitrary unions. Therefore, your notes must start at some other point. A neighborhood-space may be?
    So unless you list definitions and axioms there is no way to comment.
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    Re: Union of open sets proof

    my guess is that the thread-starter's definition of open is: a set U is open, if for every point x in U, there is a neighborhood of x that is a subset of U; or, equivalently, that every point of U is an interior point.
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    Re: Union of open sets proof

    Quote Originally Posted by Deveno View Post
    the thread-starter's definition of open is: a set U is open, if for every point x in U, there is a neighborhood of x that is a subset of U; or, equivalently, that every point of U is an interior point.
    I understand that. The difficulty with that neighborhood-spaces are widely varied from R L Moore's developable regions to Helen Cullen's neighborhood relation.
    In other words, what is a neighborhood?
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