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Math Help - Topology, complements of finite sets

  1. #1
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    Topology, complements of finite sets

    Let R be the set of real numbers, then let \omega be the set of all complements of finite sets and the empty set. Prove that (R, \omega) is a topological structure.

    I was able to prove that it's closed under finite intersections, and I was able to show that the full set is in \omega, but I'm having difficulty proving that it's closed under arbitrary unions... Any help would be appreciated. This is not for homework of anykind.
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  2. #2
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    Quote Originally Posted by Chris11 View Post
    Let R be the set of real numbers, then let \omega be the set of all complements of finite sets and the empty set. Prove that (R, \omega) is a topological structure.
    The complement of a union is the intersection of complements: \left( {\bigcup\limits_\alpha  {B_\alpha  } } \right)^\prime   = \bigcap\limits_\alpha  {\left( {B_\alpha  } \right)^\prime  }

    Surely the any intersection of finite sets is finite.
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  3. #3
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    If M is co-finite and M\subseteq N, then N is co-finite too.
    Therefore the union of a class of co-finite sets must also be co-finite.
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