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Topological property

  1. Jan 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let X be a topological space, a subset S of X is said to be locally closed if
    S is the intersection of an open set and a closed set, i.e
    S= O intersection C where O is an open set in X and C is a closed set in X

    Prove that if M,N are locally closed subsets then M union N is locally closed.

    3. The attempt at a solution

    So M = O intersection C and N = O' intersection C' where O, O' are open sets in X and
    C, C' are closed sets in X.
    It follows that M union N = (O intersection C) U (0' intersection C').
    From h ere I played with this expression a while using distributive laws but got stuck,
    somewhere I end up with the union of a closed set and an open set, i.e O U C, but I think
    we cannot said anything about this particular union. Maybe I'm missing some useful
    set-theoretical identity. Can you please help?
  2. jcsd
  3. Jan 23, 2009 #2
    Are you certain you have the problem statement written correctly, Carl? Consider [tex]\mathbb{R}[/tex] with open sets [tex]\emptyset[/tex], [tex]\mathbb{R}[/tex] and [tex](-\infty,a)[/tex] for [tex]a\in\mathbb{R}[/tex] for a counterexample.
    Last edited: Jan 23, 2009
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