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Math Help - How to show Topology and find basis

  1. #1
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    How to show Topology and find basis

    1 (a)
    Let \mathcal{C} be a collection of subset of the set X, satisfying:

    i. \phi and X contained in \mathcal{C},
    ii. finite unions of elements in \mathcal{C} are in \mathcal{C}, and
    iii. arbitrary unions of elements in \mathcal{C} are in [ \mathcal{C}

    Show that the collection

    \mathcal{T} = \{X - C|C \in \mathcal{C} \}

    is a topology on X.

    (b)
    Give an example of such a topology when X = \Re. What are the open sets in this topology?What are the closed sets? What is the basis for the topology?
    Last edited by flaming; April 8th 2009 at 11:03 AM.
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  2. #2
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    Please try to clarify how \mathcal{T} is defined.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Please try to clarify how \mathcal{T} is defined.



    That is how T is defined in the question.I am not sure what you mean by the question..sorry
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  4. #4
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    To aid in notation use this. A \in T\; \Rightarrow \;\left( {\exists A' \in C} \right)\left[ {A = X\backslash A'} \right].
    Because \emptyset  = X\backslash X\;\& \;X = X\backslash \emptyset the first two requirements are fulfilled.

    For union \left\{ {A_\alpha  } \right\} is a collection in T.
    \bigcup {\left\{ {A_\alpha  } \right\}}  = \bigcup {X\backslash A'_\alpha  }  = X\backslash \bigcap {A'_\alpha  } .

    Intersection is the same idea.
    Last edited by Plato; April 8th 2009 at 03:38 PM.
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