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Math Help - clousre

  1. #1
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    clousre

    Suppose that  \displaystyle \tau 1 \mbox  and  \quad \tau2 \mbox   \quad are  \quad \quad \quad topologies \quad on  \quad a set X  \quad with  \quad   \tau1 \subset \tau2 . \mbox  \quad if A \subset X  \quad

     \mbox  prove \quad  that :  \quad \overline{A}^2  \subset \overline {A}^1 \mbox   \quad where  \overline {A}^1 \mbox  is  \quad the  \quad clousre   \quad of A  \quad in (X, \tau1 )   \quad  \overline {A}^2  \mbox  is  \quad the  \quad clousre  \quad of A \quad  in  (X,\tau2)
    Last edited by flower3; May 28th 2009 at 10:34 AM.
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  2. #2
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    If A is an open subset in \tau_1 then A is open in \tau_2. So, if B is closed in \tau_1 then B is closed in \tau_2. Now you define the closure of A in \tau_i as the intersection of all closed subsets of (X, \tau_i) such that they contain A, but since closed sets in \tau_1 are closed in \tau_2 then the closure in \tau_2 has more elements to intersect and thus is contained in the closure in \tau_1.

    Well, that is an overview of the proof. I leave the details to you.
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  3. #3
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    Here is a different approach.
    x \in \overline A if and only if each for each open set O, x \in O\, \Rightarrow \,O \cap A \ne \emptyset .
    But O \in \tau _1 \; \Rightarrow \;O \in \tau _2 .
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