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Math Help - Closed sets

  1. #1
    Member kezman's Avatar
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    Closed sets

    Find a counterexample of:

    the union of the closure of A_i = the closure of the union of A_i

    I hope its understandable
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    Quote Originally Posted by kezman View Post
    Find a counterexample of:

    the union of the closure of A_i = the closure of the union of A_i

    I hope its understandable
    How is closure defined here?

    This is my understanding.

    Definition: Given a set S\subseteq \mathbb{R}^2 we define \bar S the "closure of S" to be the set of all points such that \forall s\in S \ \exists \delta>0 \ \ N(s,\delta)\cap S\not = \{ \}.

    Note: The notation N(s, \delta) is the open disk cented at s with radius \delta.
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  3. #3
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    Consider the set A_n  = \left( {\frac{1}{{1 + n}},1} \right] the closure of which is \overline {A_n }  = \left[ {\frac{1}{{1 + n}},1} \right].

    But consider this: \overline {\bigcup\limits_{n = 1}^\infty  {A_n } }  = \left[ {0,1} \right]\quad \mbox{but}\quad \bigcup\limits_{n = 1}^\infty  {\overline {A_n } }  = \left( {0,1} \right].
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  4. #4
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    Quote Originally Posted by Plato View Post
    Consider the set A_n  = \left( {\frac{1}{{1 + n}},1} \right] the closure of which is \overline {A_n }  = \left[ {\frac{1}{{1 + n}},1} \right].

    But consider this: \overline {\bigcup\limits_{n = 1}^\infty  {A_n } }  = \left[ {0,1} \right]\quad \mbox{but}\quad \bigcup\limits_{n = 1}^\infty  {\overline {A_n } }  = \left( {0,1} \right].
    Beautiful conterexample.

    I was trying to come up with a conterexample in \mathbb{C} using the definition I posed above (same idea) but I used finite number of sets. And I forgot the important rule of analysis (or topology here): what works for finite sets does not necessarily work for infinite sets.

    So I am guessing if S=\{A_i | i \in I\} was such that |S|<\aleph_0 then the above would actually be true. Correct? Maybe this is consequence of Heine-Borel theorem, no?
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    The example works great thanks.
    Now im trying to find a counterexample of finite unions (that was a problem of my latest exam).
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  6. #6
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    Quote Originally Posted by kezman View Post
    Now im trying to find a counterexample of finite unions.
    The nature of a standard metric topology makes it impossible to find such an example. However, I quite sure that one could construct a general (probability a finite) topological space where that property is true. I just not have thought about it in a very long time. Good Luck!
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