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Math Help - Axiom of choice.

  1. #1
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    Axiom of choice.

    Let \{U_i : i \in I\} be a collection of sets, all containing a common element x \in U_i for each i \in I.

    Do we need the axiom of choice to show that \displaystyle x \in \cap_{i\in I}U_i?

    The reason I ask is because wikipedia says that we need the axiom of choice to prove the Baire Category Theorem and in the proof it gives on the page, I can't see where this would be if it wasn't in the above.
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  2. #2
    Senior Member Shanks's Avatar
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    the definition of the intersection of a collection of sets:
    the collection of element x such that x belongs to U_i for all i in I.
    so we don't need AC to prove this definition.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Aradesh View Post
    Let \{U_i : i \in I\} be a collection of sets, all containing a common element x \in U_i for each i \in I.

    Do we need the axiom of choice to show that \displaystyle x \in \cap_{i\in I}U_i?

    The reason I ask is because wikipedia says that we need the axiom of choice to prove the Baire Category Theorem and in the proof it gives on the page, I can't see where this would be if it wasn't in the above.
    The axiom of choice, in the two most commonly used forms are "the choice function" and the "forming a set with one element from each". You do not need the AOC to say that if x\in U_i,\text{ }i\in\mathcal{I} then x\in\bigcap_{i\in\mathcal{I}}U_i, that is by definition. What you do need it for though, is to say "if x\in\bigcap_{i\in\mathcal{I}}U_i then x\in U_\jmath. blah blah blah". So, to pick a specific set from where x "came" from is what the AOC is used for.
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  4. #4
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    Hi

    In wikipedia's Baire's theorem proof, it is the axiom of choice that allows us to consider the sequence (x_n)_{n\geq 1}.

    if then
    I may have misunderstood you, Drexel28, but this does not need AC to be true.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by clic-clac View Post
    Hi

    In wikipedia's Baire's theorem proof, it is the axiom of choice that allows us to consider the sequence (x_n)_{n\geq 1}.

    I may have misunderstood you, Drexel28, but this does not need AC to be true.
    I did not actually read the proof, but instead gave what I thought may have been the issue. It is true that given x "came from" an infinite collection of sets that to specify one, and fix it requires the axiom of choice, right? Because we had to have some choice function to choose that specific set from which it came?
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  6. #6
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    Well if you just talk about one element or one set from an infinite collection, there is no problem, you can choose one and "fix" it. Problems arise when you want to fix an infinite amount of things at the same time; in such cases the axiom of choice can become necessary.
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