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Family of sets without maximal element

  1. Jun 12, 2013 #1
    I have begun to learn about maximal elements from a linear algebraic perspective (maximal linearly independent subsets of vector spaces). I have a few questions of which I have been able to get few insights online:

    1) Does every family of sets have a maximal element? How can I make a family of sets that does not have a maximal element? I have to obviously make the hypothesis of Zorn's lemma fail, but I can't quite see how to do that.

    2) Does every chain of sets have a maximal element? It seems that a chain of sets necessarily satisfies the criteria for Zorn's lemma but I am not sure.

    Thanks!

    BiP
     
  2. jcsd
  3. Jun 13, 2013 #2

    HallsofIvy

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    No, to both of those. A counter-example is [tex]A_n= [-n, n][/tex], the closed interval from -n to n.

    Zorns lemma says that if every chain in an ordered collection has a maximum, then the collection has maximal elements. If it were true that every chain has a maximum or that every family has a maximum, Zorn's lemma would be unnecessary.
     
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