Family of sets without maximal element

  • Thread starter Bipolarity
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  • #1
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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
 

Answers and Replies

  • #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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