Uni Stuff > Mathematics

Analysis

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kamil9876:
No definitely no restriction on countable sets, Zorn's lemma can be used to prove that every Vector space has a basis (and there are LOTS of different spanning sets for a given vector space).

your set (with the usual ordering of the reals)  does not satisfy the hypothesis, here is a chain:



It is not bounded above by any element in (so you can't say since it is not in )

/0:
Oh so when it says it has an 'upper bound' it means the upper bound is in Y?
I thought the upper bound didn't have to be in Y.
So should have an upper bound.

kamil9876:
Wait hold on, I think I have misunderstood your question; are you worried about the fact that has no maximal element? or are you more concerned with the existence of a maximal element in (then again you havn't told me what is)

/0:
I'm more worried about Y... X is just any old set which you get the ordered subset Y from.

kamil9876:

--- Quote ---"A partially ordered set in which any chain has an upper bound has a maximal element."
--- End quote ---

let me restate it to be more clear:

"If a partially ordered set, has the property that every chain has an upper bound. Then has a maximal element."

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