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23h
revised Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
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23h
comment Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
And thanks to both of the above users for their comments, I posted the answer mostly because I would have been surprised if it was correct given the statement of the problem and wanted to find the flaw!
23h
comment Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
@ByronSchmuland The previous conclusion was wrong, since I was using the wrong definition of chain, but even in the previous argument the minimal element would have been measurable. The problem was that if $\mathcal{F}$ is uncountable then the intersection of all elements in a chain need not be measurable.
23h
comment Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
@NateEldredge Oh yeah, thanks, I realise that I was thinking of chains as ascending chains rather than totally ordered subsets, thanks for the correction.
23h
revised Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
Corrected an error.
1d
answered Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.
Jun
11
awarded  Nice Answer
Jun
1
comment When do the Nakano identities hold?
... there's some non-trivial work going on in the background that makes the argument go through, here we need to do some work to show that some suitable taylor series converges in remark 4.2.5. I'm sorry this isn't more useful, but if you're still interested in a couple of weeks, check back and I'll see if I can remember what I thought the answer was and post it.
Jun
1
comment When do the Nakano identities hold?
@Dtseng I think I found part of the problem, I was going to post an answer once exams finish, and probably e-mail Ross about it too. The fact that not all smooth frames are holomorphic is what was causing some of the problems originally. Huybrechts refers to "Remark 4.2.5" in his book for how we can take a holomorphic frame. This is where we need the connection to be the Chern connection, since properties of the matrix A are used with the content of this remark. I don't really fully understand it, and as with much of Huybrechts...
May
13
revised What are some examples of principal, proper ideals that have height at least $2$?
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May
13
revised What are some examples of principal, proper ideals that have height at least $2$?
edited body
May
12
answered The Dimension Sequence of a Ring
May
11
awarded  Proofreader
May
11
reviewed Approve Area of a triangle from some of its parts
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9
reviewed Approve Using the Newton-Quotient
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9
reviewed Close Diffeomorphism ( differential geometry)
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9
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9
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