Is every $\sigma$-algebra of sets abstractly the Borel algebra of a topology on perhps some other set?

2012-10-14T05:56:18Z

http://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology

inspires the present question which asks for less.

Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ${\cal T}$ on perhaps some other set $Y$ such that ${\cal A}$ is isomorphic to the Borel sets determined by ${\cal T}$?

Examples contained in the answers to the quoted question indicate an answer of "not necessarily" if one also requires $X=Y$. I may be wrong, but it seems to me that a negative answer here (if appropriate) will require a new idea.

I've changed the title of my question to account for Gerald Edgar's comment. One could still ask to represent any abstract $\sigma$-algebra as a Borel field, but this isn't possible, as noted by Loomis here: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183510979 That said, the theorem Loomis proves indeed realizes abstract $\sigma$-algebras as Borel fields modulo a $\sigma$-ideal. I don't believe this settles my intended question though.

Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set? - MathOverflow

Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every $\sigma$-algebra of sets abstractly the Borel algebra of a topology on perhps some other set? - MathOverflow

Is every $\sigma$-algebra of sets abstractly the Borel algebra of a topology on perhps some other set?