Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:49:57Z http://mathoverflow.net/feeds/question/109583 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109583/is-every-sigma-algebra-of-sets-abstractly-the-borel-algebra-of-a-topology-o Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set? David Feldman 2012-10-14T05:56:18Z 2012-10-15T01:23:57Z <p><a href="http://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology" rel="nofollow">http://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology</a></p> <p>inspires the present question which asks for less.</p> <p><strong>Question:</strong> 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 ${\cal T}$?</p> <p>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.</p> <hr> <p>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: <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183510979" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183510979</a> 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.</p>