This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some time, I am
bringing it up here on mathoverflow in the hopes that it may find an answer here.
For any topological space, one may consider the Borel sets of the
space, the $\sigma$-algebra generated by the open sets of that
topology. The question is whether every $\sigma$-algebra arises in
this way.
Question. Is every $\sigma$-algebra the Borel algebra of a
topology?
In other words, does every $\sigma$-algebra $\Sigma$ on a set $X$
contain a topology $\tau$ on $X$ such that $\Sigma$ is the $\sigma$
algebra generated by the sets in $\tau$?
Some candidate counterexamples were proposed on the math.SE
question, but ultimately shown not to be counterexamples. For
example, my
answer there shows that the collection of Lebesgue
measurable sets, which seemed at first as though it might be a
counterexample, is nevertheless the Borel algebra of the topology
consisting of sets of the form $O-N$, where $O$ is open in the
usual topology and $N$ is measure zero. A proposed counterexample
of Gerald Edgar's there, however, remains unresolved. And I'm not clear on the status of a related proposed counterexample of George Lowther's.
Meanwhile, allow me to propose here a few additional candidate
counterexamples:
-
Consider the collection $\Sigma_0$ of eventually periodic subsets of
$\omega_1$. A set $S\subset\omega_1$ is eventually periodic if
above some countable ordinal $\gamma$, there is a countable length pattern which is simply repeated up to
$\omega_1$ to form $S$. This is a $\sigma$-algebra, since it is closed under
complements and countable intersections (one may find a common
period among countably many eventually periodic sets by intersecting the club sets consisting of starting points of the
repeated pattern). -
Consider the collection $\Sigma_1$ of eventually-agreeing
subsets of the disjoint union $\omega_1\sqcup\omega_1$ of two copies of $\omega_1$. That is,
sets $S\subset \omega_1\sqcup\omega_1$, such that except for
countably many exceptions, $S$ looks the same on the first copy as it does on the
second. Another way to say it is that the symmetric difference of
$S$ on the first copy with $S$ on the second copy is bounded. This is a $\sigma$-algebra, since it is closed under complement and also under countable intersection, as the countably many exceptional sets will union up to a countable set.
Please enlighten me by showing either that these are not actually
counterexamples or that they are, or by giving another
counterexample or a proof that there is no counterexample.
If the answer to the question should prove to be affirmative, but only via strange or unattractive topologies, then consider it to go without saying that we also want to know how good a topology can be found (Hausdorff, compact and so on) to generate the given $\sigma$-algebra.
Best Answer
Unfortunately, I can only provide a reference but no ideas since I don't have the paper. In "On the problem of generating sigma-algebras by topologies", Statist. Decisions 2 (1984), 377-388, Albert Ascherl shows (at least according to the summary to be found on MathSciNet) that there are $\sigma$-algebras which can't be generated by a topology.
Robert Lang (same journal 4 (1986), 97-98) claims to give a shorter proof.
As suggested by Joel, I add the ideas of Lang's example. The underlying space is $\Omega= 2^{\mathbb R}$, that is the space of all indicator functions, and the $\sigma$-algebra is $\mathcal A = \bigotimes_{\mathbb R} \mathcal P$ where $\mathcal P$ is the power set of the two element set. It is generated by the system $\mathcal E$ of the "basic open sets" of the product topology (prescribed values in a finite number of points). This generator has the cardinality $c$ of the continuum and since the generated $\sigma$-algebra can be obtained in $\omega_1$ (transfinite) induction steps the cardinality of $\mathcal A$ is also $c$. On the other hand, if $\mathcal T$ is a topology with $\mathcal A=\sigma(\mathcal T)$ then $\mathcal T$ separates points (this should follow from the "good sets principle"), in particular, for two distinct points of $\Omega$ the closures of the corresponding singletons are distinct. Hence $\mathcal T$ has at least $|\Omega|=2^c$ elements.