TITLE: Is there an example of a sigma algebra that is not a topology?
QUESTION [21 upvotes]: Is there an example of a sigma algebra that is not a topology? If this is not the case, is it possible to prove that all sigma algebras are topologies?

REPLY [21 votes]: Let $S$ be any uncountable set, and let $\mathcal{A}$ be the collection of all subsets of $S$ which are either countable or have countable complement.  
This collection is evidently closed under complementation.  If I have a countable union of elements of $\mathcal{A}$, all of which are countable, then the union is countable.  Otherwise, at least one element is cocountable, hence so is the union.  A similar argument works for intersections.  So $\mathcal{A}$ is a $\sigma$-algebra.
It is not a topology because it contains all the singleton sets but not all subsets of $S$ -- in particular, it contains no set which is uncountable with uncountable complement.