What is the difference between: ($X$: a set)
- Topology (open set system) on $X$
- Borel sigma-algebra on $X$
Both are a set of open subsets.
Both include $X$ and empty set.
Both are closed under union and intersection.
They look like the same thing, right?
Or 2. is the smallest sigma-algebra containing 1.?
Best Answer
They're not the same. If you have a topology (only the open sets) you get a Borel $\sigma$-algebra: the smallest one containing all open sets (so that contains the topology).
But this also contains all closed sets, all countable intersections of open sets (which need not be open, nor closed) etc. The Borel $\sigma$-algebra will generally be a lot bigger.
Also note that $\sigma$-algebras are closed under complements (!), countable unions and countable intersections. While topologies are closed under finite intersections and arbitrary unions. So quite different.
Any space with a topology automatically has a Borel $\sigma$-algebra (if we need it, say for measure theory), while the having a $\sigma$-algebra does not mean having a topology.