For a topological space $(X, \mathcal{T})$, the Borel $\sigma$-algebra on $X$ is defined as the smallest $\sigma$-algebra containing all the open sets. This is done by taking the intersection of all $\sigma$-algebras containing $\mathcal{T}$.
Are there other ways to do this? Is there a simple way to characterize this (e.g. in linear algebra, the smallest subspace containing a set is given as the set of all linear combinations)?
If not, is there some intuitive/logical reason why not; why can't there be a simpler description/characterization, say in terms of certain basic operations (e.g. linear combinations, in the case of subspaces of vector spaces)?
I am particularly interested in the usual Borel algebra on the real numbers (that is, $X = \mathbb{R}$ with the usual topology).
Best Answer
The intersection of all sigma algebras formulation is sort of a "top down" characterization. You can also think "bottom up": start with a basis for the topology and then add in all complements, countable unions, countable intersections, and so on. The reason this definition is sometimes avoided is that although it sounds constructive it does not give much of an idea of what an arbitrary Borel set must "look like" (and maybe there is a concern that students would form an erroneous guess about what they look like).
You may be interested in the "Borel hierarchy" of descriptive set theory, which goes into considerable detail in discussing sets in this space. https://en.wikipedia.org/wiki/Borel_hierarchy