Borelsets

Borel sets, named after Émile Borel, are the sets that can be formed from open sets of a topological space by countable unions, countable intersections, and complements. For a topological space X, the collection B(X) of Borel sets is the sigma-algebra generated by the open subsets of X; equivalently, it is the smallest sigma-algebra containing all open sets. Thus every open or closed set is Borel, and any countable union, intersection, or complement of Borel sets remains Borel.

In the real line R with its standard topology, Borel sets are the sets that can be obtained from open subsets of R by countable set operations. They include all open and closed sets, all F_sigma sets (countable unions of closed sets), and all G_delta sets (countable intersections of open sets). The rational numbers, for instance, form an F_sigma set as a countable union of singletons (each singleton is closed). In general, B(X) is closed under preimages of continuous maps: if f: X → Y is continuous and B is Borel in Y, then f^{-1}(B) is Borel in X.

Not every subset of a space is Borel. There exist non-Borel subsets of R, and the Lebesgue measure on R strictly extends the Borel sigma-algebra by its completion. In many contexts, such as in Polish spaces, Borel sets provide a canonical sigma-algebra that interacts well with measure, topology, and descriptive set theory.