Let be a topological space. Then there is a -algebra on generated by the open subsets of . Elements of are called the Borel sets (or Borel subsets, or Borel-measurable sets, etc) of , and itself is called the Borel -algebra on .
The preceding abstract definition can be made concrete (and predicative, at least over , although most predicative mathematicians don't accept ) as follows:
So we need an uncountable number of steps, not just two.
This is only the beginning of descriptive set theory?; our are their —except that for some reason they start with instead of —, and the subject continues to higher values of the superscript.
(To remember the other symbols, you need to know two languages: French and German. The ‘’ comes from French ‘fermé’ for ‘closed’, while ‘’ is simply the next letter; the ‘’ and ‘’ are from German ‘Summe’ for ‘union’ and ‘Durchschnitt’ for ‘intersection’ and are commonly used for countable such.)
Note that countable choice is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union. But the full axiom of choice is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the axiom of determinacy? instead.
Sometimes one does measure theory with -rings or even -rings, which are more general than -algebras. However, even the -ring generated by a topology is in fact a -algebra.
Nevertheless, on a locally compact Hausdorff space, we may define the Borel -ring and the Borel -ring to be the -ring or -ring generated by the compact subsets of . When is compact (such as the unit interval), then these both agree with the Borel -algebra; when is -compact (a countable union of compact subsets, such as the real line), then the Borel -ring still agrees with the Borel -algebra.