Borel system of sets

-system, generated by a system of sets

The smallest -system of sets, , containing . The sets belonging to are called the Borel sets (cf. Borel set), or -sets, generated by the system . For each ordinal number , where is the initial ordinal number of cardinality , the Borel classes are defined as follows: ; consists of the unions if is odd, and it consists of the intersections if is even, of sequences of sets belonging to , . In such a case . The same Borel system of sets is obtained if the union and intersection above are interchanged. A Borel set belongs properly to a class if it belongs to the class but does not belong to if . (Sometimes one takes non-intersecting classes, i.e. then the system is called a class.)