Infinity-Borel set
In set theory, a subset of a Polish space is ∞-Borel if it can be obtained by starting with the open subsets of , and transfinitely iterating the operations of complementation and wellordered union. This concept is usually considered without the assumption of the axiom of choice, which means that the ∞-Borel sets may fail to be closed under wellordered union; see below. Formal definition We define the set of ∞-Borel codes and the interpretation function below. A ∞-Borel set is a subset of which is in the image of the interpretation function . The set of ∞-Borel codes is an inductive type generated by functions , and for each ; the interpretation function is defined inductively as , and . Here denotes the Hartogs number of : a sufficiently large ordinal such that there is no injection from to . Restricting to unions of length below doesn't affect the possible unions (as any union of length can be replaced by one of length by removing duplicates), but ensures that the ∞-Borel codes form a set, not a proper class. This can be phrased more set-theoretically as a definition by transfinite recursion as follows: For every open subset , the