In set theory, a subset of a Polish space X {\displaystyle X} is ∞-Borel if it can be obtained by starting with the open subsets of X {\displaystyle X} , and transfinitely iterating the operations of complementation and well-ordered 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 well-ordered union; see below.
Formal definition We define the set of ∞-Borel codes C {\displaystyle C} and the interpretation function ‖ − ‖ : C → P ( X ) {\displaystyle \left\|-\right\|:C\to {\mathcal {P}}(X)} below. A ∞-Borel set is a subset of X {\displaystyle X} which is in the image of the interpretation function ‖ − ‖ {\displaystyle \left\|-\right\|} . The set of ∞-Borel codes is an inductive type C {\displaystyle C} generated by functions o p e n : O ( X ) → C {\displaystyle {\mathtt {open}}:{\mathcal {O}}(X)\to C} , c o m p : C → C {\displaystyle {\mathtt {comp}}:C\to C} and u n i o n α : C α → C {\displaystyle {\mathtt {union}}_{\alpha }:C^{\alpha }\to C} for each α < Ξ {\displaystyle \alpha <\Xi } ; the interpretation function is defined inductively as ‖ o p e n ( U ) ‖ = U {\displaystyle \left\|{\mathtt {open}}(U)\right\|=U} , ‖ c o m p ( c ) ‖ = X ∖ ‖ c ‖ {\displaystyle \left\|{\mathtt {comp}}(c)\right\|=X\setminus \left\|c\right\|} and ‖ u n i o n α ( c → ) ‖ = ∪ β < α ‖ c β ‖ {\displaystyle \left\|{\mathtt {union}}_{\alpha }({\vec {c}})\right\|=\cup _{\beta <\alpha }\left\|c_{\beta }\right\|} . Here Ξ {\displaystyle \Xi } denotes the Hartogs number of P ( X ) {\displaystyle {\mathcal {P}}(X)} : a sufficiently large ordinal such that there is no injection from Ξ {\displaystyle \Xi } to P ( X ) {\displaystyle {\mathcal {P}}(X)} . Restricting to unions of length below Ξ {\displaystyle \Xi } doesn't affect the possible unions (as any union of length ≥ Ξ {\displaystyle \geq \Xi } can be replaced by one of length < Ξ {\displaystyle <\Xi } 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:
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