1F.3] 1F. Countable operations ... 35 and for each ordinal number 1, let Σ0 = ¬( Σ0). Unscrambling this, P is in Σ0 is there are pointsets P0,P1,... with each Pi in some Σ0, , such that P = i (X \ Pi). We call Σ0 the Borel pointclass of order . The dual and ambiguous Borel pointclasses are defined in the obvious way,(8,9) Π0 = ¬Σ0, ∆0 = Σ0 ∩ Π0. For finite this definition yields the pointclasses Σ0 as we know them, so there is no conflict in notation. It is very easy to extend the basic properties of the finite Borel pointclasses to all Borel pointclasses and we will leave this for the exercises. We only state here the basic characterization of the pointclass B of Borel sets,(8) B = Σ0. 1F.3. Theorem. For each product space X the class B X of Borel subsets of X is the smallest collection of subsets of X which contains the open sets and is closed under complementation and countable union similarly, B X is the smallest collection of subsets of X which contains the open (or the closed) sets and is closed under countable union and countable intersection. Proof. If P is Borel, then P is in Σ0 for some , so ¬P = X \ P ∈ Σ0 +1 , in particular, ¬P is Borel. Also, if Pi is Borel for every i, Pi ⊆ X, then Pi ⊆ Σ0 i for some i , so ¬Pi ∈ Σ0 i +1 and taking = supremum{ i +2 : i = 0,1,2,...}, we have P ∈ Σ0,since P = i Pi = i ( X \ (X \ Pi) ) . Thus the class of Borel subsets of X is closed under ¬ and . Conversely, if S is any collection of subsets of X which is closed under ¬ and , then S clearly contains all open subsets of X and an easy induction on shows that P ⊆ X,P ∈ Σ0 =⇒ P ∈ S. For the second assertion, notice first that B X is easily closed under countable intersection, since i Pi = X \ ( i (X \ Pi) ) . Conversely, if S contains all the opensubsets of X and is closedunderboth countable union and countable intersection, then each P ⊆ X which is either Σ0 or Π0 is in S byatrivialinductionon becauseclosedsetsarecountableintersectionsofopensets and in general P ∈ Σ0 =⇒ P = i Pi with each Pi ∈ Π0 i , i , P ∈ Π0 =⇒ P = i Pi with each Pi ∈ Σ0 i , i .

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