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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