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Unformatted text preview: ction) and a countable union of closed sets an Fσ set (from
the French ferm´ for closed and somme for union).
The following theorem characterizes open subsets of R and will occasionally be of use. 5–1 Theorem 5.2. If E ⊆ R is an open set, then there exist at most countably many disjoint
open intervals Ij , j = 1, 2, . . ., such that
Ij . j =1 Proof. The trick is to deﬁne an equivalence relation on E as follows. If a, b ∈ E , we say
that a and b are equivalent, written a ∼ b, if the entire open interval (a, b) is contained in
E . This equivalence relationship partitions E into a disjoint union of classes. We do not
know a priori that there are at most countably many such classes. Therefore, label these
classes Ij , j ∈ J , where J is an arbitrary index set. Note that Ij is, in fact, an interval for
the following reason. If aj , bj ∈ Ij , then aj ∼ bj so that the entire open interval (aj , bj ) is
contained in Ij . As aj , bj were arbitrary we see that Ij is, in fact, an interval. The next step
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