E the following theorem characterizes open subsets of

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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). e 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 E= ∞ ￿ 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 is...
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This document was uploaded on 04/07/2014.

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