Intro to Analysis in-class_Part_22

Intro to Analysis in-class_Part_22 - 3.2 Open sets and...

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Unformatted text preview: 3.2 Open sets and closed sets 49 Definition 3.2.7. The interior of A ⊆ R is int A := { x ∈ A . . . x ∈ ( a,b ) ⊆ A, for some a,b } . x is an interior point of A iff x ∈ int A . It is obvious that every set contains its interior points. For a set to be open, it means that every point is an interior point. Theorem 3.2.8. x = lim x n iff every neighbourhood of x contains all but finitely many points of { x n } . Proof. Exercise: this is basically the same as the first theorem in this section. Definition 3.2.9. x is a limit point of A iff every neighbourhood U of x contains a point of A , other than x itself. For U open, A := { x . . . ∀ U open nbd, { x } ( U ∩ A } . Write A for the set of limit points of A . If a sequence has only a finite number of repetitions, this coincides with the defn of limit point of a sequence: { 5 , 5 , 5 , 5 , 5 ,... } has 5 as a limit point of the sequence, but not the set. (This is because of the requirement { x } ( U ∩ A .) See HW § 3.2.3 #2.3....
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Intro to Analysis in-class_Part_22 - 3.2 Open sets and...

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