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4130hw5 - (b Show that every open set can be expressed as a...

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4130 HOMEWORK 5 Due Tuesday March 9 (1) A subset I of R is called an interval if for all x, y I and all z R , if x < z < y then z I . Show that if I is a bounded interval, then (inf I, sup I ) I . Using this, show that I must be one of the following four intervals: (inf I, sup I ) [inf I, sup I ) (inf I, sup I ] [inf I, sup I ] . (2) For each of the following S R ∪ {±∞} , state whether there is a sequence whose set of limit-point is S . If there is, find one. If not, give a reason. (a) S = { 0 } . (b) S = {∞ , -∞} . (c) S = { 1 n : n N } (d) S = N . (e) S = N ∪ {∞} . (3) Let U be the following collection of subsets of R . U = { ( q, r ) : q, r Q , q < r } . (a) Show that U is countable.
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Unformatted text preview: (b) Show that every open set can be expressed as a union of intervals from U . (c) Let X denote the set of all open subsets of R . Show that |X| = | R | . (Hint: recall that | R | = |P ( N ) | .) (d) Show that the set of all closed subsets of R also has cardinality | R | . [Remark: Let Y denote the set of all subsets of R that are either open or closed. One can now show that |Y| < |P ( R ) \ Y| . In other words, loosely speaking, “most” subsets of R are neither open nor closed.] 1...
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