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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 Spring '08
 FLANAGAN
 Work, Empty set

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