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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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This note was uploaded on 03/29/2010 for the course PHYS 3318 taught by Professor Flanagan during the Spring '08 term at Cornell University (Engineering School).
- Spring '08