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# p5 - #4 The boundary of a subset E of a metric space X...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #5 (due on Wednesday, October 12). 50 points are divided between 5 problems, 10 points each. #1. Show that for an arbitrary sequence E 1 ,E 2 ,... of sets, the set n =1 ( k = n E k ) is contained in n =1 ( k = n E k ) . #2. Let E be a nonempty open subset of a metric space ( X,d ), which is dense in X . Is it true in general that E = X ? Either prove that E = X , or give a counterexample. #3. Let E be a nonempty subset of R 1 , which is both open and closed in R 1 . Show that E = R 1 . Hint. Use Theorem 2.28.
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Unformatted text preview: #4. The boundary of a subset E of a metric space X , ∂E := { p ∈ X : ∀ r > both E ∩ N r ( p ) and E c ∩ N r ( p ) are nonempty } , where N r ( p ) := { q ∈ X : d ( p,q ) < r } . Show that ∂E is closed in X . #5 (exercise 18 on p.44). Is there a nonempty perfect set in R 1 which contains no rational number? Hint. This problem is not so easy. You can try to adjust the construction of the Cantor set in 2.44 on p.41....
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