Math521Assignment3Answers - Math 521 Lecture 2 Hints for Assignment 3 Problem 1 Let X be a metric space and let E X be a subset The interior of the set

Math521Assignment3Answers - Math 521 Lecture 2 Hints for...

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Math 521, Lecture 2 Hints for Assignment # 3 Problem 1 Let X be a metric space, and let E X be a subset. The interior of the set E is the set E o = x E there exists r > 0 so that B ( x, r ) E . The closure of E is the set E = E E where E is the set of limit points of E . (a) Prove that E o E is always an open set, and that E E is always a closed set. Let x E o . By hypothesis, there exists r > 0 so that B ( x, r ) E . To show that E o is open, it suffices to show that B ( x, r ) E o . Let y B ( x, r ) . We showed in class that B ( x, r ) is open. Thus there exists δ > 0 so that B ( y, δ ) B ( x, r ) . But since B ( x, r ) E , this shows that B ( y, δ ) E , and hence y E o . Thus B ( x, r ) E o . To show that E is closed, it suffices to show that every limit point of E is contained in E . Let x be a limit point of E . Then for each n N there exists a point y n E E such that y n = x and d ( x, y n ) < 1 n . It follows (why?) that x is either a limit point of E or a limit point of E . If x is a limit point of E , then by definition it belongs to E which is a subset of E . If x is a limit point of E , then by a problem on Assingment #2, it follows that x E . Thus in either case, x E . (b) Prove that E o is the largest open set contained in E and that E is the smallest closed set containing E . (Part of the problem is to figure out what “largest” and “smallest” should mean.) By part (a), E o is an open set contained in E . On the other hand, suppose that V is any open set contained in E . Let x V . Then since V is open, there exists r > 0 so that B ( x, r ) V . But since V E , it follows that B ( x, r ) E . Hence x E o , and consequently, V E o . Thus E o is an open set contained in E and it contains every open set contained in E . In this sense it is the largest open set contained in E . Also by part (a), E is a closed set containing E . Suppose W is a closed set containing E . Let x be any limit point of E . Then x is also a limit point of W since E W . But since W is closed, it follows that x W . Thus W contains E . Since by hypothesis it contains E , it contains E E = E . Thus we have shown that if W is any closed set containing E , then W contains E . It is in this sense that E is the smallest closed set containing E .
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  • Fall '14
  • Math, Topology, Metric space, α, Closed set, Δj

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