1873_solutions

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Unformatted text preview: e conclude that L AÞB  L A ÞL B . 7. Is it true that if A and B are subsets of a metric space X then LA B LA LB? What if A and B are closed? What if A and B are open? What if A and B are intervals in R? The answers are no, no, no and no. Look at the following example: 129 A  0, 1 and B  1, 2 These two sets are closed in R and LA B L 1  while LA L B  0, 1 Now look at the following example: A  0, 1 and In this case LA B L and LA L B  0, 1 1, 2  1 . B  1, 2 .  1, 2  1 . 8. Is it true that if D is a dense subset of R then L D  R? The assertion is true. Suppose that D  R. We know that whenever a and b are real numbers and a  b there must be members of D lying between a and b. Now suppose that x is a real number. To show that x is a limit point of D, suppose that   0. Since there must be members of D in the interval x, x   we conclude that the set x , x   D x is nonempty. 9. Is it true that if D is a dense subset of a metric space X then L D  X? No! Consider, for example, a metric space like 1, 2, 3 that has no limit points at all. The set 1, 2, 3 is dense in this space. 10. Is it true that if D is a dense subset of a connected metric space X then L D  X? Yes, as long as the space X contains at least two different points then if X connected and D is dense in X we must have L D  X. As a matter of fact, we can do better than this. The condition that X be connected says that X has no open closed subsets other than and X. All we actually need to know is that no singleton in X can be an open set. Suppose that X is a metric space in which no singleton is open. Suppose that D is a dense subset of X. Suppose that x X and, to show that x is a limit point of D, suppose that   0. Since the ball B x,  is open and x is not open we know that B x,  x , in other words, B x,  x . Furthermore, since B x,  x  B x,  X x which is open, the set B x,  x , being a nonempty open subset of X, must intersect with the set D. Thus B x,  D x . 11. Is it true that if X is a metric space and L X  X then for every dense subset D of X we have L D  X? The answer is yes. See the solution to Exercise 10. 12. Given that a set S of real numbers is nonempty and bounded above but that S does not have a largest member, prove that sup S must be a limit point of S. State and prove a similar result about inf S. To show that sup S is a limit point of S, suppose that   0. Since sup S   sup S and since sup S is the least upper bound of S the number sup S  fails to be an upper bound of S. Choose a member x of S such that sup S   x. Since x sup S and since sup S does not belong to S we have x  sup S. We conclude that sup S , sup S   S sup S . 13. Given that S is a closed subset of a metric space X and that every infinite subset of X has at least one limit point, prove that every infinite subset of S must have at least one limit point that belongs to S. Every infinite subset of S, being an infinite subset of X, must have a limit point somewhere in X....
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## This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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