1873_solutions

That g r and for this purpose all we have to show is

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Ask a homework question - tutors are online