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Problem 3.23. Let X = { | n e Z+}U{0} c R. (1) What is the relationship between the subspace topology on X and the order topology on X? (2) Give X the order topology. What are the limit points of X?

Problem 3.23. Let X = { | n e Z+}U{0} c R. (1) What is the relationship between the subspace topology on X and the order topology on X? (2) Give X the order topology. What are the limit points of X?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
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Solve 3.23 in detail please 

3.2. Order Topology. Definition 3.15. Let X be a nondegenerate set with linear order <. The order topology on X is the topology generated by the subbasis S consisting of all positive and negative open rays; that is, S= {(-x,b) | b E X}U{(a, ∞) | a E X}. Exercise 3.16. Let X be a nondegenerate ordered set with the order topology. Show the following: (1) every open interval is open, (2) every closed interval is closed, (3) every open ray is open, and (4) every closed ray is closed. Problem 3.17. Suppose X has no least nor greatest element. Describe the basis B for the order topology generated by the subbasis S. Show that B\ S is also a basis for the order topology on X. Problem 3.18. Suppose X = [a,w] has both a least and a greatest element. Describe the basis B for the order topology generated by the subbasis S. Is B\S also a basis for the order topology on X ? Problem 3.19. For X = N with the natural order <, describe the order topology on N. Have we seen this topology on N before? Find a minimal basis for the order topology on N. Remark 3.20. For the real numbers R with the natural order <, the basis described in Problem 3.17 is called the standard basis for the standard topology on R. For the unit interval [0, 1] with the natural order < (inherited from the order on R), the basis described in Problem 3.18 is called the standard basis for the standard topology on (0, 1].
Transcribed Image Text:3.2. Order Topology. Definition 3.15. Let X be a nondegenerate set with linear order <. The order topology on X is the topology generated by the subbasis S consisting of all positive and negative open rays; that is, S= {(-x,b) | b E X}U{(a, ∞) | a E X}. Exercise 3.16. Let X be a nondegenerate ordered set with the order topology. Show the following: (1) every open interval is open, (2) every closed interval is closed, (3) every open ray is open, and (4) every closed ray is closed. Problem 3.17. Suppose X has no least nor greatest element. Describe the basis B for the order topology generated by the subbasis S. Show that B\ S is also a basis for the order topology on X. Problem 3.18. Suppose X = [a,w] has both a least and a greatest element. Describe the basis B for the order topology generated by the subbasis S. Is B\S also a basis for the order topology on X ? Problem 3.19. For X = N with the natural order <, describe the order topology on N. Have we seen this topology on N before? Find a minimal basis for the order topology on N. Remark 3.20. For the real numbers R with the natural order <, the basis described in Problem 3.17 is called the standard basis for the standard topology on R. For the unit interval [0, 1] with the natural order < (inherited from the order on R), the basis described in Problem 3.18 is called the standard basis for the standard topology on (0, 1].
Problem 3.22. Let X = [0, 1] CR. On the one hand, X has a subspace topology T, induced by the standard (order) topology on R. On the other hand, X has a natural order induced by the natural order on R. This order induces an order topology T2 on X. How are Ti and T2 related? Problem 3.23. Let X = {; |n € Z+}U {0} C R. (1) What is the relationship between the subspace topology on X and the order topology on X? (2) Give X the order topology. What are the limit points of X ? 4 J. C. MAYER Problem 3.24. Let X = (0, 1] U {2} C R. On the one hand, X has a subspace topology T1 induced by the standard (order) topology on R. On the other hand, X has a natural order induced by the natural order on R. This order induces an order topology T2 on X. (1) How are Ti and T2 related? (2) Is 1 a limit point of X? (3) Is 2 a limit point of X? Problem 3.25. Redo Problem 3.24 with X = [0, 1) U {2}.
Transcribed Image Text:Problem 3.22. Let X = [0, 1] CR. On the one hand, X has a subspace topology T, induced by the standard (order) topology on R. On the other hand, X has a natural order induced by the natural order on R. This order induces an order topology T2 on X. How are Ti and T2 related? Problem 3.23. Let X = {; |n € Z+}U {0} C R. (1) What is the relationship between the subspace topology on X and the order topology on X? (2) Give X the order topology. What are the limit points of X ? 4 J. C. MAYER Problem 3.24. Let X = (0, 1] U {2} C R. On the one hand, X has a subspace topology T1 induced by the standard (order) topology on R. On the other hand, X has a natural order induced by the natural order on R. This order induces an order topology T2 on X. (1) How are Ti and T2 related? (2) Is 1 a limit point of X? (3) Is 2 a limit point of X? Problem 3.25. Redo Problem 3.24 with X = [0, 1) U {2}.
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