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?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2Fe17480b2-076f-4b7e-800e-eddc8b846f0f%2F49oh66_processed.jpeg&w=3840&q=75)
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}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2Fe17480b2-076f-4b7e-800e-eddc8b846f0f%2Fvcyp6a_processed.jpeg&w=3840&q=75)
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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