Solve problem 3.21 in detail please 3.2. Order Topology.Definition 3.15. Let X be a nondegenerate set with linear order <. The order topology onX is the topology generated by the subbasis S consisting of all positive and negative openrays; that is,S = {(-∞, b) | b E X}U{(a, ∞) | a E X}.Exercise 3.16. Let X be a nondegenerate ordered set with the order topology. Show thefollowing: (1) every open interval is open, (2) every closed interval is closed, (3) every openray 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 theorder topology generated by the subbasis S. Show that B\ S is also a basis for the ordertopology on X.Problem 3.18. Suppose X =basis B for the order topology generated by the subbasis S. Is B\S alsotopology on X ?[a, w] has both a least and a greatest element. Describe thebasis for the orderProblem 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 inProblem 3.17 is called the standard basis for the standard topology on R. For the unitinterval (0, 1] with the natural order < (inherited from the order on R), the basis describedin Problem 3.18 is called the standard basis for the standard topology on [0, 1].Problem 3.21. Give R the standard (order) topology. Show that every point of R is a limitpoint of R. Is every point of R a sequential limit point?Problem 3.22. Let X = [0, 1] CR. On the one hand, X has a subspace topology T, inducedby the standard (order) topology on R. On the other hand, X has a natural order inducedby the natural order on R. This order induces an order topology T2 on X. How are T1 andT2 related?Problem 3.23. Let X = {; | n E Z+}U {0}C R. (1) What is the relationship betweenthe subspace topology on X and the order topology on X? (2) Give X the order topology.What are the limit points of X?

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Description: Solve problem 3.21 in detail please 3.2. Order Topology.Definition 3.15. Let X be a nondegenerate set with linear order <. The order topology onX is the topology generated by the subbasis S consisting of all positive and negative openrays; that is,S

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Problem 3.21. Give R the standard (order) topology. Show that every point of R is a limit point of R. Is every point of R a sequential limit point?

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Chapter2: Second-order Linear Odes

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Solve problem 3.21 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 = {(-∞, 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 =
basis B for the order topology generated by the subbasis S. Is B\S also
topology on X ?
[a, w] has both a least and a greatest element. Describe the
basis for the order
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 3.21. Give R the standard (order) topology. Show that every point of R is a limit
point of R. Is every point of R a sequential limit point?
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 T1 and
T2 related?
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?

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