Final Exam Solutions
Topology I (Math 5853)
1. Let R be the set R with the topology given by the basis B = cfw_[a, b) | a < b and a, b Q.
Determine the closures of the following sets in R :
(a) A = (0, 2)
(b) B = ( 2, 3)
Solution.
(a) A = [0, 2]. We clai
Math 5853 homework solutions
36. A map f : X Y is called an open map if it takes open sets to open sets, and is called
a closed map if it takes closed sets to closed sets. For example, a continuous bijection is a
homeomorphism if and only if it is a close
Math 5853 homework solutions
57. Let A be a closed subset of a normal space X. Let f : A A X be continuous, where each X is homeomorphic either to R or to a closed interval in R. Prove that f extends to X. For each , the Tietze Extension Theorem gives an
Math 5853 homework
Instructions: All problems should be prepared for presentation at the problem sessions. If a
problem has a due date listed, then it should be written up formally and turned in on the
due date.
32. (10/5) Let a0 , a1 , a2 and b0 , b1 , b
Math 5853 homework solutions
50. Let X be a locally compact Hausdor space. Prove that X has a basis consisting of sets
whose closures are compact.
Let B = cfw_V X | V is open and V is compact. To show that B is a basis, it suces
to check the hypotheses of
Math 5853 homework
Instructions: All problems should be prepared for presentation at the problem sessions. If a
problem has a due date listed, then it should be written up formally and turned in on the
due date.
22. (due 9/21) Let X = (R, L), the reals wi
Math 5853 homework solutions
32. Let a0 , a1 , a2 and b0 , b1 , b2 be two anely independent sets in R2 . Prove that the formula
M ( i ai ) =
i bi in barycentric coordinates denes an ane homeomorphism M , which is the
unique ane homeomorphism taking ai to
Mathematics 5853
Name (please print)
Examination II
November 9, 2004
Instructions: Give brief, clear answers.
I.
(10)
Prove that every compact subset of a Hausdor space is closed.
II.
(10)
Dene what it means to say that a space X is locally compact. Dene
Math 5853 homework
Instructions: All problems should be prepared for presentation at the problem sessions. If a
problem has a due date listed, then it should be written up formally and turned in on the
due date.
1. (due 8/31) Dene f : R R by f (x) = 0 if
Mathematics 5853 Examination I September 28, 2004 Instructions: Give brief, clear answers.
Name (please print)
I. Let X = R and let T = cfw_U X | M R, (M, ) U cfw_ (where (M, ) means cfw_r R | r > M ). (10) 1. Prove that T is a topology on X (you do not
Mathematics 5853
Name (please print)
Final Examination
December 13, 2004
Instructions: Give brief, clear answers.
I.
(10)
Let f : X Y be a function between topological spaces. Prove that f is continuous if and only if for
every x X and every open neighbor
Mathematics 5853
Name (please print)
Examination I
September 28, 2004
Instructions: Give brief, clear answers.
I.
Let X = R and let T = cfw_U X | M R, (M, ) U cfw_ (where (M, ) means cfw_r R | r > M ).
(10)
1. Prove that T is a topology on X (you do not
Math 5853 homework solutions 53. Let (X, d) be a metric space. Dene d : X X R by d(x, y) = d(x, y) when d(x, y) 1 and d(x, y) = 1 when d(x, y) 1. 1. Prove that d is a metric on X. First, we have d(x, y) = 0 if and only if d(x, y) = 0 if and only if x = y.
Exam II
Topology (Math 5853)
October 25, 2005
1. Let X be the set of real numbers with the nite complement topology (complements of nite
sets are open).
(a) Does X satisfy the T1 axiom?
(b) Is X Hausdor?
(c) To what point or points does the sequence x n =
Exam I
Topology (Math 5853)
October 14, 2013
1(a) State the axioms for B to be a basis.
(b) Dene the topology T generated by B.
(c) Suppose B1 and B2 are bases generating the topologies T1 and T2 respectively on a set X.
State a necessary and sucient crit
Exam II
Topology (Math 5853)
November 20, 2013
Choose four problems. If you do ve, please say which four you want graded.
1(a) Show that every closed subspace of a compact space is compact.
(b) Show that every compact Hausdor space is regular.
2. Let p: X
Exam I Solutions
Topology (Math 5853)
1(a) State the axioms for B to be a basis.
(b) Dene the topology T generated by B.
(c) Suppose B1 and B2 are bases generating the topologies T1 and T2 respectively on a set X.
State a necessary and sucient criterion i
Exam II solutions
Topology (Math 5853)
Choose four problems. If you do ve, please say which four you want graded.
1(a) Show that every closed subspace of a compact space is compact.
(b) Show that every compact Hausdor space is regular.
Solution.
(a) Let Y
Exam III
Topology (Math 5853)
November 22, 2005
1. Let X be the quotient space obtained from C cfw_0, 1 by identifying z 0 with z 1 for every
complex number z with |z| > 1.
(a) Does X satisfy the T1 axiom? Why or why not?
(b) Is X Hausdor? Why or why not?
Final Exam
Topology (Math 5853)
December 15, 2005
1. (a) Show that every closed subspace of a compact space is compact.
(b) Show that every compact Hausdorff space is regular.
(c) Show that every compact Hausdorff space is normal.
2. Let r: S1 >
Final Exam
Topology I (Math 5853)
December 11, 2013
1. Let R be the set R with the topology given by the basis B = cfw_[a, b) | a < b and a, b Q.
Determine the closures of the following sets in R :
(a) A = (0, 2)
(b) B = ( 2, 3)
2. Prove the Tube Lemma: C
Math 5853 homework
Instructions: All problems should be prepared for presentation at the problem sessions. If a
problem has a due date listed, then it should be written up formally and turned in on the
due date.
9. (due 9/7) Prove that R with the lower li
Math 5853 homework
Instructions: All problems should be prepared for presentation at the problem sessions. If a
problem has a due date listed, then it should be written up formally and turned in on the
due date.
29. Verify the following facts about the is
Exam I
Topology (Math 5853)
September 27, 2005
1(a) State the axioms for B to be a basis.
(b) Dene the topology T generated by B.
(c) Show that if B is a basis for a topology on X, then the topology generated by B equals the
intersection of all topologies