Assignments
Weekly assignments are due Wednesdays at 11am. Occasionally, as on the first day of class, you will be given one problem due at the start of the next class.
Diagostic exercises are indicated by DX; these should not be submitted. All other exercises require written solutions. Check the errata for exercises marked *.
Assignment 1, due F., 9/2 [Do not discuss with others.]
Assignment 2, due W., 9/7
DX: 1.1, 1.3, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.15
Required: 1.2, 1.4, 1.5, 1.12, 1.13 (1.12, 1.13 delayed until Asst 3)
Required: Prove that the interval [0,1) is uncountable using Cantor's diagonalization argument. (Do not look this up, unless you've tried it for a long time!)
Assignment 3, due W., 9/14
DX: 1.17, 1.25, 1.28, 1.29, 1.30, 1.31
Required: 1.12, 1.13, 1.14, 1.16, 1.19, 1.20 (defining a subbasis), 1.21, 1.27, 1.34*, 1.35
Required: Show that the separation axioms obey: T4 implies T3 implies T2.
Assignment 4, due W., 9/21
DX: 1.38, 2.4, 2.5, 2.6, 2.7, 2.8
Required: 1.22, 1.34*, 1.35, 1.39, 1.44, 2.1, 2.3, 2.10
Required: Show that the separation axioms obey: T4 implies T3 implies T2.
Assignment 5, due F., 9/23. We will present the proof of Theorem 2.15 (exercise 2.28) in class. There are 12 parts, so most of you will be randomly chosen to present. If you present, your grade is based upon that. If you don't, you will submit 3-4 parts in class to be graded.
Assignment 6, due W., 9/28
DX: 2.26, 2.27, 2.31
DX: Let T and T' are two topologies on X, with T coarser than T'. Show, for A a subset of X, that the interior of A under T is contained in the interior of A under T'. Similarly show that the closure of A under T contains the closure of A under T'.'
Required: 2.13, 2.14*, 2.18, 2.19*, 2.23, 2.24,
Challenge Problem: (Munkres 17.21) Consider the power set P(X) of X. The operations
closure and complement may be viewed as maps P(X)->P(X).
(a) Show that, starting from a given set A, one can form no more than 14 distinct sets by applying
these operations.
(b) Find a subset of the reals for which the maximum of 14 distinct sets is achieved.
Assignment 7, due W., 10/12
DX: 3.1-3.6, 3.9, 3.12, 3.13, 3.14, 3.18, 3.20, 3.23, 3.25, 3.28
Required: 2.29, 2.35, 3.7, 3.15, 3.17, 3.22, 3.26, 3.29, 3.30, 3.33
For 2.29, assume all intervals [c,d] have positive length, i.e., c≠d.
Assignment 8, due W., 10/19
DX: 3.40, 4.1, 4.2, 4.7
Required: 3.44, 4.3, 4.4, 4.5, 4.10, 4.12, 4.18, 4.19, 4.20, 4.21
Challenge problem: Describe the configuration space C3(S1). What familiar space is it equivalent to?
Hint (3.44): The bonding angle of a water molecule is constant, roughly 104.5 degrees.
Assignment 9, due M., 10/26
Assignment 10, due F., 11/11
DX: 4.38, 5.1, 5.2, 5.4, 5.5, 5.6, 5.10, 5.13, 5.16, 5.22, 5.26, 5.27, 5.30, 5.33, 6.1, 6.2, 6.5
Required: 5.3, 5.12, 5.14*, 5.24, 5.29**, 5.31, 5.35, 5.37, 5.41, 6.7b, 6.9, 6.18, 6.20, 6.27, 6.30
Problem [replacing 5.29b]. Show that for all c1>0, for all c2 satisfying 0< c2 ≤ c1(b-a), there exists f in C[a,b] s.t. ρM(f,0)=c1 and ρ(f,0)=c2.
Required: Read section 4.3. What is "gimbal lock"? How did NASA encounter and address it 40-50 years ago?
Attend either the colloquium on 11/10 and write a 1-page reaction, or do problem 6.24.
Assignment 11, due W., 11/16
Assignment 12, due Tu., 11/22
DX: 7.1, 7.2, 7.3, 7.6, 7.9
Required: 6.41, 6.43, 6.44, 6.45, 7.5 (you must use open covers), 7.11
Assignment 13, due Tu., 12/6
Assignment 14, due F., 12/9