Tips for not losing points on homework:

• Write your answers in complete sentences.
• When in doubt, explain more, not less. It is a good idea to target your answer towards someone who knows less about the problem than you, not more.

There will occasionally be challenge problems on homework. These problems are not required and will not count toward your grade. They are there just to give you something extra to think about if the homework does not provide enough of a challenge.

• Homework 12, due Wed. Dec 5:
  • 1) Problem 6.6
  • 2) Problem 6.9(b)
  • 3) Problem 6.18(a,b,d)
  • 4) Problem 6.25
  • 5) Show that if f:X->Y is a continuous bijection, X is compact, and Y is Hausdorff, then f is automatically a homeomorphism.
  • Challenge Problem: Problem 7.8
• Homework 11, due Wed. Nov 28:
  • 1) Problem 5.1
  • 2) Problem 5.5
  • 3) Problem 5.14
  • 4) Problem 5.30
  • 5) Problem 5.34
  • Challenge Problem: Let X and Y be metric spaces. Suppose that for every convergent sequence x_n in X, the sequence f(x_n) in Y also converges. Prove that f is continuous.
• Homework 10, due Fri. Nov 9:
  • 1) Problem 3.30.
  • 2) Problem 3.33(c,d,e,f,g).
  • 3) Problem 3.36(a).
  • 4) Problem 3.37.
  • Challenge Problem: 3.38.
• Homework 9, due Wed. Oct 31:
  • 1) Problem 3.15.
  • 2) Problem 3.20.
  • 3) Problem 3.22.
  • 4) Problem 3.25.
  • 5) Problem 3.27.
  • Challenge Problem: 3.28.
• Homework 8, due Wed. Oct 24:
  • 1) Problem 4.24 from the text.
  • 2) Problem 4.29 from the text.
  • 3) Problem 4.32(b) from the text.
  • 4) Problem 3.12 from the text.
  • 5) Problem 3.13 from the text.
• Homework 7, due Fri. Oct 19: (Optional)
  Exam 1 correction instructions:
  • 1) Submit original exam
  • 2) Write complete, detailed solutions on any (numbered part of a) problem on which you would like some points back.
  • 3) For True/False problems, justification is now required.
  • 4) On Problem 3, justification is now required. Only include items marked wrong.
• Homework 6, due Wed. Oct 3:
  • 1) Problem 3.4 from the text.
  • 2) Problem 3.5 from the text.
  • 3) Problem 3.6 from the text.
  • 4) Problem 4.1 from the text.
  • 5) Problem 4.5(a) from the text.
  • Challenge Problem: Problem 3.11.
• Homework 5, due Wed. Sept 26:
  • 1) Problem 2.27 from the text.
  • 2) Show that the alternating sequence x_n = (-1)ⁿ does not converge in the cofinite topology on R.
  • 3) Show that the sequence 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ... converges to 1 (and only to 1) in the cofinite topology on R.
  • 4) Problem 2.13 from the text (do parts (a)-(d)).
  • 5) Problem 2.15 from the text.
  • Challenge Problem: Parts (h) and (i) of 2.13. Problem 2.18 is also a goodie.
• Homework 4, due Wed. Sept 19:
  • 1) Problem 2.5 from the text.
  • 2) Problem 2.9 from the text (only the statement about the closure, not the one about the interior).
  • 3) Problem 2.10 from the text.
  • 4) Problem 2.12 from the text.
  • 5) Problem 2.24 from the text (do parts (a)-(d)).
  • Challenge Problem: Parts (h) and (i) of 2.24.
• Homework 3, due Wed. Sept 12:
  • 1) Problem 1.15 from the text.
  • 2) Problem 1.16 from the text.
  • 3) Problem 1.25 from the text.
  • 4) Problem 1.26 from the text.
  • Challenge Problem: Show that the arithmetic progression topology is Hausdorff.
• Homework 2, due Fri. Sept 7: (Office hours for the week of Labor Day will move to Wednesday)
  • 1) Problem 1.2 from the text.
  • 2) Problem 1.3 from the text.
  • 3) Problem 1.6 from the text.
  • 4) Problem 1.8 from the text.
  • 5) Problem 1.9 from the text.
  • Challenge Problem: Find all 29 topologies on the set X={a,b,c}.
• Homework 1, due Wed. Aug. 29:
  • 1) Show that [0,1] is not open in R
  • 2) Prove the first of de Morgan's laws (Theorem 0.9), that
    A - (B∪C) = (A-B)∩(A-C)
  • 3) Consider the set
    X = { 1/n | nε Z₊}.
    Is X open in R? Is X closed?
  • 4) Show that the unit square
    { x ε R² | 0 < x₁ < 1 and 0 < x₂ < 1 }
    is open in R².
  • Challenge Problem: The empty set ∅ and the set R are both open and closed subsets of R. Are there any others? Why/why not?