**Instructor:**Olga Plamenevskaya, office 3-107 Math Tower, e-mail:`olga@math.sunysb.edu`**Office hours:**Monday 12:45-2:15 in 3-107, Wednesday 12:45-2:15 in P-143, or by appointment.**Class meetings:**MWF, 11:45am-12:40pm, Physics P116.**Grader:**Anant Atyam, office hours Monday 4-5pm in

The course will cover a range of topics. We will look at some weird pictures and play with the famous Mobius band and Klein bottle. We'll also make a connection to analysis classes and work with limits and continuous functions. We'll finally learn what those epsilon and delta really mean!As this is an upper-level class, familiarity with proofs is expected. Indeed, you will have to write careful proofs in your homework.

- L. Christine Kinsey,
*Topology of surfaces*.

** Final Exam: Wednesday, Dec 15, 5:15-7:45pm in P-116. (Our usual room.)
**
The final exam is cumulative and covers everything we studied during semester.

** Midterm I: Wednesday, Oct 13, in class. **

** Midterm II: Friday, Nov 19, in class. ****
Note change of date!! ** Checklist of topics for exam II is here. The exam focuses on material covered since the first midterm;
although you will not be tested on the Exam I material, you are still responsible for basic concepts
learned earlier in the semester.

The first exam covers the material we've learned so far. (including things that are not in the textbook.) A checklist of topics is here.

** Important: ** For each homework problem,
please give a proof or detailed explanation as appropriate (unless otherwise stated). Please
write up your solutions neatly, be sure to put your name on the first page and staple all pages.
Illegible homework will not be graded.
You are welcome to collaborate with others and to consult books,
but your solutions should be written up in your own words,
and all your
collaborators and sources should be listed.

** Week 1 ** (08/30 – 09/03) sections 1.1, 2.1 (up to Definition 2.10).

** Homework 1, due Sept 8: ** please do the following exercises from
the book.

1.1 (use your intuition about continuous deformations; no explanations required).

2.1 and 2.3 for the sets E, F and G only (i.e. parts 4,5 and 6).
In addition, find isolated points for each of these sets.
Explain your answers.

2.6, 2.9, 2.11 (give proofs). Some solutions

Also, determine which of the following identities are true:

(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)

(A ∩ B) ∪ C = (A ∩ C) ∪ (B ∩ C)

(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C). Give proofs or counterexamples to support your answer.

** Week 2 ** (09/08) section 2.1 (more on open and closed sets)

** Homework 2, due Sept 15: ** please do the following exercises
from
the book.

2.7, 2.8, 2.12, 2.15, 2.16. Prove everything. Some solutions

** Week 3 ** (09/14 – 09/17) sections 2.2, 2.3

** Homework 3, due Sept 22: ** Do the following exercises
from
the book, as well as two additional questions below. Prove everything.

2.19, 2.20, 2.22, 2.25, 2.26 ab. Some solutions

-- Give an example of sets A, B, such that B consists of a single point, B ⊂ A ⊂ R^{2}, B ≠ A,
and B is both open and closed relative to A.

-- Which of the following are true? Here, f : X → Y is a function, A, B subsets of X, C, D subsets of Y,
and f^{ -1} (C) stands for pre-image of the set (not for the inverse function, which might not exist.)

f (A ∩ B) = f (A) ∩ f (B)

f^{ -1} (C ∩ D) = f^{ -1}( C) ∩ f^{ -1} (D)

f^{ -1} (C ∪ D) = f^{ -1} (C) ∪ f^{ -1} (D). Prove or give counterexamples to support your answers.

** Week 4 ** (09/20 – 09/24) sections 2.3, 2.5

** Homework 4, due Sept 29: ** pdf Some solutions

** Week 5 ** (09/27 – 09/31) sections 2.5, 2.6

** Homework 5, due Oct 6: ** pdf Solutions

** Week 6 ** (10/3 – 10/8) section 2.4 (+extra material)

** Homework 6, due Oct 13: ** pdf Solutions

** Week 8 ** (10/18 – 10/22) section 3.2, part of section
3.1 (we took Thm 3.5 as a definition of topology)

** Homework 7, due Oct 27: ** pdf Solutions

** Week 9 ** (10/25 – 10/29) Metric space and their topology

** Homework 8, due Nov 3: ** pdf Solutions

** Week 10 ** (11/1 – 11/5) Surfaces: 4.1, start 4.3

** Homework 9, due Nov 10: ** pdf Solutions

** Week 11 ** (11/8 – 11/12) Classification of surfaces:
4.3,
4.5

** Homework 10, due Nov 17: ** pdf Solutions

Exam on Nov 19

** Week 12 ** (11/21 – 11/23) Idea of the Euler
characteristic (Chapter 5). Platonic solids (pp.105-107)

** Week 13 ** (11/29 – 12/3) The Euler
characteristic, topological invariance (5.2-5.4)

** Homework 11, due Friday, Dec 3 : ** pdf

** Homework 12, due Friday, Dec 10 : ** pdf

**Students with Disabilities: **If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 632-6748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.