**Instructor:**Olga Plamenevskaya, office 2-112 Math Tower, e-mail:`olga@math.stonybrook.edu`**Office hours:**Tuesday 1-3pm in Math 2-112, or by appointment. A Zoom office hour is tentatively scheduled for Wednesdays 2-3pm, but please drop me an email (by noon Wednesday) if you intend to connect to my Zoom. You are also welcome to make an appointment for Zoom at other times.**Class meetings:**TuTh, 3:00pm-4:20pm, Earth & Space 079.**Course webpage:**http://www.math.stonybrook.edu/~olga/mat364-fall22/.All course information will be posted on the course webpage (not on Blackboard). Blackboard will be used for grades only.

**Grader:**Mohamad Rabah, email:`mohamad.rabah@stonybrook.edu`, office hours: here.**Textbook:**Colin Adams, Robert Franzosa,*Introduction to Topology: Pure and Applied*.This is a required text. We will not cover all of it, and sometimes we'll be doing things in a different order, but it discusses key notions in a lot of detail and offers more examples that we can cover in class. You should read this book when you are doing homework or preparing for exams and quizzes.

**Exams:**there will be two midterm exams and a final exam.**Midterm 1 will be held on Thursday, Sept 29.****Homework:**weekly homework and reading assignments will be posted on this page.**You are responsible for the MAT 200 material on sets and functions. This includes using terminology and notation correctly.**The notation used in the textbook (such as unions and intersections of sets, the empty set, images, preimages, composition of functions, etc) is the standard notation in mathematics. The same notation is used in MAT 200 and in other math courses at Stony Brook. Some of you may have seen different notation elsewhere (for example, in courses on computer science); in this course, you must understand and use the standard notation accepted in mathematics.**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. Although you are welcome to work with others to understand how to solve the problems, you have to write all the solutions on your own, in your own words. (See Academic Integrity Statement at the bottom of the page.) You are not allowed to look for solutions online. A major part of the homework and quizzes portion of your grade will depend on the homework-based in-person quizzes.**Reading assignments**for each week are to be taken very seriously -- even if you understood the material in class, being able to follow a proof in a book is an important skill. You will often be required to "read ahead" on some material, both to prepare for class and to sharpen your math reading skills.**Week 1**(08/22 – 08/26)**Read section 1.1. Also read section 0.1 for an intuitive introduction.****Review the MAT 200 material (sets and functions).**You can use sections 0.3 and 0.5 in the textbook (except sequences at the end of 0.5) or any other sources such as your notes from MAT 200. You should be able to perform operations with sets correctly, quickly, and confidently, and be able to give proofs of simple statements about sets and functions.

**Learning Goals for Week 1:**Know the definition of a topological space and of a topology on a given set, as well as the notion of open sets (in a given topology). Be able to determine whether a given collection of subsets of a given set forms a topology. Given a topology on X, determine whether a given subset is open. Know the definitions of the trivial topology and the discrete topology on a set. Understand the notion of comparison of topologies; given two topologies on X, be able to determine whether one of them is finer/coarser than the other (or not comparable).There will be a

**quiz**on MAT 200 prerequisites (sets and functions) on**Tuesday, Aug 30.**The quiz will cover

- basic properties and operations with sets (such as distributive laws, DeMorgan laws);

- functions between sets, their composition, inverse functions, bijective, surjective, injective functions;

- taking an image f(A) or an inverse image (aka preimage) f^{ -1}(B) for a function f: X → Y, A ⊂ X, B ⊂ Y; properties of images and preimages (such as taking unions and intersections)

The quiz will *not* include any of the new material on topological spaces.**Homework 1**due Thursday, Sept 1, in class.**Week 2**(08/29 – 09/02)**Read section 4.1**(skip Theorem 4.3, Examples 4.5 and 4.6, Theorems 4.5 and 4.8, and Lemma 4.10). We'll come back to this skipped material as well as Chapters 1-3, but it is useful to get an idea of continuous functions early on.**Learning Goals for Week 2:**Know the definition of a continuous function between topological spaces. Be able to determine if a given function is continuous (in simple examples). Know and be able to prove basic properties (such as: composition of continuous functions is continuous).**Homework 2**due Thursday, Sept 8, in class.**Week 3**(09/6 – 9/9)**Read section 1.2; in section 4, read Theorem 4.3 and Examples 4.5 and 4.6.****Learning Goals for Week 3:**Know the definition of a basis and of the topology generated by the given basis. Be able to determine whether a given collection of subsets of X forms a basis; be able to determine whether a given subset V is open in the topology generated by the basis. Be able to determine if given function is continuous by inspecting preimages of the basis sets on the target (Theorem 4.3).

Caution: to be open, V doesn't have to belong to the basis! A set is open if it can be represented as a (possibly infinite) union of sets from the basis. Theorem 1.9 is often very helpful (easier to apply than finding unions of sets).

We have briefly discussed homeomorphisms and homeomorphic spaces (spaces that are "the same" from the topology viewpoint), section 4.2. This material will be revisited later; until then, homeomorphisms will not appear on any quizzes.**Homework 3**due Thursday, Sept 15, in class.There will be a

**quiz**in class on**Tuesday, Sept 13.**The quiz covers the material of Weeks 1 and 2; the emphasis will be on continuous functions.

**Week 4**(09/12 – 9/16)**Read section 1.3 (up to definition 1.18); read section 2.1****Learning Goals for Week 4:**Know the definitions of closed sets and of interior and closure. Be able to work with definitions to prove their properties. In a given topological space (standard topologies on R and R^{2}, as well as other topological spaces), be able to determine whether a given set is closed, and to find interior and closure of a given set.**Homework 4**due Thursday, Sept 22, in class.**Week 5**(09/12 – 9/16)**Read section 2.3; read section 3.1****Learning Goals for Week 5:**Know the definition of the boundary of a subset in a topological space, be able to use this definition to prove properties of the boundary and to find boundary of a set in simple examples. (Theorems: boundary of a set is always closed; boundary is empty if and only if the set is both open and closed.) Know the definition of subspace topology, be able to check that it is a topology, determine open and closed sets in subspace topology, work with continuous functions in subspace topology.**Homework 5**due Thursday, Sept 29, in class.**Midterm 1 will be held on Thursday, Sept 29.**The midterm covers sections 1.1, 1.2, 1.3, 2.1, 2.3, 3.1, 4.1. Topics omitted from class discussion so far (such as Hausdorff spaces and homeomorphisms) are not included. Refer to "learning goals" for each week for a detailed description of the material. You may be asked to state definitions and some of the key facts, and prove some of the statements that we discussed in class. You are not required to memorize a zillion properties of the interior, closure, and boundary given in the textbook, but may be required to prove similar simple statements about these sets.**Course material:**the course will cover a range of topics, in particular- Intuitive idea of topology
- Abstract point-set topology, open and closed sets; continuous functions
- Point-set topology in the Euclidean space; relation to analysis
- Topological properties (compactness, connectedness)
- Surfaces: definition, examples
- Cut-and-paste arguments for surfaces
- Idea of classification (with some proofs if time permits)
- Euler characteristic

More specific descriptions of topics, skills, and learning goals will be posted on this page for each week as the course progresses.

As this is an upper-level class, familiarity with proofs is expected. You will have to write careful proofs in your homework and tests.

**Grading Policy:**The course grade will be determined from (a) homework and quizzes, 25%; (b) two midterms, 20% each; (c) final exam, 30%; (d) class participation, 5%.The above grading policy assumes that the course is given in the in-person format. If there are any changes in the COVID-related University policies and class format, the instructor reserves the right to make the necessary adjustments to the grading policy.

No make-up exams will be given for midterms. If a student misses a midterm exam for a well-documented medical reason or other similar circumstances beyond the student's control, the student may be excused from the exam, with the final grade determined from the other exams, homework, and class participation. The same policy applies to missed quizzes and missed homework, but all absences (including Covid-related) must be properly documented to qualify for an exemption. For the final exam, make-ups will be given ONLY in cases of properly documented medical reasons or other similar circumstances, at the instructor's discretion.

**Academic Integrity Statement**Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology and Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.**Critical Incident Management**Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.**Student Accessibility Support Center Statement**If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.