All course information will be posted on the course webpage (not on Blackboard). Blackboard will be used for grades only.

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.

The final exam will be held on Tuesday, December 13, 11:15am -- 1:45pm, in our usual classroom Earth & Space 079.

The final exam is cumulative. "Learning goals" posted for each week give a summary of the course material that you are responsible for.

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).

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.

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.

Week 5 (09/19 – 9/23) 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.

Week 6 (09/26 – 9/30)

Week 7 (10/3 – 10/7) Read section 4.2 and section 6.1 up to Theorem 6.6 (including this theorem).

Learning Goals for Week 7:
Homeomorphisms: Know the definition. Be able to show that being homeomorphic is an equivalence relation on topological spaces. Be able to construct simple homeomorphisms (eg to show that an open interval is homeomorphic to the real line). In simple examples only, be able to prove that two spaces are not homeomorphic (at this point, we do not have a lot of tools).
Connected topological spaces: know the definition (several equivalent versions). Be able to check if a space is connected (in simple examples). Connected subsets (in subspace topology). Theorem: if f is a continuous function on X and X is connected, then f(X) is connected; know statement and proof. Theorem: a subset A of the real line is disconnected if A contains two points p and q but does not contain a point r between p and q.

Week 8 (10/10 – 10/14) Read section 6.1 at least through Thm 6.9 (reading the entire section is recommended but optional). Read Theorem 6.17 and example 6.11 in section 6.2 (connectedness of the interval). Read section 6.3 (at least up to example 6.20; the rest optional). Optional reading: section 6.4.

Learning Goals for Week 8: Intermediate value theorem for general connected spaces: know statement and proof. Theorem: the real line is connected (understand proof). Other connected spaces: intervals, rays, R^n, circle, spheres, etc. Use connectedness as a tool to prove that two spaces are not homeomorphic (eg a circle and an interval; (a,b) and [a,b]; other examples).

Week 9 (10/17 – 10/21) Read section 7.1 through Thm 7.7 (skip 7.6). Recommended but optional: 7.9, 7.10 in 7.1. Read section 7.2 through example 7.7.

Learning Goals for Week 9: Compactness: know definition, be able to work with open covers to determine if a given topological space is compact. Theorem: a closed subset of a compact space is compact (know statement and proof). Theorem: a closed interval in R is compact. A subset in Rⁿ is compact if and only if it is closed and bounded. (You should know this statement very well, be able to use it, and understand proof). Theorem: if X is compact, and f is a continuous function on X, then f(X) is compact (know statement and proof).

There will be a quiz on compactness on Thursday, Oct 27.

Week 10 (10/21 – 10/28) Read section 5.1 and Theorems 5.12 and 5.13 in section 5.2 The rest of 5.2 is optional but recommended. Read the second of section 1.3 (about Hausdorff spaces).

Learning Goals for Week 9: Compactness: extreme value theorem for compact spaces (know statement and applications, undertsand proof). Metric spaces: know definition/axioms, be able to work with the triangle inequality; metric topology. Theorem: a compact set in a metric space must be closed and bounded; proofs are the same as in R^n. (caution: converse is not true). Continuity in metric spaces (epsilon-delta definition). Hausdorff spaces: know the definition, be able to determine if a topological space is Hausdorff. Theorem: metric spaces are Hausdorff. Be able to prove simple properties of Hausdorff spaces.

Week 11 (10/30 – 11/4) Midterm 2 will be held on Thursday, Nov 3. It will focus on the material of weeks 7-10.
On Tuesday, Nov 1, we will discuss further properties of Hausdorff spaces: compact sets in Hausdorff spaces are always closed (Theorem 7.8); in a Hausdorff space, any compact set K and a point a have disjoint neighborhoods. We will also discuss convergence of sequences briefly: in Hausdorff spaces, limits are unique (Theorem 2.12); this is not true in general. In compact metric spaces, every sequence has a convergent subsequence (Theorem 7.16). The Tuesday topics will not be on the test but they will reinforce the previous material.

Week 12 (11/7 – 11/11) Read section 3.3 (Quotient Topology) and Section 13.1 (Graphs) up to Theorem 13.9. Also, read Section 14.1 (Manifolds) up to Definition 14.4.

Lecture notes on Graphs

Learning Goals for Week 12: understand the definition of quotient topology both informally ("gluing") and formally via preimages, be able to describe topology on quotient statements in simple examples. Understand the notions of graphs and manifolds; be able to prove simple statements about graphs, arguing with vertices and edges.

Week 13 (11/14 – 11/18) Read section 3.4 (More examples of quotient topology) and Sections 14.1 and 14.2. . This completes our course material for the semester (some of it will be discussed in the remaining lectures, and some parts, like triangulations, may be omitted).

Learning Goals for Week 13: understand the notion of a surface and be able to check whether a given space is a surface. Know statement of the classification theorem for surfaces. Be able to compute Euler characteristic of a given surface.

• 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.

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.