Math 440: Topology, Fall 2022

MWF 11:00-11:50am in GFS 223

Teaching Staff

Instructor Joj Helfer
email jhelfer (at) usc (dot) edu
Office Hours After class in KAP 464-B

You may be looking for the course notes, the problem sets, the lecture plan, the practice exam, or the project assignment.

Announcements +open all -close all

Course announcements made on Blackboard will appear here as well.

Finals graded Thursday, Dec. 1

Hi all,

The finals have been graded and can be viewed on Gradescope. The solutions have been posted on the course webpage.

If you have any questions or concerns about your graded final, please let me know as soon as possible.

Thanks again for a very enjoyable semester.


Final on December 7 at 11am-1pm Thursday, Dec. 1

Hello everyone,

Here is some information regarding the final. If you have any questions or concerns, please let me know as soon as possible.

  1. As indicated on the final will be on Wednesday, December 7 at 11am-1pm in our usual classroom GFS 223. It will be two hours long.
  2. No materials besides pen and paper (notes, books, calculators, phones, etc.) will be permitted during the exam.
  3. On the exam, you will be allowed to use any results that were proven in class, or that were stated in homework problems.
  4. A practice final has been posted on the course webpage, with solutions. Try to do the problems yourself before looking at the solutions.
  5. There will be a review session on Monday.
  6. The examinable topics include everything covered in class up to and including Friday, November 18. On these topics, you are expected to be able to solve all of the problems which were given in homework assignments. In addition, you should know the definitions of (at least) the following concepts and statements of the following theorems, in addition to the list of things that was given in the announcement for the midterm (you are also advised to consult the course notes!):
    • Convergent sequence. (And the criterion in metric spaces for continuity, openness, and closedness and terms of seqeuences.)
    • Hausdorff space.
    • Limit point.
    • Connectedness and connected component. (And that the union of connected spaces with a point on common is connected.)
    • Path connectedness and path components.
    • Compactness (And that the image of a compact set under a continuous map is compact, that a closed subspace of a compact space is compact, and that a compact subspace of a Hausdorff space is closed, and the criterion for compactness in Rn.)
    • Quotient topology and quotient maps. (And the criterion for continuous maps out of a quotient.)
    • Embeddings.
    • The disjoint union topology.
    • The compact-open topology. (And the criterion for continuity of a map into the space of functions with the compact-open topology.)
    • Locally compact Hausdorff space. (And that the evaluation map on the space of functions from X to Y with the compact-open topology is continuous when X is locally compact Hausdorff.)


Project draft due today Monday, Nov. 21

Hi all,

I may have misspoken in class about the due date of the first draft for the project.

As stated in the project assignment, the first draft is due today, November, 21st.


Correction in HW 10 Friday, Nov. 4

Hello everyone,

There was a missing assumption in Problem 2 of Homework 10: that each initial segment of the total order is countable. The problem set has been corrected.


Project assignment Wednesday, Nov. 2

Hello all,

The project assignment has been posted on the webpage.

Please take a look at it and start thinking about which topic you would like to choose. You should send me your chosen topic (and your group) by the end of the day on Friday (November 4).


Errors in HW8 Friday, Oct. 21

Hi all,

There were some typos in HW8. They have been corrected. Please use the new version.


Midterm on Wednesday, October 12 and remote class on Monday, October 10 Wednesday, Oct. 5

Hello everyone,

On Monday, October 10, the lecture will be given remotely over Zoom. Attendance is mandatory, as usual. You will be able to find the link for the Zoom meeting on the Blackboard page for the course.

Here is some information on the midterm, which will in class be on Wednesday, October 12. If you have any futher questions, please let me know.

  1. The exam will take place during class (i.e., from 11am to 11:50am in GFS 223) and will be 50 minutes long. It will start right at the beginning of class, so make sure you are there on time. If for any reason you don’t think you will be able to make it on time, please let me know in advance.
  2. A practice exam will be posted on the course webpage, with solutions.
  3. This will a closed book, closed notes exam, with no calculators or other electronic aids permitted.
  4. If you are having trouble studying any of the topics for the exam, please (ask your peers for help and) come ask me about it in office hours.
  5. There will be no homework due Wednesday, October 12. Homework 6 is due on October 5, and Homework 7 will be assigned on October 12 and due October 19.
  6. The examinable topics include everything covered in class up to and including Wednesday, October 5. On these topics, you are expected to be able to solve all of the problems which were given in homework assignments. In addition, you should know the definitions of (at least) the following concepts (you are also advised to consult the course notes!):
    • The basic set-theoretic notions: subset, intersection, union, ordered pair, product, relation, function, etc.
    • Equivalence relations and quotients. (And the relationship between equivalence relations and partitions.)
    • Partially and totally ordered sets.
    • The axioms of the real numbers (i.e. of a complete Archimedean ordered field).
    • The product of a family of sets.
    • Finite, countable, and uncountable sets.
    • Metric space and topological space.
    • The metric topology.
    • Continuous maps and homeomorphisms. (And the criterion for continuity of maps bewteen metric spaces.)
    • The subspace topology.
    • Basis and subbasis of a topological space.
    • The product topology. (And the criterion for continuity of maps into a product.)
    • Closure, interior, and boundary.
    • Convergent sequence.


HW 5 is graded Monday, Oct. 3

Hi all,

Homework 5 has been graded and the solutions are posted on the course webpage.


Correction in HW 5 Monday, Sep. 26

Hi everyone,

There were two typos in Problem 6 (iv) of Homework 5. (It said RxR instead of R2xR2, and the variables were mixed around in the definition of the ordering.)

The corrected version is posted on the website.


Correction in HW 3 Monday, Sep. 12

Hello everybody,

A mistake in Problem 1 (h) of Homework 3 was pointed out to me (the assumption a<b was missing).

The problem set on the course webpage has now been corrected. Please use the new version.

Also, please continue letting me know about any mistakes you find in the homework!


Correction in the definition of a field Wednesday, Sep. 7

Hi everyone,

I forgot a very important condition today in the definition of a field. In addition to the conditions I stated, we also require: 1≠0.

Also, I made a small mistake in the definition of complete Archimedean ordered field: I said that any bounded-above subset has a least upper bound, but I should have said that any non-empty bounded-above subset has a least upper bound.

The course notes contain the correct definitions in both cases.

Also, Homework 3 is now posted on the course webpage.


Correction in HW 2 Friday, Sep. 2

Hi everyone,

It was brought to my attention that there was a mistake in Problem 3 of Homework 2 (the definition of the set Y was missing).

A corrected version is now posted on the course webpage. Please use the corrected version.

Also, please continue to alert me as soon as you find any issues with the problem sets.


Welcome! Friday, Aug. 19

Hello everyone and welcome to Math 440! I’m looking forward to meeting you all on Monday.

I encourage you to check out the course website at, though we will be going over the information there together on Monday.

One thing, however, that I would like to draw your attention to, is that classes will be in person and attendance is mandatory.

In particular, if you have any concerns about being able to make it on Monday -- or if there is anything else you would like to ask me about -- please email me at jhelfer (at) usc (dot) edu.


Overview, textbook, and prerequisites

Topology provides the language of modern analysis and geometry. This course is an introduction to point-set topology, which formalizes the notion of a shape (via the notion of a topological space), notions of “closeness” (via open and closed sets, convergent sequences), properties of topological spaces (compactness, completeness, and so on), as well as relations between spaces (via continuous maps). We will also study many examples, and see some applications.

For a list of topics to be covered, please see the tentative lecture plan below.

Textbook and notes

The official course text is:

Topology (2nd edition) by James R. Munkres.

Within this text, we will focus on Part I, particularly Chapters 1-3 (and other portions on an as-needed basis).

There are also

which will be updated regularly.


Math 440 is very proof-oriented and requires a certain level of mathematical sophistication. It is recommended that you take at least one other upper level mathematics course before Math 440 (i.e., a MATH course labeled 400 or above), or otherwise that you have some comfort or familiarity with writing proofs.

Math 440 will emphasize a rigorous, proof-intensive development of topics. There is some overlap with the topics covered in first semester real analysis (Math 425a), particularly with the notion/study of metric spaces, but the overall emphasis is rather different.


Class attendance is mandatory! If you have any scheduling conflicts, or you think you might be unable to make it to any given class, please inform the instructor as soon as possible.

Classes will be in person. Certain classes may be given remotely over Zoom, in which case you will be notified in advance, and you will be able to access the Zoom meeting via the Blackboard page.

This page, Blackboard, Gradescope, etc.

All course information will be available on this webpage.

We will make limited use of other platforms as follows:

Grading scheme

Homework assignments

Homework assignments will be posted here each week.

Homework is to be submitted on Gradescope (you should see the course there if you are enrolled; if you have any problems, please inform the instructor). When you submit on Gradescope, please don’t forget to match your scanned pages with the problems.

Due dates

Homework must be submitted by the posted due dates. If you expect to have issues submitting the homework on time, or if you are having difficulties with gradescope, please write to the instructor as soon as possible, and attach a scanned copy of your submission (this is a wise practice for any class).

Problem sets:

Due date Assignment
Wed, Aug. 31 Homework 1. (Solutions)
Wed, Sep. 7 Homework 2. (Solutions)
Wed, Sep. 14 Homework 3. (Solutions)
Wed, Sep. 21 Homework 4. (Solutions)
Wed, Sep. 28 Homework 5. (Solutions)
Wed, Oct. 5 Homework 6. (Solutions)
Wed, Oct. 19 Homework 7. (Solutions)
Wed, Oct. 26 Homework 8. (Solutions)
Wed, Nov. 2 Homework 9. (Solutions)
Wed, Nov. 9 Homework 10. (Solutions)
Wed, Nov. 16 Homework 11. (Solutions)
Mon, Nov. 28 Homework 12. (Solutions)

Mathematical writing

We are looking not just for valid proofs, but also readable, well explained ones (and indeed, you will be partly graded on readability). This means you should try to use complete sentences, insert explanations, and err on the side of writing out "for all" and "there exist", etc. symbols if there is any chance of confusion.

For further advice on writing your homework (and project paper), see:


Both exams are closed book, closed notes exams, with no calculators or other electronic aids permitted. The final exam will cover all topics from the semester, but will have greater emphasis on topics developed after the midterm.


A portion of your class grade will be based upon a project exploring an aspect of topology beyond the topics covered in class. Concretely, with a small group of 2-3 students, you will be asked to write a short expository article (around 4-6 pages, typed), and give an in-class 20 min presentation. The topic of study will be chosen in consultation with the Instructor.

Here is the project assignment.

Office hours

Office hours are directly after class, and by appointment; i.e., if you want to attend office hours on a given day, send an email beforehand, or just let the instructor know at the end of class.

Students with Disabilities

Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. If required, please make sure that the DSP letter (for approved accomodations) is delivered to me as early in the semester as possible. For more details, see the DSP web site here; in particular contact information is here.

Academic Integrity

The instructor strongly adheres to the University policies regarding principles of academic honesty and academic integrity violations, and will strictly enforce these rules. You are encouraged to review those, for instance in SCampus, the Student Guidebook (see e.g., University Governance, Section 11.00 and Appendix A).


This syllabus is not a contract, and the Instructor reserves the right to make some changes during the semester.

Tentative schedule

Note: this schedule is tentative and will be continuously updated to adapt to the pace of the course. Please check back regualrly for updates.

Week Date Material References
Mon, Aug. 22 Overview of topology. Concept of geometric properties invariant under an equivalence relation. Concept of homeomorphism. Distinction between algebraic/geometric topology and point-set topology. Distinction between “nice point-sets” (manifolds), general point-sets, and “abstract spaces” (metric spaces and topological spaces). Some big names: Cantor, Poincaré, Fréchet. The “historical” texts referenced above.
1 Wed, Aug. 24 Everyone gets their very own topological space. Axioms of set theory. Empty set and sets with specified elements (“{a,b,c}” notation). Basic operations (intersection, union, difference). Some basic properties of the operations. Munkres Ch.1, §1.
Fri, Aug. 26 Power set. Intersection and union of a set of sets. Ordered pairs. Cartesian product. Relations. Functions. Munkres Ch.1, §§1-2.
Mon, Aug. 29 More on functions: restriction, composition, identity function, injections, surjections, bijections, images and preimages. Equivalence relations and quotients. Munkres Ch.1, §§2-3.
2 Wed, Aug. 31 Equivalence relations vs. partitions. Partial and total orderings. The Dedekind-Peano axioms. Munkres Ch.1, §§3-4.
Note that we will treat the naturals and reals in a somewhat different way from Munkres. One reference with a similar approach is Edmund Landau’s Foundations of Analysis.
Fri, Sep. 2 Dedekind systems. The recursion theorem. Addition, multiplication, exponetiation, and ordering on the natural numbers. Any two Dedekind systems are isomorphic. Pure/hereditary sets. Same as above.
Mon, Sep. 5 Labor day, no class!
3 Wed, Sep. 7 The von Neumann natural numbers. Complete Archimedean ordered fields. Beginning of the construction of the real numbers. Same as above.
Fri, Sep. 9 Construction of the real numbers. Uniqueness of the real numbers up to isomorphism. Russell’s paradox. The axiom of restricted comprehension. Same as above.
Mon, Sep. 12 Indexed families of elements and sets. Sequences and tuples. Indexed unions, intersections, products, and disjoint unions. Finite and infinite sets. Countable and uncountable sets. Cardinality. Axiom of choice. Munkres Ch. 1, §§5-11.
4 Wed, Sep. 14 Metric spaces. Euclidean metric. Taxicab metric. Discrete metric. Standard product metric. Open and closed balls. Open and closed sets. Open sets include the whole space, the empty set, and are closed under arbitrary union and finite intersection. Munkres Ch. 2, §20.
Fri, Sep. 16 Continuous maps and homeomorphisms between metric spaces. A map is continuous if and only if the preimage of each open set is open. Munkres Ch. 2, §§18,21.
Mon, Sep. 19 Topological spaces. The metric topology and metrizability. The discrete and indiscrete topologies. Continuous maps and homeomorphisms between topological spaces. The subspace topology. Munkres Ch. 2, §§12,16,18,20.
5 Wed, Sep. 21 Bases for a topology. Munkres Ch. 2, §13.
Fri, Sep. 23 The product topology. Continuity at a point. Munkres Ch. 2, §§15,18.
Mon, Sep. 26 Subbases. The product topology on an arbitrary product. Restricting the domain or codomain of continuous functions. Munkres Ch. 2, §§13,18,19.
6 Wed, Sep. 28 The basic operations on the real numbers are continuous. Constant functions are continuous. The inclusion function from a subspace is continuous. Topological spaces and continuous functions form a category. Functions into a product are continuous if and only if each component is. Munkres Ch. 2, §18.
Fri, Sep. 30 The pasting lemma. Closures, interiors, and boundaries of subsets. Characterization of open and closed sets in terms of boundary. Munkres Ch. 2, §17-18.
Mon, Oct. 3 Topology pageant.
7 Wed, Oct. 5 Convergence of sequences. Characterization of open and closed sets and continuous maps in terms of sequence. Munkres Ch. 2, §17.
Fri, Oct. 7 Separation axioms: Hausdorff, T1-T4 Munkres Ch. 2, §17.
Mon, Oct. 10 Limit points. Connectedness. The union of connected subsets containing a point in common is connected. Connected components. Munkres Ch. 2, §17 and Ch. 3, §§23,25.
8 Wed, Oct. 12 Midterm in class.
Fri, Oct. 14 Fall recess, no class!
Mon, Oct. 17 Given an open partition 𝒰 of X, any connected subset A⊂X is contained in some U∈𝒰. Connected components form a partition. The phrase “Without Loss Of Generality” (WLOG). The connected subsets of ℝ are exactly the convex subsets. Munkres Ch. 3, §§24,25.
9 Wed, Oct. 19 The intermediate value theorem. A finite product of connected topological spaces is connected. ℝn is connected. Path-connectedness. Path-connected implies connected. ℝn is path-connected. Munkres Ch. 3, §§23,24.
Fri, Oct. 21 Open and closed balls are path-connected. The punctured ℝn is path-connected (for n≥1). The sphere is path-connected. The topologist’s sine curve is connected but not path-connected. Path components. Local path connectedness. A locally path-connected space is connected if and only if it’s path-connected, and its connected components are the same as the path components. Munkres Ch. 3, §§24,25.
Mon, Oct. 24 Compactness. Some example of compact and non-compact spaces. Characterization of compact subspaces. The image of a compact space under a continuous map is compact. The extreme value theorem for compact spaces. Closed intervals are compact. Munkres Ch. 3, §§26,27.
10 Wed, Oct. 26 Closed intervals are compact. Finite product of compact spaces are compact. The Tube Lemma. A subset of ℝn is compact if and only if it’s closed and bounded. A closed subset of a compact space is compact. Munkres Ch 3, §27.
Fri, Oct. 28 Any closed subset of a compact space is compact. Any compact subset of a Hausdorff space is closed. In a Hausdorff space, a compact set and a point have disjoint open neighbourhoods. A compact Hausdorff space is T3. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Limit point compactness and sequential compactness. Compact spaces are limit point compact. For metric spaces, the three notions of compactness are equivalent. Munkres Ch 3, §27,28.
Mon, Oct. 31 Lebesgue covering lemma. ε-net lemma. Continuous maps on compact metric spaces are uniformly continuous. Quotient topology. Munkres Ch 3, §27,28 and Ch 2, §22.
11 Wed, Nov. 2 Characterization of continuous maps out of quotients. Equivalence relation generated by a relation. The quotient space of a square obtained by identifying opposite sides is homeomorphic to a torus. Munkres Ch 2, §22.
Fri, Nov. 4 Surface of genus g as quotient of a 4g-gon. Quotient of a space obtained by contracting a subspace to a point. A disk with the boundary circle contracted to a point is homeomorphic to a sphere. Quotient maps. Open maps and closed maps are quotient maps. Munkres Ch 2, §22.
Mon, Nov. 7 Example of non-Hausdorff quotient spaces. Embeddings. Disjoint union topology. Glueing of spaces along a subspace. Connected sum of surfaces. Any space is a glueing of a closed cover along their intersection. Function spaces. The topology of pointwise convergence. Munkres Ch 2, §§16,22 and Ch 7, §46 and Weeks
12 Wed, Nov. 9 Convergence in the topology of pointwise convergence is pointwise convergence. Convergent sequences are continuous functions out of the extended natural numbers. The supremum metric. Munkres Ch 7, §46
Fri, Nov. 11 Veterans Day. No class!
Mon, Nov. 14 Extended real numbers and extended metrics. The topology uniform convergence. Continuous functions are closed in the topology of uniform convergence. Munkres Ch 7, §46
13 Wed, Nov. 16 The compact-open topology. On the space of functions from a compact space to a metric space, the compact-open topology agrees with the uniform topology. Munkres Ch 7, §46
Fri, Nov. 18 A map into a function space is continuous if the corresponding two-parameter map is. If the domain is locally compact Hausdorff, then the converse also holds. Also, the evaluation map on the space of functions is continuous if the domain space is locally compact Hausdorff. Munkres Ch 7, §46
Mon, Nov. 21 Closed subspaces of Rn are locally compact Hausdorff. Homotopy is an equivalence relation. Homotopy rel endpoints is an equivalence relation. Simple-connectedness. Convex subspaces of Rn are simply connected. Munkres Ch 9, §51.
14 Wed, Nov. 23 Thanksgiving. No class!
Fri, Nov. 25 Thanksgiving. No class!
Mon, Nov. 28 Groups. Homomorphisms and isomorphisms of groups. The fundamental group. Independence of the fundamental group on the base point. The fundamental group is a topological invariant up to isomorphism. A connected space is simply connected if and only if the fundamental group is trivial. Statement of the fundamental group of spheres, punctured Euclidean spaces, and products. Applications: R2 is not homeomorphic to Rn for n>2; the sphere and the torus are not homeomorphic. Munkres, Ch 9.
15 Wed, Nov. 30 Project presentations.
Fri, Dec. 2