Topology Spring 2020

Prof. Dr. Alessandro Carlotto
Course Assistant
Giada Franz
Teaching Assistants
Luca De Rosa, Francesco Fournier Facio, Yannick Krifka, Miguel Moreira, Alexandru Paunoiu
Mon 09-10 / HG F 3
Wed 13-15 / HG F 3
Exercise classes
Mon 10-12
Office hours
Mon 15-17
First lecture and first exercise class
Course Catalogue
401-2554-00L Topology


The content of the first-year courses in the Bachelor program in Mathematics. In particular, each student is expected to be familiar with notion of continuity for functions from/to Euclidean spaces, and with the content of the corresponding basic theorems (Bolzano, Weierstrass etc..). In addition, some degree of scientific maturity in writing rigorous proofs (and following them when presented in class) is absolutely essential.


An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.

Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.

Info course online

As you should know, starting from Monday, March 16th, all our teaching activities will be transferred online. We are trying to find the best solution for lectures, exercise classes, grading, office hours, etc. We will keep you updated by email and here on the website. Information about the online resources below are marked with the symbol .


Primary references

We will follow these, freely available, standard references by Allen Hatcher:

Extra references

Diary of the lectures

The lectures recordings are available here: lectures recordings. The lectures will be recorded at the usual times (i.e. Mon 9-10 and Wed 13-15) and will be available in the platform within 48 hours.

117.02. Introduction to the course: resources, key objectives and the roadmap in front of us. Some advice on taking a Math course. [GT] pp. 1-3 Study Guide
219.02. Topological spaces: definition and examples. Partial order structure on the class of topologies on a given set. Open and closed sets. Interior, closure and boundary (via two equivalent perspectives). [GT] pp. 3-7 Interior, closure, boundary
324.02. Bases of a topological space: definition and examples. A criterion for a collection of subsets to generate a topology. Application: metric spaces. Different distances may determine different bases of the same topology. [GT] pp. 7-10
426.02. Topological subspaces, definition and equivalent characterisations. Metric subspaces. Discrete subspaces. Subspaces and topological operations. Continuous maps and continuity criteria. Homeomorphisms. Some examples. [GT] pp. 10-13
502.03. Topological products, (equivalent) definitions and examples. Projections, injections, and a continuity criterion for maps with target a product space. [GT] pp. 13-16
604.03. Connectedness: definition and examples. A subset of the real line is connected if and only if it is an interval. The continuous image of a connected set is connected. Path-connectedness: definition and examples. Path-connectedness implies connectedness (the converse is false). The continuous image of a path-connected set is path-connected. Cut points. Application: certain pairs of spaces cannot possibly be homeomorphic (counting cut points). [GT] pp. 18-21
709.03. Path-connected components, definition and basic facts. Examples of connected spaces that are not path-connected. [GT] pp. 21-23
811.03. Connected components, definition and basic facts. Review of the key example. Connected + locally path-connected implies path-connected. [GT] pp. 23-25
Compactness: definition and examples. Three results: a closed subspace of a compact space is compact; the continuous image of a compact space is compact; the product of two compact spaces is compact (weak Tychonoff). Three equivalent notions of compactness for subdomains of Euclidean spaces. [GT] pp. 30-33
916.03. Proof of the Heine-Borel theorem. Separation axioms: the Hausdorff property. Examples, basic facts. Compact subspaces of a Hausdorff space are closed. A criterion to ensure that a continuous bijection is a homeomorphism. [GT] pp. 33-36Notes - L09
1018.03. Normal spaces. A Hausdorff space that is not normal. Two classes of normal spaces: i) compact Hausdorff spaces, ii) metric spaces. The Lebesgue number of a cover. Compact metric spaces have a Lebesgue number. Application: uniform continuity for functions from/to metric spaces. [GT] pp. 36-39Notes - L10
1123.03. Infinite products. The box topology and its defects. The product topology and its virtues. Proof of the Tychonoff compactness theorem in the case of infinite products. [GT] pp. 39-41Notes - L11
1225.03. Quotient maps and quotient spaces. Examples. A continuity criterion for maps whose domain is a quotient space. The Klein bottle. The real projective plane. [GT] pp. 44-52Notes - L12
1330.03. The problem of openness for quotient maps. An example. The saturation of a set and an openness criterion. -Notes - L13
1401.04. Topological manifolds: definition, examples and basic facts. The problem of classifying n-dimensional compact topological manifolds: resolution of the case n=1, and presentation of the case n=2. [GT] pp. 53-55Notes - L14
1506.04. Some heuristics about categories and functors. Homotopy of maps (possibly relative to a set); the special case of paths with fixed endopoints. Homotopic equivalence of topological spaces. Contractible spaces. [AT] pp. 1-4 and pp. 21-26Notes - L15
1608.04. Paths and concatenation. Based loops. Construction of the fundamental group of a topological space. Some examples. Role of the basepoint. Functorial properties. Homeomorphic spaces have isomorphic fundamental groups. [AT] pp. 26-28 and pp. 34-35Notes - L16
1720.04. Homotopic invariance of the fundamental group. Contractible spaces are simply connected; some examples. [AT] pp. 36-38Notes - L17
1822.04. Covering maps, basic definitions and examples. Constacy of the cardinality of the fibers, degree of a covering. Lifts, uniqueness and existence. The fundamental group of the circle. The fundamental group of product spaces, and application to the case of tori. [AT] pp. 29-30 and pp. 34Notes - L18
1927.04. Proof of the existence of lifts of paths and homotopies. Corollary: the monodromy theorem. [AT] pp. 30 and pp. 60Notes - L19
2029.04. The free product of groups, basic facts. Free group over n generators. The universal property of free products. Statement of Van Kampen's Theorem, special cases. The fundamental groups of spheres in any dimension. The fundamental group of the wedge of circles. [AT] pp. 40-43Notes - L20
2104.05. Retracts and deformation retracts. The disc does not retract onto its boundary circle, and the Brouwer fixed point theorem. The topological wedge of an arbitrary collection of topological spaces and calculation of its fundamental group. [AT] pp. 31-32, pp. 36 and pp. 43Notes - L21
2206.05. Factorisation of loops, subordinate to an open cover. Equivalence of factorisations. Proof of Van Kampen's Theorem. [AT] pp. 44-46Notes - L22
2311.05. A covering map induces an injection of the fundamental groups. The index of the image subgroup equals the degree of the covering. Application: the fundamental group of real projective spaces. Definition of normal coverings. [AT] pp. 56-61Notes - L23
2413.05. The general criterion for lifting maps between arbitrary topological spaces. A description of the Galois correspondence between covers of a given space and subgroups of its fundamental group. Construction of universal covers of an arbitrary topological space. [AT] pp. 62-65Notes - L24
2518.05. Existence of covers (of an arbitrary topological space) with assigned fundamental group. [AT] pp. 66Notes - L25
2620.05. A uniqueness/classification result for covering spaces. Universal cover. Deck transformations, definition and basic examples. Normal covering spaces and their deck transformations. An example of a non-normal covering space. [AT] pp. 67-68 and pp. 70-71Notes - L26
2725.05. The geometric effects of conjugacy and the fundamental theorem on deck transformations for normal coverings. [AT] pp. 71Notes - L27
2827.05. An introduction to group actions, some examples. [AT] pp. 71-74Notes - L28
The fundamental group of graphs and an algebraic application. [AT] pp. 83-85

Final exam

The duration of the written exam (in any session) will always be 120 minutes. During the exam no written aids nor calculators or any other electronic device are allowed in the exam room. Mobile phones must be switched off and stowed away in your bag during the whole duration of the exam.

We organized three additional office hours sessions during the semester break (end of July / beginning of August, see precise schedule in the section Office hours) as substitution to Ferienpräsenz. Please feel free to exploit this resource (as well as the Forum) to clarify your doubts at due course, possibly well in advance with respect to the exam date.


In order to easily interact, we set up a forum for our course at the link Topology (Spring 2020) - forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc. Use it!

Exercise classes

Exercise classes will be recorded on Mondays 10-12 (only one session per week) on ETH Zoom. You can follow them live with the link and credentials you have received by email. Video recordings and notes of the exercise classes will also be available here on the website. During the exercise classes we will give examples and further explanations about the theory covered during the lectures and we will discuss the questions you asked on the forum; hence it is important that you use this platform to communicate with us.

Please register and enroll for a teaching assistant in myStudies. The enrolment is not needed to attend the exercise class (since there is only one unified exercise class per week), but to hand in your homework.

18.03.Review of compactness with examples and exercises. Construction and properties of the Cantor set. Discussion of problems 4.7, 4.8 and 4.9. Giada FranzVideo and notes on Cantor set
25.03.Convergence in topological spaces, sequential continuity, countability axioms. Two new examples: the co-countable topology and the lower limit topology. Some hints on exercises 5.5, 5.6, 5.7, 5.9. Francesco Fournier FacioNotes - EC02
01.04.Relationship between topological, metric and normed spaces. Arzelà-Ascoli theorem. Remarks about the difference between finite and infinite dimensional spaces. Hints on exercises 6.2, 6.5, 6.6. Luca De RosaNotes - EC03
08.04.Remarks on topological manifolds, example of a locally Euclidean but not Hausdorff space. The torus as a quotient space. The One-Point Compactification and its basic properties, the Alexandroff Compactification. Luca De RosaNotes - EC04
20.04.Discussion of exercises 7.1, 7.2, 7.3, 7.4 and 7.9. Visualisation of the Klein bottle. Alexandru PaunoiuVideo - EC05
Notes - EC05
27.04.Discussion of exercises 8.2, 8.3, 8.6, 8.7 and 8.8. Difference between contractible and admitting a deformation retraction. Yannick KrifkaVideo - EC06
Notes - EC06
04.05.Common mistakes on pset 8. Recall of covering spaces. Discussion of exercises 9.3, 9.4, 9.7 and 9.10. Francesco Fournier FacioVideo - EC07
Notes - EC07
11.05.Discussion of problems 10.1, 10.7, 10.5, 10.8 and 10.9. Fundamental group of graphs. Miguel MoreiraVideo - EC08
Notes - EC08
18.05.Discussion of problems 11.2, 11.3, 11.6, 11.7 and 11.8. Miguel MoreiraVideo - EC09
Notes - EC09
25.05.Discussion of exercises 12.3, 12.4, 12.7 and 12.8. Alexandru PaunoiuVideo - EC10
Notes - EC10

Problem sets

Every Monday, a new problem set is uploaded here. You have one week time to solve the problems. If you have difficulties understanding or solving certain tasks and you want clarifications or hints, you can ask your questions on the forum or during office hours. Some of the questions will also be discussed at exercise class.

Homework collection and delivery: You may hand in your homework for grading to the assistant of the exercise class you enrolled for. Please send a pdf-file with the solutions to your assistant by email. The deadline is on Mondays 10am. The graded homework sheets are returned by email within one week.

How to write your solutions: We would really appreciate solutions written in LaTex, but you can send your solutions handwritten and scanned / photographed. However, make sure that everything you send is clearly readable. It is better to focus on fewer exercises rather than providing sloppy solutions to all of them. Please help us making the grading possible by taking care more than before of the style of your solutions!

Every problem is marked with one of the following symbols:

Apply what you learn in basic situations.
Construct examples and give full proofs.
Challenging problems. It is recommended that you start working on them only after you have reviewed the weekly material and carefully solved all other exercises in the assignment. We will not provide solutions for these problems. However, you can write your solutions (separately from the other exercises) and hand them in to Giada Franz (by email). The best solution to each problem will be uploaded to the website.

Assignment dateDue dateProblem setSolutionChallenge problem solved byBest solution
Wed 19.02. Wed 26.02. Problem set 1 Solutions 1 Jonathan Clivio, Gabriel Dettling, Gabriel Frey and Kevin Zhang, Meilin Gong, Markus Krimmel, Kevin Lucca, Elia Mazzucchelli, Aurelio Sulser, Ana Marija Vego, Raphael Zhiang Wu First prize to Aurelio Sulser
Second prize to Gabriel Dettling
Wed 26.02. Wed 04.03. Problem set 2 Solutions 2 Jonathan Clivio, Gabriel Dettling and Ruben Skorupinski, Kevin Lucca, Silvan Suter, Ana Marija Vego, Johann Wenckstern First prize to Johann Wenckstern
Wed 04.03. Wed 11.03. Problem set 3 Solutions 3 Partial solutions: Jonathan Clivio, Niklas Dahlmeier and Ruben Skorupinski, Julian Huber, Aurelio Sulser, Silvan Suter, Kevin Zhang First prize to Silvan Suter
Note: Injective sub-paths
Wed 11.03. Fri 20.03. Problem set 4 Solutions 4 Kevin Zhang First prize to Kevin Zhang
Fri 20.03. Mon 30.03. Problem set 5 Solutions 5 Gabriel Frey, Silvan Suter, Kevin Zhang First prize to Silvan Suter
Mon 30.03. Mon 06.04. Problem set 6 Solutions 6 Adrian Müller, Raul Rao, Aurelio Sulser, Silvan Suter First prize to Raul Rao
Mon 06.04. Mon 20.04. Problem set 7 Solutions 7 No challenge problem (unofficial midterm)
Mon 20.04. Mon 27.04. Problem set 8 Solutions 8 Aurelio Sulser, Kevin Zhang First prize to both Aurelio Sulser and Kevin Zhang
Mon 27.04. Mon 04.05. Problem set 9 Solutions 9 Ruben Skorupinski, Kevin Zhang First prize to Ruben Skorupinski
Mon 04.05. Mon 11.05. Problem set 10 Solutions 10 Kevin Lucca, Silvan Suter, Johann Wenckstern, Kevin Zhang First prize to Kevin Zhang
Second prize to Kevin Lucca
Mon 11.05. Mon 18.05. Problem set 11 Solutions 11 Aurelio Sulser, Kevin Zhang First prize to Aurelio Sulser
Mon 18.05. Mon 01.06. Problem set 12 Solutions 12 Keving Zhang
Partial solution: Anna Knörr and Florian Meier
First prize to Kevin Zhang

Office hours

For the office hours we will use the online platform ETH Zoom (precise instructions will be sent to you by email), in which you will be free to ask questions and hints. The schedule is as follows (up to possible short-term changes, please check for updates).

Thu 20.02. 15-17 HG G 28 Giada Franz
Thu 27.02. 15-17 HG J 16.5 Yannick Krifka
Thu 05.03. 15-17 HG G 66.4 Francesco Fournier Facio
Mon 09.03. 15-17 HG J 16.1 Miguel Moreira
Mon 16.03. CANCELLED
Mon 23.03.15-17ETH Zoom Luca De Rosa
Mon 30.03.15-17ETH Zoom Alexandru Paunoiu
Mon 06.04.15-17ETH Zoom Yannick Krifka
Mon 20.04.15-17ETH Zoom Francesco Fournier Facio
Mon 27.04.15-17ETH Zoom Miguel Moreira
Mon 04.05.15-17ETH Zoom Luca De Rosa
Mon 11.05.15-17ETH Zoom Alexandru Paunoiu
Mon 18.05.15-17ETH Zoom Yannick Krifka
Mon 25.05.15-17ETH Zoom Francesco Fournier Facio
Mon 20.07.15-17ETH Zoom Francesco Fournier Facio
Mon 27.07.15-17ETH Zoom Alexandru Paunoiu
Mon 03.08.15-17ETH Zoom Yannick Krifka