Topology Spring 2019

Alessandro Sisto
Luca De Rosa

Goal of the course

The course is a first introduction to Topology. The aim of the course is to provide basic notions and main properties of topological spaces and continuous maps. The topics coved in the lecture include: connectedness, compactness, product and quotient topology, separation axioms, homotopy, fundamental group, covering spaces.


Here you can find the last version of the course summary.

New exercise sheets will be posted here on Wednesdays. On Mondays there will will be exercise classes, in which you are welcome to ask questions and present your doubts.

Please hand in your solutions by 13:00 on Wednesdays in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68. Please be aware that many solutions, as well as the course summary, may contain typos and mistakes. Read and use such material in a mindful and critical way.


It is possible to hand in exercises in teams of at most three people. The teams are self-organized and may vary from one exercise sheet to the other.

exercise sheet due by solutions notes
Exercise sheet 1 February 27, at 13:00 Solutions 1
Exercise sheet 2 March 6 Solutions 2 Ex. 6b): (R\{0})x{0} instead of R2\{0}
Exercise sheet 3 March 13 Solutions 3
Exercise sheet 4 March 20 Solutions 4
Exercise sheet 5 March 27 Solutions 5
Exercise sheet 6 April 3 Solutions 6
Exercise sheet 7 April 10 Solutions 7
Exercise sheet 8 April 17 Solutions 8
Exercise sheet 9 April 30 Solutions 9
Exercise sheet 10 May 8 Solutions 10
Exercise sheet 11 May 15 Solutions 11
Exercise sheet 12 May 22 Solutions 12 Ex.2 a), need \(X\) and \(Y\) to be path connected.
Exercise sheet 13 Solutions 13
Mo 10-12CHN C 14Maxim Gerspachen
Mo 10-12CHN D 48Subhajit Janaen
Mo 10-12ETZ F 91Nadir Bayoen
Mo 10-12ETZ H 91Niclas Kupperen
Mo 10-12ETZ J 91Francesco Fournier Facioen
Mo 10-12HG E 33.1Luca De Rosaen

For 2019 Summer examinations, we have organized office hours, so that you can come and ask questions and discuss doubts. You can find here the schedule. Please register to the session you are interested in via the doodle form.

Insert here your comments, what you particularly liked or disliked, or any suggestions. It can regard the overall lecture, or the exercise classes as well. If your feedback refers to a precise teaching assistant, please mention it. Please keep in mind that any input is completely anonymous. NOTE: Feedback facility is working during exam preparation period as well.

The exam consists of an oral examination of 20 minutes.

As a general rule, the questions that will be asked at the exam will be taken from the two documents 'List of Questions' and 'Review Tables'. Notice however, that you might get asked to clarify your answers, leading to further questions not technically from the above mentioned documents.

List of Questions

Review Tables

Regaring the file "Review Tables", it contains two tables.

-In the first one, for a given topological operation and a given topological property, you can see whether or not that property is preserved under that operation (e.g., whether a product of connected spaces is connected).

-In the second table, you can see which "standard" topological spaces have which properties.

The tables were made by Francesco Fournier Facio.

At the exam, you might get asked about entries of the table (e.g., "is property X preserved by operation Y?"), and to give a proof or a counterexample, as appropriate. We absolutely don't expect you to memorise all the entries of the table (we didn't!); we will give you some time to think about it and figure out the correct answer.

Please refer to the introductory text in "Review Tables" for the full detailed description of how to use them.

It is important to emphasise that we are NOT expecting you to have prepared answers for the questions of the two documents. In fact, if in the process of trying to find the correct answer on the spot we see that you understand all the relevant concepts and you're able to handle them, you will get full credits, regardless of you getting to the correct answer or not! On the contrary, you will not get full credits if you give us a prepared answer, and on asking a clarification you're not able to answer.

To sum up, we recommend to prepare for the exam by studying the theory and doing exercises from the exercise sheets rather than by preparing specifically for the questions in this document. In other words, use the two files as a guideline for your preparation, and not as a 'checklist'.

Comments and Corrections

- In 'List of Questions', at the question 37, the map f is continuous.

- In the 'contructions table' solutions, in 'green circles' section, it is written "A finite product of locally path-connected spaces is path-connected", while it should be "A finite product of locally path-connected spaces is LOCALLY path-connected." Moreover, at the beginning of the solution, it is written "Let X1, ..., Xn be locally compact spaces", it should be "Let X1, ..., Xn be locally PATH CONNECTED spaces".

- In 'List of Questions', at the question 13, the topology on X is the product topology.

- In question 60, the map f is f(x,y) = (x, -y), and not f(x,y) = (-x,-y) .

- In 'Review Tables' at some point it is claimed that an uncountably infinte space with discrete topology is second countable. This is wrong. X has to be countable.

Additional Office Hours

In order to provide some additional support, we have scheduled for 18th of July some extra office hours, from 13pm till 15:30 in HG J43 (Prof. Sisto Offce). Please feel free to come ask questions and share your doubts. If you are interested, you can drop an anonymous message on the feedback box, so that we can have a rough estimate of how many people expect, but it is not necessary (you can also decide at the last moment and just pass by).