- First of all: the file with the marks (the students that did the retake appear with a white background).

- I would like to talk separately (just for 2 minutes) to the following: Mark Kamsma, Joren Harms, Ivor van den Hoog. Can you please drop by my office (Math building, 8th floor) on Tuesday April 7, or Monday April 13, or Tuesday April 14, at 2PM (or after if you cannot at 2PM)? If you are not around (i.e. in the Uithof) any of those days, then do not come just to talk to me, but try to drop by my office some other time.

- I would also like to talk to a larger group of people:

Tim Baanen,

Lukas Zwaan,

Roy Smitshoek,

Sharon Steen,

Babette de Wolff,

Berend Ringeling,

Mees Verheije,

Esther Visser,

Stijn Nugteren,

Steyn van Leeuwen,

Roald Neuteboom.

Can you please come to the Mathematics Building (Freudenthal) on Tuesday, April 14, at 1PM, in front of 611 (sixth floor)? If you are not around that day/time, can you please look for me some otehr time when you are in the Uithof?

- First of all: the file with all the marks (homeworks, the exam, the various averages and the final mark); and here are the results the are missing in the first file.

- I hope the table is clear. Anyway: H1-H7 are the marks for the homworks, Homew is the average, Av1 is the average of the Homework with the Exam in which the exam counts more, Av2 is the one for which it counts less, Max-av is the maximum of the two averages, and then, you find the final result (rounded).

- In many many cases, you lost points in a "silly way" (e.g. at easy exercises while doing the more difficult ones, or by not being careful). In particular, many of the marks that are between 4 and 6 are obtained by students that, normally, should get above 8. Even some students with marks around 3 clearly understand a lot more. For all those: I strongly encourage you to do the retake.

- By the way: I am going to ignore the rules that impose conditions on who is allowed take the retake (but please do not "just try your luck" i.e., after receiving a mark below 4 at the exam and not preparing accordingly for a retake!).

- for those of you that want to take a look at your exam: I will be available on Tuesday, February 24th, between 13:00 and 15:00, in room 611 of the mathematics building (Hans Freudenthal building).

In particular, there are some of you that I really have to talk to. First of all, there are some students that I have to talk to in order to clarify a few aspects related to the exam and homeworks:

- Ragnar Groot Koerkamp

- Marc Houben

- Djurre Tijsma

- Jori Hoencamp

- Robert Christian Subroto

- Damian vd Heisteeg

- Bjarne Kosmeijer

- Esther Visser

- Joren Harms

- Mathijs Henquet

- Alex Ben Hassine

- Linde FrĂ¶lke

For these students: can you please drop by my office (Math building, 8th floor) on Monday (February 9th), somewhere between 13:00 and 16:40? (hopefully not all at once!).

Then I have a few comments for the following students (with the very strong recommendation that they do the retake in order to get an (even) better mark), which I hope to meet on Tuesday:

- Timon Knigge

- Tim Baanen

- Marion Snijders

- Richard Schoonhoven

- Mark Kamsma

- Bart van Hoek

- Rik vd Stelt

- Kabirdas Henry

- Monique Huveneers

- Roy Smitshoek

- Rutger Hartog

- Lukas Zwaan

- Babette de Wolff

- Davide Berend Ringeling

For these students: can you please drop by my office (Math building, 8th floor) on Tuesday (February 10th), somewhere between 13:00 and 17:00? (hopefully not all at once!). If some of you would have to travel from other cities (Amsterdam? Eindhoven?), then please do not come just for this; instead, get in touch with me, so that we arrange a meeting in a day that you are in Utrecht anyway.

Please be aware of the correct date, time and location for the final exam:

Wednesday January 28, 9.00-12.00, in the EDUCTHEATRON

NEW: the assignment for our honours programme: read/understand properly Chapter 9 of the lecture notes (including the material the is used in this chapter and was not presented in the class, if any). Then contact me for a discussion on the blackboard (some kind of "oral exam").

This is the web-site for the course "Inleiding Topologie" for the year 2014-2015 (blok 2, Fall 2014). Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc. You can also have a look at the similar site from the last year .

THE LECTURES:

------ Mondays: 17:15 - 19.00, ANDRO, C101.

------ Wednesdays: 9.00 - 10.45, ANDRO C101.

LECTURER: Marius Crainic.

TEACHING ASSISTANTS: Dana Balibanu (danabalibanu(tra-la-la)gmail.com), Davide Alboresi (D.Alboresi(tra-la-la)uu.nl), Thomas Blom (T.Blom1(tra-la-la)uu.nl) and Remie Janssen (R.Janssen2(tra-la-la)uu.nl). The students are divided in two groups.

THE WERKCOLLEGES:

------ Mondays: 11.00 - 12.45 in BBG 83(Groep 1) and BBG 315/317(Groep 2).

------ Wednesdays: 11.00 - 12.45 in BBLG 161 (Groep 1) and BBG 165 (Groep 2).

As a general rule: werkcollege number n is devoted to: solving the problems related to lecture number n-1 (announced at the end of the lecture and/or on the web-page).

HOMEWORKS:

Each Wednesday you will receive a homework that you have to hand in one week later, before the start of the next lecture (!!!). If you are not present at the lecture, please make sure that a colleague of yours brings your homework (again: before the start of the lecture!), or that you give it beforehand to the teaching assistants.

Note: please do not send your homework by email to me (the lecturer), but to the TAs. Thanks.

- hand in exercises, once per week (every Wednesday). You will receivce the exercise at the end of the werkcollege. You have to hand it in one week later (on Wednesday), at (or before) the beginning of the lecture. The everage of the marks for all the homeworks will give one mark HW (maximum 10).

- final exam, for which you will receive a mark E (maximum 10).

Note: you are allowed to bring with you, and use during the exam, three sheets of A4 papers (= six pages) containing definitions, theorems, etc from the course.

** Final mark:** The final mark will be obtained by combining E (exam mark) and HW (homework mark), by the formula:

max{(7 E+ 3 HW)/10, (17E+ 3H)/20}

- .pdf file.

- .ps file.

- .dvi file.

Note that some of the pictures use colours.

Please be aware that the lecture notes still contain typos. So, if there is something that you do not understand, please ask it at the werkcollege; also, if you find typos, please communicate them to the teaching assistants. This will be of great help to improve the lecture notes and make them into a regular "dictaat" that can be printed for students.

**WEEK 46/Lecture 1 (November 10):** Key-words of Topology (in analogy with group theory): continuity and convergent sequences; main objects of topology: topological spaces; main "functions" of Topology: the conitnuous ones; "isomorphisms" in Topology: functions $=f: X---> Y which are bijective, f continuous, the inverse of f continuous.

For now, "space": any X on which it is clear (intuitively!!!) what convergence means (i.e., for a sequence in X, it is clear intuitively what the convergence of the sequence means).

Examples of "spaces": metric spaces (X, d); the notion of metric space is also a first attempt/first candidate for the notion of "topological space" but, as we shall see, it is not quite the best one. E.g. I mentioned that two metrics on the Euclidean space, namely the Euclidean and the square metric, although they are very different from each other, they induce the same notion of convergence (i.e. they induce "the same topology"). More to come.

I also mention some central questions in Topology.

Then we looked at some examples: intervals, the unit circle, circles; the cylinder, the Moebius band, the torus and the Klein bottle- obtained by starting from a square and performing some gluings, showing the result by a picture (except for the bottle!).

Exercises for the werkcollege of the next Wednesday: Exercises 1.1- 1.8, and 1.13.

**WEEK 46/Lecture 2 (November 12):** stereographic projection, gluings, examples of gluings, including the projective space.

Exercises for the werkcollege of the next Monday: 1.30, 1.32, 1.29, 1.14, 1.15, 1.16, 1.9, 1.10.

Homework (to be handed in by the next Wednesday, November 19, before the lecture): 1.31 and 1.25.

**WEEK 47/Lecture 3 (November 17):** reminder on gluings and examples, with the explicit models for the Moebius band and the torus. Then the notion of topology, topological space, opens and closed subsets and firts examples.

Exercises for the werkcollege of the next Wednesday: 1.17, 1.18, 1.20, 1.21, 1.27, 2.1, 2.2.

**WEEK 47/Lecture 4 (November 19):** Reminder on the notion of topological spaces, with first examples (trivial, discrete, co-finite, co-countable topologies on any set; metric topologies, with the proposition that the metric topology is a topology which is "the best one", a summary on the Euclidean topology on the real line and then, based on the obvious analogy: the lower limit topology on the real line; then subspace topologies). Then the basic definition from the next two subsections: continuity, neighborhoods, convergence, continuity at a point and sequential continuity + a few remarks and the statement that explains the relationship between: (1) continuity, (2) continuity at each point, (3) sequential continuity.

Exercises for the werkcollege of the next Monday: 2.15, 2.16, 2.13, 2.2, 2.23, 2.3, 2.31, 2.40, 2.41, 2.5, 2.29, 2.30, 2.8.

Homework: 2.19.

**WEEK 48/Lecture 5 (November 24):** reminder on the continuity and convergence, with more remarks/examples, the proof of the theorem we stated at the end of the previous lecture (equivalence between the various notions of continuity). Also defined the notion of topology basis and the 1st countability condition (with example metric topologies) and the notion of Hausdorffness (with metric topologies still as an example) and the characterization of Hausdorfness in terms of uniqueness of limits. Then defined the notion of interior and closure.

Exercises for the werkcollege of the next Wednesday: make sure you are done with the exercises from the previous list, then do as many as you can from: 2.30, 2.33, 2.9, 2.7, 2.60, 2.48, 2.49, 2.53, 2.58, 2.43.

**WEEK 48/Lecture 6 (November 26):** some of the left overs from this chapter: the criteria/lemma regarding interiors/closures, 2nd countability, the notion of manifold, normal spaces. I also mentioned a few things that will only show up in the later parts of the course (e.g., when discussing topological properties, I mentioned connectedness and, more importantly, compactness and the way this can be used to handle homeomorphisms/embeddings). If you did not attend the lecture, it is a good idea that you look at the notes taken by a colleague that was there.

Exercises for the werkcollege of the next Monday: 2.52 (try to write down the complete proof!!! If you are not sure you do it properly, please ask the teaching assistants to check it!!), 2.35, 2.38, 2.56, 2.12, 2.48, 2.56, 2.58, 2.12, 2.11.

Homework (to be handed in by Monday, December 8): 2.53.

**WEEK 49/Lecture 7 (December 1):** We started the new chapter- the one on "constructions of topological spaces": quotients (with some examples, including the projective space); group actions.

Exercises for the werkcollege of the next Wednesday: make sure you have done all the exercises for the previous werkcollege. In particular, make sure that you have filled in all details for 2.56. Then read Example 3.5, then do: 3.2, 3.10, 3.9, 3.1.

**WEEK 49/Lecture 8 (December 3):** quotients modulo group actions (ex: real line modulo Z, plane modulo Z^2, the sphere modulo Z_2); the proof that the quotient of a Hausdorff space modulo the action of a finite group is Hausdorff. Then product topology (explaining why the product topology is definied as it is). Then other special quotients: collapsing a subspace to a point, the cylinder, the cone and the suspension of a space.

Exercises for the werkcollege of the next Monday: 3.16, 3.17, then do Exercise 3 from the exam from the last year, 3.21, 3.4, 3.22, 3.23, 3.11, 3.20, 3.18.

Homework: do exercises 1.11 and then prove that the quotient D^2/S^1 (obtained from the closed disk by collapsing the boundary circle) is homeomorphic to the sphere S^2 (hint: for the proof, do not forget to use Corollary 3.4 (and remember what was discussed during the lectures) and also remember the use of "compactness" that was mentioned to you during the lectures- but was not in the first chapters of the lecture notes).

**WEEK 50/Lecture 9 (December 8):** Reminder on metric topologies, the Euclidean topology on the real line, the lower limit topology on the real line, the product topology. Inspired by this, we discussed the notion of "topology basis on a set X" (not to be confused with "the basis of a topological space"), it properties and the lemma that compares two topologies induced by two bases. Then we discussed connectedness and path connectedness: definition, main properties, corollaries (leaving the proof of a lemma and a proposition for next time). Also, I have mentioned several examples of how to show that two spaces are not homeomorphic using connectedness (and the removal one-point trick); since these examples are not in the lecture notes, if you did not follow the lecture, have a look at the handwritten notes of one of your colleagues that was present.

Exercises for the werkcollege of the next Wednesday: 3.29, 4.22, 4.12, 4.17, 4.13, 4.14, 4.15, 4.20, 3.33, 4.23.

IMPORTANT CHANGE (December 8th, 19:47): FOLLOWING SEVERAL DEMANDS, THE HOMOEWORK THAT SHOULD BE HANDED IN THIS WEEK ON WEDNESDAY IS POSTPONED TO THE NEXT MONDAY (HENCE YOU DO NOT HAVE TO HAND IN ANYTHING THIS WEDNESDAY, DECEMBER 10TH, BUT ONLY ON MONDAY, DECEMBER 15TH). HOWEVER, YOU DO GET A NEW HOMEWORK TO HAND ON THE WEDNESDAY OF THE NEXT WEEK (DECEMBER 17TH): DO PARTS a), b), c) AND d) OF EXERCISE 1 FROM THE EXAM OF THE LAST YEAR.

**WEEK 50/Lecture 10 (December 10):** (tentative): finnish with connectedness (the proof of the lemma and of the proposition; then connected components); I also spent a bit more time with some examples and some topological invariants (this is not in the lecture notes- so please have a look at the notes of a colleague that was present at the lecture) and then gave an idea of how to prove that Euclidean spaces of higher different dimensions are not homoemorphic (this is extra, not required for the exam, just to give you an indication of what "Algebraic Topology" (from the course "Meetkunde en Topologie") is about. Then I started with compactness: definition, some examples of non-compact spaces, then I stated the main properties (without proofs) and explained how, using them, one derives the fact that a subspaces of an Euclidean space is compact if and only if it is closed and bounded.

Exercises for the werkcollege of the next Monday: finnish the exercises for the last wercollege. Then do: 4.3, 4.24, 4.11, 4.18, 4.25, 4.26, 4.28, 4.21.

Homework: as mentioned above, for the next Wednesday (December 17th), DO PARTS a), b), c) AND d) OF EXERCISE 1 FROM THE EXAM OF THE LAST YEAR

**WEEK 51/Lecture 11 (December 15):** Compactness- main properties: statements and proofs.

Exercises for the werkcollege of the next Wednesday: make sure you gave finnished the exercises related to compactness, from the previous werkcollege. Then do: 4.29, 4.38, 4.39, 4.27, 4.33, 4.35.

**WEEK 51/Lecture 12 (December 17):** (tentative): a bit more on compactness; then local compactness and the one-point compactification.

Exercises for the werkcollege of the Monday, January 5: 4.40, 4.41, 4.42, 4.43, 4.45, 4.47, 4.49.

Homework: 4.46 (this is an exercise given at a previous exam; it is a good idea to first solve some of the previous exercises, and only after that do this one).

**WEEK 2/Lecture 13 (January 5):** Uryshon's lemma (Theorem 5.21 in the lecture notes); Uryshon's metrizability theorem (Theorem 7.1 iun the lecture notes) + the consequences (but some were derived differently than in the notes, because we only discussed the first metrizability theorem).

Exercises for the werkcollege of the next Wednesday: make sure that you have done all the exercises from the last couple of werkcolleges. If that is the case, please read the last part of the proof of Theorem 7.1 (Claim 2 in the proof, from which we only proved one part on the blackboard).

**WEEK 2/Lecture 14 (January 7):** we will move to the algebra of real-valued continuous functions, Stone-Werierstrass density theorem and the Gelfand-Naimark theorem (these will keep us busy for the next 2-3 lectures).

Exercises for the werkcollege of the next Monday: 8.2, 8.3, 8.4, 8.5, 8.6.

Homework: no homework.

**WEEK 3/Lecture 15 (January 12):** Gelfand-Naimark.

Exercises for the werkcollege of the next two werkcolleges (nex Wednesday and Monday): 8.10, 8.11, 8.14, 8.16,8.15, 8.12, 8.13. Then you can try 8.18, but there is a mistake in that exercise- at point (iii) (you can try to figure out what goes wrong; I will try to post a corrected version one of this days).

**WEEK 3/Lecture 16 (January 14):** Gelfand-Naimark.

Exercises for the werkcollege of the next Monday: see above.

Homework: 8.17, to be handed in at the last lecture.

**WEEK 4/Lecture 17 (January 19):** Finite partitions of unity+ application: any compact manifold can be embedded in some Euclidean space; also started discussing infinite partitions of unity.

Exercises for the werkcollege of the next Wednesday: 5.1, 5.4, 5.5, 5.13. You can also try some of the similar exercises from previous exams (e.g. 10.7 on pp. 133, 10.25 on pp 138, 10.42 on pp 142). If you have more time, have a look at the exercises from the previous werkcolleges that you did not finnish.

**WEEK 4/Lecture 18 (January 21):** End the discussion on partitions of unity.

**WEEK 5, January 28 (Wednesday), from 9.00 to 12.00, in the EDUCTHEATRON:** Final exam.

Also: Extra-werkcollege: BBG-061 from 11 to 12.45 a.m. on Monday January 26th!

Note: you are allowed to bring with you, and use during the exam, three sheets of A4 papers (= six pages) containing definitions, theorems, etc from the course.

**Woensdag 11 maart van 08.30-11.30 in Educatorium ALFA:** Retake exam.

**Enjoy the sphere ** (and not only).