As you may notice in the list above, there was a very large number of students with marks below 5 at the exam (most of which have failed). It is my strong impression/belief that many of those students are "good students"- which worked quite a bit during the year, who did understand the material when it was presented, etc etc. When correcting the exams, I often came accross cases in which points were lost because of some silly mistake or confusion on the easy part of an exercise, while a more difficult part was done correctly. This (and the types of confusions/mistakes I encountered) indicate that those students just need a bit more time to assimilate all the concepts/abstractions. So, I encourage those students not to be disappointed, but to try to act accordingly and, at the end, pass the exam or just improve the mark. Note that this is the first time since 2005 that this phenomena occurs. If you have any explanations for this, please let me know (m.crainic at uu.nl).

Also, I would like the following students to drop by my office (office 802, 8th floor, math building, for a couple of minutes) one of these days (I am around the end of the lunch time, all days of the week, but probably not on Monday, February 24):

- Maarten Mol

- Stefan Korenberg

- Laurens Stronks

- Jan-Willem van Ittersum

- R.J. Doornenbal

- Lars van den Berg

- Aldo Witte

- Sven Bosman

- Rik Voorhaar

- Thijs van der Gugten

- Alexander Gietelink Oldenziel

- Jette van den Broeke

- Mark Raaijmakers

- Lois van der Meijden

- Franziska Gerken

- Piet Lammers

- Joris Ketelaars

- Thomas Blom

- Tom van Overbeeke

- Daniel Kroes

- Jetze Zoethout

(written on December 2nd).

(written on November 25th).

This is the web-site for the course "Inleiding Topologie" for the year 2013-2014 (blok 2, 2013). 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: 9.00 - 10.45, MIN 211.

------ Wednesdays: 9.00 - 10.45, MIN 211.

LECTURER: Marius Crainic.

TEACHING ASSISTANTS: Valentijn Karemaker, Joao Nuno Mestre Fernandes da Silva, Sjoerd Boersma, Bas Nieraeth, Koen van Woerden. The students are divided in two groups (check here to check the group that you belong to; sorry, the link does not work yet).

THE WERKCOLLEGES:

------ Mondays: 11.00 - 12.45 in BBL 169 (Groep 1) and BBL 079 (Groep 2).

------ Wednesdays: 11.00 - 12.45 in BBL 169 (Groep 1) and BBL 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).

- answering questions regarding the theory of lecture n.

HOMEWORKS:

Each Wednesday you will receive a homework that you have to hand in one week later, before the start of the next lecture (!!!). This rule is made in order to avoid situations that happened in the past, in which some students were busy during the lecture with solving/writing down the solution for the homework. If you are not present at the lecture, please make sure that a colleague of yours brings your homework, 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).

- ocasionally there may also be "bonus exercises" (worth 0.25 or 0.50 points), which are not compulsory. With them, you may a certain number of points, call it B.

**New: ** There will also be several quizzes during the werkcollege (15 minutes, in the second hour of the werkcollege).
It is not compulsory to do them, but they may be helpful. You can take part in a quizz and decide whether you hand it in
for correction or not. If you hand it in, then it will count for the final mark (see below).

For each quizz you will get a mark and the average of all the quizes you handed it gives a quizz mark, call it Q and call k the number of quizes you handed in.

** Final mark:** The final mark will be

min{10, (3E+ HW)/4+ B+ k(Q-5)/25}.

where E, HW, B, k, Q are as above (the only condition in order to pass is that the final mark is at least 6).

As examples: if you did (and handed in) one quizz only (k= 1), then:

- if you did it perfectly (hence Q= 10) then the quizz component will add 0.2 points to the final mark.

- if you did it completely wrong (Q= 0), it will subtract 0.2 points.

Or, if you did (and handed in) k= 10 quizzes, all perfect (hence Q= 10), then the quizz component will add 2 points to the final mark. However, if you did k= 10 quizes, all wrong (hence Q= 0), then it will subtract 2 points.

Of course: no quizzes (k= 0) will not contribute to the mark at all.

For the re-examination (herkansing): you get the chance to improce E (but not HW) and we apply the same formula as above.

- .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 11):** Short description
of the key-words of topology (spaces, continuous maps, homeomorphisms). The first attemtp to defines "spaces": metric
spaces. Then examples, revealing several reasons that using metrics is not the way to go. Also mentioned some of the
main questions of topology. As examples: intervals, circles (embedded in various ways in the plane or space), some funny faces,
cylinders. Then very briefly the torus and the Moebius band.

Exercises for the werkcollege of the next Wednesday: Exercises 1.1- 1.8, and 1.13. Also, given in the class: show that, for a map between two metric spaces, the usual notion of continuity (defined with epsilon and delta) is equivalent to the continuity as described in the class (that the function preserves limits).

**WEEK 46/Lecture 2 (November 13):** Reminder; various way
to think of the 2-sphere; gluings (= abstract equivalence relations and quotients); then we discussed the torus in full detail,
in an order slightly different than in the notes: first the abstract torus, then the models T_{R, r} (parametric and
implicit), then as a product of the circle with itself. Then we passed to the Moebius band, mentioning that it works like for the torus
and indicating how to get the equations for concrete models sitting in the space.

Exercises for the werkcollege of the next Monday: 1.8, 1.32, 1.10, 1.12, 1.16, 1.17, 1.18. You should try all of these (continue them at home if you do not finish them at the werkcollege). If you want more exercises, I advise you to continue with: 1.11, 1.30, 1.33, 1.25, 1.20.

Homework (to be handed in by the next Wednesday, November 20, before the lecture): Exercise 1.23 from the notes (note: in particular, you also have to do 1.22).

**WEEK 47/Lecture 3 (November 18):**
Short reminder on quotients, one more example with the abstract quotients that is not in the notes (horizontal lines in the plane),
very briefly the other examples that we have see; then, in more detail, the projective plane. Then moved
to general topological spaces: after briefly looking at metric spaces for inspiration, we defined the notion of topology
and topological spaces, made some comments on the axioms (also the fact that a space could be defined using the collection
of all closed subsets instead of that of opens). Then the first examples (trivial, discrete, co-finite and co-countable
topologies- defined on any set) and we showed that the topology induced by a metric is indeed a topology and I
explained that a subset is an open in the metric topology if and only if it can be written as a union of balls (with respect
to the metric).

Exercises for the werkcollege of the next Wednesday: 1.26, 1.27, 1.36, 1.35, 2.3, 2.13, 2.14, 2.15, 2.16.

**WEEK 47/Lecture 4 (November 20):** Reminder on topological spaces,
metric topologies, induced topologies. Then the notions of continuity, homeomorphisms, embeddings. Then neighborhoods,
basis of neighborhoods, 1-st countability, convergence, continuity at a point, sequential continuity: definitons,some
explanations
and I mentioned the relationship between the three notions of continuity: continuity is equivalent to continuity at
each point and, these also
imply sequential continuity; the converse of the last implication holds if the domain is 1-st countable. Next time we will return
to these implications and give some proofs.

Exercises for the werkcollege of the next Monday: 2.5, 2.30, 2.18, 2.35, 2.40, (2.19 and 22.31 and 2.41), (2.39 and then look again at 1.4).

Homework: Exercise 2.27 (please motivate your answers). Note about the marking process: in total there are 10 questions to be answered; you get 1 point per question if answered completely (and less if you do not motivate the answer).

**WEEK 48/Lecture 5 (November 25):** Reminder on neighborhoods,
convergence, continuity, basis of neighborhoods and 1st countability; then we discussed the proofs of the results that
relate the three notion of conitnuity. Then Hausdorffness, proved it for metric spaces then gave the characterization
in therms of uniqueness of limits; I poointed out that simple gluings may give rise to non-Hausdorff spaces. Then
basis of topological spaces and the 2nd countability axiom and the notion of topological manifold; pointed out that
2nd countability implies 1st countability, I mentioned that metric spaces may fail to be 2nd countable (although they all
are 1st countable) and then pointed out that subsspaces of Euclidean spaces, with the Euclidean topology, do satisfy the
2nd countability axiom. I also commented on the reasons that 2nd countability is required in the definition of topological
manifdolds. Then we moved to "Inside a topological space", giving the definition of interiors, closures and boundaries
of subsets of a topological space, stating the Lemma 2.37 (pointing out that it is important for computing, "in practice",
interiors and boundaries). The proof of the lemma will be discussed in the next lecture.

Exercises for the werkcollege of the next Wednesday: 2.44, 2.59, 2.60, 2.44, 2/45. 2.27, 2.19, 2.52 (part of which will be discussed on the lackboard), show that the n-sphere is a topological manifold, then do 2.12. If you have more time, do al;so: 2.22 and 2.50. If these are too "boring" and you want a more difficult exercise, you can try 2.63.

Bonus exercise: there is also a bonus exercise given this week: 2.62. Recall that this is NOT COMPULSORY; it is actually a rather difficult exercise (you should try it only after the werkcollege). If you solve it, you get extra-points on the very final mark (but to get that, you have to fully solve the exercise). You have two weeks to solve this one and hand it in (hence the deadline is Monday, December 9).

**WEEK 48/Lecture 6 (November 27):**
The proof of Lemma 2.37, examples. Then quotients with examples such as the projective space and the abstract torus
(with a comment on how compactness can be used to conclude that, in many cases, the continuous injections
that we get are automatically embeddings). Then very fast went over group actions and quotients modulo group actions.

Exercises for the werkcollege of the next Monday: 2.47, 2.48, 2.56, 2.50, 2.51, 2.61, 3.10, 3.2. Then you should try a very nice (but not so easy) exercise: 2.58.

Quizz number 2: the questions; and now with answers.

Homework: 2.53 (recall that the similar exercise 2.52 was explained, at least partially, to you on the blackboard. Please try to give similar arguments, with all details, nicely explained). In other words, in this homework, we value not only the solution but also the way it is explained (a proper mathematical proof). Correction: in the last part of the exercise, the correct definition of local finiteness is the one between the brackets. In other words, instead of "any point x in X admits a neighborhood V which intersects all but a finite number of A_i's (i.e. such that ...", please read "any point x in X admits a neighborhood V which intersects only a finite number of A_i's (i.e. such that ...".

**WEEK 49/Lecture 7 (December 2):** Quotients modulo group actions
(examples: the circle and the torus), product topology, collapsing a subspace to a point, cylinders, cones and suspensions of
a space.

Exercises for the werkcollege of the next Wednesday: Read 4.2 (pp 53) and 3.15 (pp. 57-58) of the lecture notes. Then: 3.11, then finish 3.10 from last time if you did not do it already, 3.14, 10.3 part (i), 10.35 part (i) (you can try also (ii), but then you should do first 3.13), 3.18, 3.23, 10.11 part (iv), 3.26.

Bonus exercise: show that the cone of the open interval (0, 1) cannot be embedded in any Euclidean space.

**WEEK 49/Lecture 8 (December 4):** Motivated by the
construction of metric topologies, the lower limit topology and the product topology, we introduced/discussed the notion
of topology basis. Then "spaces of fucntions" were discussed.

Exercises for the werkcollege of the next Monday: finish what you did not do from the last homework; then do: 3.29, then (2.16+ 2.17+ 3.30), 3.31, 3.34, 3.35, 3.36.

Quizz number 2: the questions; and now with answers.

Homework: 3.33.

**WEEK 50/Lecture 9 (December 9):** The lecture will
be given by David Martinez Torres. The plan is to do section 1 of chapter 4 (connectedness).

Exercises for the werkcollege of the next Wednesday: 4.17, 4.13, 10.44, 4.15, then do 2.39 again, then 4.12, 4.20, 4.24.

Quizz number 3: the questions; and now with answers.

**WEEK 50/Lecture 10 (December 11):**
The lecture will be given by David Martinez Torres. The plan is to do a large part of section 2 of chapter 4 (compactness).

Exercises for the werkcollege of the next Monday: 4.29, do again 4.12 (can you find some new proofs now?) and then 4.39, 4.26, 4.28. 4.27.

Quizz number 4: the questions; and now with answers.

Homework: 10.60.

**WEEK 51/Lecture 11 (December 16):** Finished
compactness (up to sequential compactness, including it) then we gave the definition of local compactness and of
1-point compactifications (mentioning orally the uniqueness of the 1-point compactification).

Quizz number 5: the questions.

Exercises for the werkcollege of the next Wednesday: If you did not do 4.28, 4.27 last time, do them now. Then continue with: 4.30, 4.38, 4.33, 4.36, 4.31.

**WEEK 51/Lecture 12 (December 18):** Finnish with local compactness
and 1-point compactifications, and then state/prove the Urysohn lemma (Theorem 5.21, pp. 93-94 in the notes).

Exercises for the werkcollege of the Monday, January 6: 4.41, 4.42, 4.45, 4.47; if you have more time, try also 4.48 and 4.49.

Homework: 4.46.

**WEEK 2/Lecture 13 (January 6):**
Metrizability theorems.

Quizz number 6: the questions.

Exercises for the werkcollege of the next Wednesday: look back at the exercises you did not do yet (for the other parts of the course).

**WEEK 2/Lecture 14 (January 8):** Continuous
functions: algebraic structures (vectro space, algebra, *-algebra), algebraico-topological structures (metric, normed,
Banach space, Banach algebra, C^*-algebra); mentioned the Gelfand-Naimark theorem (to be continued in more detail next time).
Then explained the statement and gave the proof of Stone-Weierstrass theorem (with more details than in the notes).

Quizz number 7: the questions.

Exercises for the werkcollege of the next Monday: 8.2, 8.5, 8.6, 8.3, 8.8.

Homework: 8.4.

**WEEK 3/Lecture 15 (January 13):**
Plan: Gelfand-Naimark Theorem.

Exercises for the werkcollege of the next Wednesday: all from Gelfand-Naimark.

**WEEK 3/Lecture 16 (January 15):**
First part: review of the spectrum of an algebra + another way to prove the last part of the main theorem (a way that is not in the notes
and is useful for exercises). Second part: started with partitions of unity: main definitions and statements + explanations
and motivations.

Exercises for the werkcollege of the next Monday: 5.7, 5.14.

Homework: no homework.

**WEEK 4/Lecture 17 (January 20):** Continuation of finite
partitions of unity: recalled the statement, then gave an application that is not explained in the notes (which implies
in particular that any compact manifold can be embedded in some Euclidean space), then the proof (incl. the shrinking lemma).
Then we moved to arbitrary partitions of unity, defining local finitness, paracompactness and stating the theorem.
Plan: Partitions of unity II.

Exercises for the werkcollege of the next Wednesday: 5.4, 5.5.

**WEEK 4/Lecture 18 (January 22):**
Finish with partitions of unity, then answer questions.

**WEEK 5, January 29:** Final exam.

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.

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