"Inleiding Topologie" 2015/2016 (blok 2)

Change: the werkcollege groups will be merged: group 2 will be absorbed by group 1. Hence the werkcolleges will takes place in the rooms assigned to group 1. This change will be implemented starting with Monday, December the 7th.

The results of the retake exam!
Here are the exam grades and final grades (plus some comments). For those of you that have passed, but I do advise to do the retake: the retake cannot lower your mark (I will take the maximum between the first and the second). Note also that I plan to communicate the marks to the amdinistration (hence be placed in Osiris) only after the retake. So, if for some reasons you need your mark in Osiris before that, please let me know.

Here is the new set of .lecture notes, to be used for the entire course (posted on November 10, 2015). Note that some of the pictures use colours.

This is the web-site for the course "Inleiding Topologie" for the year 2015-2016 (blok 2, Fall 2015). 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 .

When, where, etc:


------ Mondays: 17:15 - 19.00, KBG, room: ATLAS.

------ Tuesdays: 17:15- 19:00, RUPPERT, room: PAARS.

LECTURER: Marius Crainic.

TEACHING ASSISTANTS: The students are divided into two groups (this is relevant for the werkcolleges and to know to whom you should hand in the exercises). Group 1: Davide Alboresi (D.Alboresi(tra-la-la)uu.nl) and Maarten Mol (m.mol1(tra-la-la)uu.nl). Group 2: Francesco Cattafi (f.cattafi(tra-la-la)uu.nl) and Ilja Nelen (i.f.m.m.nelen(tra-la-la)students.uu.nl).


------ Mondays: 15.15 - 17.00.

------ Wednesdays: 9.00 - 10.45.

but please be aware that the location depends on the group you belong to and on the week; please look at the rooster in osiris. 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).


Each Tuesday you will receive a homework that you have to hand in one week later, when you arrive at the werkcollege on Monday (hence: you are not allowed to do the homework during the werkcollege!!!). If you are not present at the werkcollege, 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, unless you aggree with the TAs that you can do that. In any case: please never send the homework to me (the lecturer) by email!!!

Marks: there will be:

- hand in exercises, once per week (every Monday). You will receivce the exercise at the end of the class on Tuesday. You have to hand it in one the next Monday, when you arrive at the werkcollege. 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}

Lecture notes: Here is the new set of .lecture notes, to be used for the entire course (posted on November 10, 2015). Note that some of the pictures use colours.

Please be aware that the lecture notes may 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.

The schedule week by week (here we will add, after each lecture, a description of what was discussed in the lectures + the exercises from the lecture notes that you are supposed to do during the werkcollege):

- WEEK 46/Lecture 1 (November 9): Key-words of topology; the inuitive notion of space (with the precise versions of metric spaces, but arguing that metric spaces are not satisfactory). Homeomorphisms. Various (homeomorphic) models of the circle.

Exercises for the werkcollege (for the coming Wednesday and Monday): 1.1, 1.2, 1.19, 1.17, 1.13, 1.18, 1.25, 1.27.

- WEEK 46/Lecture 2 (November 10): Decided to recall first the precise notion of equivalence relations, defined the notion of "a quotient" of X modulo a (given) equivalence relation, and then the abstract quotient (all of these with intuitive/geometric examples). Then we returned to the torus, models for it, mentioned the Moebius band, Klein bottle and then, at the very end, discussed intuitively how the projective plane can be obtained from a square by performing a certain gluing similar to the other examples."

Exercises for the werkcollege (for the coming Wednesday and Monday): 1.15, 1.29, 1.31, 1.32, 1.28, 1.11, 1.36, 1.37.

Homework: 1.30.

- WEEK 47/Lecture 3 (November 16): Introduced the notion of topological space + remarks on the definition+ various examples: the trivial topology, the discrete topology, the co-finite topology, the co-countable topology (all these on any set $X$!), the topology metric topology (associated to a metric), the Euclidean topology, then looked more carefully at the Euclidean topology on $R$ and introduced also the lower limit topology on $R$ (which appear in the notes only in the exercises). Also: the induced topology on a subspace of a topological space. Then the definition of continuity.

Exercises for the werkcollege (for the coming Wednesday and Monday): make sure you finish 1.36, 1.37 from last time. Then do 2.16, 2.18, 2.19, 2.27, 2.26, 2.29, 2.31, 2.21

- WEEK 47/Lecture 4 (November 17): Neighborhoods, convergence of sequences, continuity, continuity at a point, sequential continuity, basis of neighborhoods, 1st countability + the relationship between the three types of continuity. Then defined the interior and the closure of a subspace of a topological space (next time we will start with the very very important criteria given in Lemma 2.37).

Exercises for the werkcollege (for the coming Wednesday and Monday): 2.31, 2.45, 2.46, 2.30, 2.40, 2.49, 2.39, 2.5, 2.8.

Homework: Exercise 2.44 (the removing one point trick).

- WEEK 48/Lecture 5 (November 23): interior, closure + the criteria from Lemma 2.37, Hausdorffness, second countability, topological manifolds, the stereographic projection.

Exercises for the werkcollege (for the coming Wednesday and Monday):2.30, 2.50, 2.53, 2.58, 2.62, , 2.67, 2.68.

- WEEK 48/Lecture 6 (November 24): Quotient topology; how to handle "concrete models"+ examples. In particular, I mentioned the main properties of compactness, for now just as a tool-box to use to prove that certain injective continuous mpas are actually embeddings (if you missed the lecture, please have a look at thenotes of one of your colleaues; it will really make some proofs easier!). Than I introduced the product of two topological spaces (trying to explain the definition); I will return to this next time. ยง

Exercises for the werkcollege (for the coming Wednesday and Monday): 3.13, 3.17, 3.34, 3.29, 3.27.

Homework: Exercise 3.41.

- WEEK 49/Lecture 7 (November 30): Recalled metric topology, lower limit topology and then (in analogy with these) we finished with the product topology. Then we looked at special classes of quotients: I-group actions, II-collapsing a subspace to a point, cones, suspensions.

Exercises for the werkcollege (for the coming Wednesday and Monday): start with 3.4, 3.35 and make sure that you did 3.34 from the previous werkcollege. Then do again 1.11 and concldude that the space obtained from the disk by collapsing its boundary circle to a point is homeomorphic to the 2-sphere; then 3.15, 3.20, 3.23 and 3.24. If you have time, look also at 3.26.

- WEEK 49/Lecture 8 (December 1): Constructions of topologies starting with a topology basis on a set. Then we looked at several examples/exercises (from the previous exams). At the end I also discussed subsection 8.1 from the lecture notes.

Exercises for the werkcollege (for the coming Wednesday and Monday): 2.37, 2.38. Thene there are a few long exercises on topology bases, from the previous exam. I recommend you to do: 3.42, 3.44, 3.45.

Homework: Exercise 3.43, parts a, b, e and h.

- WEEK 50/Lecture 9 (December 7): Connectedness.

Exercises for the werkcollege (for the coming Wednesday and Monday): 4.11, 4.12, 4.13, 4.14, 4.15, 4.17, 4.22.

- WEEK 50/Lecture 10 (December 8): Compactness: definition + stated all the main theorems, and then starting proving them. Left to prove: that [0, 1] is compact and the product of two compacts is compact.

Exercises for the werkcollege (for the coming Wednesday and Monday): there are three different types of exercises to look at:

- 4.28, 4.53, 4.51, 4.52

- 4.45, 4.46, 4.47

- 4.30, 4.31, 4.41, 4.39

Please make sure that you do at least two from each type.

Homework: 4.50.

- WEEK 51/Lecture 11 (December 14): reviewd the main theorems regarding compactness from last time, then we finished the proofs. Then also discussed sequential compactness + a bit on 1-point compactifications.

Exercises for the werkcollege (for the coming Wednesday and Monday): please make sure that you finish all the exercises from the previous werkcolleges.

- WEEK 53/Lecture 12 (December 15): locally compact spaces; 1-point compactification.

Exercises for the werkcollege (for January 4th): 4.59, 4.63, 4.64, 4.69, 4.65, 4.62.


- WEEK 1/Lecture 13 (January 4): Discussed the "space of observables" on a compact Hausdorff space $X$, i.e. the set $C(X)$ of scalar-valued continuous functions on $X$, where the field of scalars was either the real or the complex one. We discussed the various structures present on $C(X)$: algebraic (vector space, algebra and *-algebra when working over complex numbers), topological (the sup-metric) and the combinations of the two (normed vector space/Banach space, Banach algebra and $C^*$-algebra). Then we passed to the Stone-Weierstrass theorem, giving the statement, explaining it and discussing 1/2-2/3 of the proof (to be continued tomorrow).

Exercises for the werkcollege (for the coming Wednesday and Monday): 8.1, 8.2, 8.9, 8.8, 8.3, 8.4.

- WEEK 1/Lecture 14 (January 5): Finish the proof of the Stone-Weierstrass then we started with Gelfand-Naimark, up to the point of defining what characters are (however, what I explained inthe class is more than what is in the lecture notes so, if you missed the class, it may be a good idea to ask your colleagues for their notes).

Exercises for the werkcollege (for the coming Wednesday and Monday): make sure that you are done with all the exercises from compactness, local compactness and 1-point compactifications and the stone-Weierstrass.

- WEEK 2/Lecture 15 (January 11): Recalled the discussion on Gelfand-Naimark, then we finnished the proof of the main theorem. The presentation was a bit different than in the lecture notes- in the sense that we made some separate renarks that are very useful also for exercises. Then we did a few exercises/examples of computing the topological spectra of various algebras: of $R$, of $R[t]$, of $R[t, s]$, then I mentioned the general construction of the algebra of polynomials modulo a polynomial and we discussed $R[t}/(t^2-1)R[t]$, $R[t}/t^2R[t]$, $R[t}/(t^2+1)R[t]$, $R[t, s]/(t^2+ s^2- 1) R[t s]$.

Exercises for the werkcollege (for the coming Wednesday and Monday): 8.12, 8.13, 8.14, 8.23,

- WEEK 2/Lecture 16 (January 12): Urysohn lemma and Urysohn metrization theorem.

Exercises for the werkcollege (for the coming Wednesday and Monday): finish all the exercises from Stone-Weierstrass and Gelfand-Naimark. Then do 5.17.

- WEEK 3/Lecture 17 (January 18):

- WEEK 4/Lecture 18 (January 19):

Exam: January 25th (Monday), 09.00 - 12.00, Gebouw: BEATRIX, Zaal: 7E ETAGE.


Enjoy the sphere (and not only).