Topology (Inleiding Topologie)
Blok 2, 2024/2025
This is the web-site for the course "Inleiding Topologie" (Wisb243) given in blok 2, Winter 2024/2025.
We will try to keep here all the practical informations about the course, updates/changes that take place during the year, extra-material etc.
But, before we move to the course, please reflect about the possibility of adding an educational minor to your bachelor!!!! It takes only half a year extra (but students can do it in their 'profileringsruimte') and it comes with quite a few advantages such as:
- With that in hand, you always have the possibility to consider a job as a math teacher, which may be a lot of fun (anything but boring, making a really impact, with a lot of room for new ideas and for doing things better and differently). Furthermore, math teachers are always in demand, hence finding a nice job (even just to try it out for a year) should be relatively easy.
- It would be great if some of the high school teachers (even if just doing it temporarily) would know what a topological space is. And what a group is etc etc!
- And, even if already know that you will never want to become a teacher (...), you certainly gain new skills that are appreciated on the job market (e.g., in presenting).
If you are interested you can have a look at the page Educatieve minor bèta.
UPDATES:
November 7, 2024: unfortunately, there seems to be a delay with printing the new dictaat. The last promise that I heard is that they will be ready on November 11.
September 30, 2024: the new updated dictaat has been sent to the printer, and can also be downloaded here. It consists of two parts, part 1, with the theory, and part 2, with exercises.
September 30, 2024: This page was launched!
THE LECTURES:
- Tuesdays: 11:00-12:45. (RUPPERT - PAARS)
- Thursdays: 17:15-19:00 (RUPPERT - PAARS except for December 5th, when it is in KBG-Pangea).
LECTURER: Marius Crainic.
THE WERKCOLLEGES :
- Tuesdays: 09.00 - 10.45; at various locations, depending on the date and on the group (please see below and/or consult your mytimetable).
- Thursdays: 15:15 - 17:00; at various locations, depending on the date and on the group (please see below and/or consult your mytimetable).
TEACHING ASSISTANTS:
Group 1: Anna Fokma and Jelle Bloemendaal.
Group 2: Anouk Eggink
Group 3: Maite Carli
Group 4: Douwe Hoekstra
WERKCOLLEGE ACTIVITIES:
Tuesday morning quizzes: each Tuesday werkcollege starts with a 10 min quiz followed by 15-20 min discussion of the quiz. The quiz checks the understanding of some of the most basic concepts from the material of the week before. They are meant to help the students detect their weak points and remedy them. Therefore the quizzes will not be graded be the TA, only discussed with the TAs. On the other hand we will keep track of the students that do the quizzes, and that may play a role in rounding the marks, in the borderline cases (see below).
Thursday homework discussion: each Thursday starts with 15-30 min discussion of the homework that you have just handed in (see below).
Mind maps: after lecture 6 and after lecture 12, in the Tuesday werkcolleges that follow those lectures you will do, together with your colleagues and TAs, a small recap of the main concepts in the interactions between them.
Ordinary werkcollege: the remaining time of the werkcollege (at least 1x45 min per session) it will be ``ordinary werkcollege'': you solve exercises that were assigned to you and you can ask the TA for feedback and/or help. The exercise assigned to each werkcollege will be announced, on this web-page, at the end of the lecture preceding the werkcollege.
HOMEWORKS:
Each Thursday afternoon you will receive a homework that you have to "hand in" by the Thursday that follows, before the werkcollege.
In principle, the homeworks will consists of simpler exercises that should make sure you understood the basic concepts. The emphasis of the homework may differ from homework to homework- e.g. in some cases some points will be awarded especially for the style and the properness of the explanations (but you will be announced in advance).
Passing the course: that will be subject to some minimal requirements, and the final marks will depend on several items:
- Hand in exercises- see above. The average of the marks for all the homeworks will give one mark HW (maximum 10).
- Bonus exercises: there may be a couple of Bonus exercises, which are considerably harder exercise for which, if solved correctly, you can receive a number of points that will be added to your homework marks (in each case we announce it more precisely how it counts).
- Final exam: there will be a "final exam", that will give you one mark, E (maximum 10).
- Minimal requirements: at least 5 for the average of the Homeworks, at least 5 for the Final Exam and the usual requirement that the final mark (see the formula below) is at least 6. Please be aware that, for the requirement of "at least 5", there will be no automatic rounding up for marks below 5 (e.g. 4.95 will not be a passing mark). On the other hand, in principle, for marks above 5, the rounding will go to the closest integer or integer.5. For instance, 6.72 is rounded to 6.5, while 6.77 to 7. In some limit cases, the presence at the exercise classes will be decisive in how the rounding is done.
- Final mark: is computed using the average of the homework marks HW (and we add the bonus points there as well), the mark E for the final exam , according to the formula min{10, max{(7 E+ 3 HW)/10, (17E+ 3H)/20}}?)
- Retake: the same rules apply for the retake.
LECTURE NOTES: We will be using the freshly upgraded version of the lecture notes: part 1, with the theory, and part 2, with exercises.
THE SCHEDULE WEEK BY WEEK (I will try to 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 12): An introduction to the keywords of the course, and first attempts to make precise the concept of "spaces". And some concrete examples: circles, cylinder, the Moebius band, etc, in various incarnations ("models"). A lot of hand-waving.
Blackboard photos of Lecture 1.
Werkcollege rooms for this day:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
For todays's werkcollege: read pages 3-4 (basic definitions on metric spaces, to refresh your memory), Exercise 1.4, Exercise 1.7, Exercise 1.8, Exercise 1.11, then read/understand the explicit formulas for the torus (1.6.1) and the Moebius band (1.5.1) and then do 1.15 and 1.16.
Coming next:
- The plan for the next lecture (November 14) is: to give the precise definition of topological spaces and look at first examples.
- For the next werkcollege (November 14) the exercises are: finish what is left over from the previous time, then move on to 1.9, 1.14, 1.18 (and maybe 1.19 as well), then 1.48, 1.49, 1.47.
- The rooms for the next werkcollege (November 14) are:
- Group 1: KBG - 208
- Group 2: BBG - 165
- Group 3: DALTON 500 - 2.08
- Group 4: BBG - 205
♣ WEEK 46/Lecture 2 (November 14): Definition of topology, topological spaces, opens/closed/open neighborhoods and mentioned in the passing the concepts of convergence and continuity (to be discussed in more detail later on) and then examples: the trivial, discrete, co-finite and co-countable topologies on any set $X$, metric topologies (any metric induces a topology), various topologies on $R$ such as the Euclidean topology and the lower limit topology, the Euclidean topology on subspaces of $R^n$, and induced topologies.
Blackboard photos of Lecture 2.
Here is the Homework 1 (compulsory, to be handed in by Nov 21 before the werkcollege) as well as a Bonus-to-homework 1 (now with the correct link)(not compulsory, to be handed in by Nov 24 in the evening). You can upload your solutions via the blackboard (we will create the necessary blackboard environments before the deadline). Person in charge: Anna Fokma for the homework and Jelle Bloemendaal forthe bonus one.
Coming next:
- The plan for the next lecture (November 19) is: continuity and convergence.
- The exercises for the next werkcollege (November 19) are: 10.1, 10.3, 10.6, 2.12, 2.16, 2.14, 10.15, 10.8 (try to do as many as you can, make sure you do at least 10.3, 10.6, 2.16 and 10.8 or 10.15)
- The rooms for the next werkcollege (November 19) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 47/Lecture 3 (November 19): continuity, continuity at a point, convergence, sequential continuity, the theorem relating all of these. Bases of neighborhoods, 1st countability, then completing the theorem with "if and only if" in all directions. Various examples along the way. See the blackboards.
Blackboard photos of Lecture 3. But please be ware that the order for reading the blackboards is not linear. You see on some pages some blue numbers: those say in which order to read what: first blue 1, then blue 2 (on another blackboard), then jump to blue 3, etc etc.
Coming next:
- The plan for the next lecture (November 21) is: TBA
- The exercises for the next werkcollege (November 21) are: 10.17, 10.20,
10.16 where I recall that for subsets of $\mathbb{R}^n$, of we do not specify what the topology we use, it means we use the Euclidean topology (hint for this: do not forget Example 2.20),
10.34, 10.35, 10.21, 10.36,
10.66 (points b and c),
10.23. Please keep in mind: for metric spaces $(X, d)$ we have seen that continuity and convergence in the sense you learned in analysis is equivalence to the new notions that we introduced, applied to the topology $\mathcal{T}_d$ associated to $d$. In particular, when looking at subsets of Euclidean spaces ($X\subset R^n$) endowed with the induced (= Euclidean) topology, you can just use what you know about continuity and convergence from Analysis. For instance, polynomial or other elementary (standard) functions from $R^n$ to $\R$ are continuous..
- The rooms for the next werkcollege (November 21) are:
- Group 1: KBG - 224
- Group 2: BBG - 165
- Group 3: DDW - 1.32
- Group 4: BBG - 205
♣ WEEK 47/Lecture 4 (November 21):: homeomorphisms, topological properties (like 1st countability of metrizability), embeddings, mentioning along the way three central (but hard) questions:
- how can we decide whether two given spacxes are homeomorphic or not
- how to tell when a space is metrizable?
- how to tell when a space can be embedded in some Euclidean space?
Then moved on to bases and 2nd countability, with some examples. Then the notion of interior and closure inside a topological space.
Blackboard photos of Lecture 4.
Homework: here is the Homework 2. Person in charge of this one: Maite Carli.
Coming next:
- The plan for the next lecture (November 26) is: TBA
- The exercises for the next werkcollege (November 26) are (try to do as many as possible, in this order): 10.53, 10.24, 10.25,
10.55,10.56, 10.39, 10.43, 10.30, 10.47. Then choose one of the properties the discussed from the following list: "it is metrizable, "it is Hausdorff'',"t is 1st countable'', "it is second countable'' and prove that it is a topological property. If you have more time do one more, then look at and 10.65, 10.28.
- The rooms for the next werkcollege (November 26) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 48/Lecture 5 (November 26): Quotient topologies and topological quotient maps, product topologies, topology bases. I.e. 3.1, 3.6 and 3.7 from the dictaat.
Blackboard photos of Lecture 5.
Coming next:
- The exercises for the next werkcollege (November 28) are: 11.31, 11.30 the question about Hausdorff, 11.39. If you have more time (in the werkcollege, or at home, or later in the year if you want to revise this part): also 11.32, then the part about 11.30 on metrizability. If you want more for home: look at 11.41, 11.33.
- The plan for the next lecture (November 28) is: generating topologies (part of 3.8), spaces of functions (part of 3.9) then move to equivalence relations and quotients (3.2).
- The rooms for the next werkcollege (November 28) are:
- Group 1: KBG - 208
- Group 2: KBG - 224
- Group 3: DALTON 500 - 3.09
- Group 4: BBG - 205
♣ WEEK 48/Lecture 6 (November 28): Reminder on the topology associated to a topology basis and the comparison lemma (Lemma 3.48), then generated topologies (i.e. the first part of 3.8), then spaces of functions (section 3.9) but without discussing in detail the Peano curve. Then a reminder on quotient topologies and topological quotient maps (from last time, section 3.1) then a very brief advertisement of what will come next week: gluing and quotients, special quotients, etc, i.e., sections 3.2, 3.3, etc.
Blackboard photos of Lecture 6.
Homework: Homework 3.
Coming next:
- The plan for the next lecture (December 3) is: gluing and quotients from section 3.2 and special quotients from section 3.3. .
- The exercises for the next werkcollege (December 3) are:
- going to the discussion of the topology $\mathcal{T}_{cp}$ of compact convergence, prove that the collection of all sets of type $B_K(f, \epsilon)$ (denoted $\mathcal{B}_{cp}$ on page 5 of the blackboard of the last lecture) is, indeed, a topology basis.
- Then do 11.48, 11.1.
- Then do the following: show that if $f: X\to Y$ is both a topological quotient map as well as a bijection, then it is a homeomorphism.
- And if you have more time look at 11.42.
- The rooms for the next werkcollege (December 3) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 49/Lecture 7 (December 3): Recap on topological quotient maps and quotient topologies, then equivalence relations $R$ as a way to encode gluing, the concept of ``a quotient of $X$ modulo $R$" and its topological version, with some examples, then the abstract (topological) quotient, then the prop that they are all "the same". With some examples. Then briefly about some special quotients (collapsing a subset to a point, cone, suspension). I.e. section 3.2 and very fast 3.3 (to which I will return a bit next time).
Blackboard photos of Lecture 7.
Coming next:
- The plan for the next lecture (December 5) is: recap some of the stuff we went fast through last time, then group actions and quotients modulo group actions (section 3.4).
- The exercises for the next werkcollege (December 5) are: recap some of teh discussion from last time by reading Examples 3.16, 3.21 and 3.24 (all about Moebius). Then prove that the map from Example 3.18 is a topological quotient map. Then 11.3, 11.6. You can also have some fun with 11.4 (playing on the picture). If you have more time: 11.5.
- The rooms for the next werkcollege (December 5) are:
- Group 1: KBG - 224
- Group 2: BBG - 165
- Group 3: BBG - 319
- Group 4: BBG - 322
♣ WEEK 49/Lecture 8 (December 5): Basically just section 3.4 plus a couple of examples. In a bit more detail: some groups, then actions of groups, quotients modulo group actions, with various examples and, at the end, one examples with "proper proofs" (I described two possible proofs). Please make sure you understand those two proofs: read them/work them out until they are clear, then put everything aside, take a piece of paper and try to do them on your own!
Blackboard photos of Lecture 8..
Homework: Homework 4.
Coming next:
- The plan for the next lecture (December 10) is: a reminder on quotients, maybe with 1-2 more examples, then we move to the next chapter, namely connectdedness (section 4.1).
- The exercises for the next werkcollege (December 12) are: first do what I mentioned in the description of the course (see above). Then do 11.15 and 11.16. If you have more time: look at 11.25.
- The rooms for the next werkcollege (December 10) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 50/Lecture 9 (December 10): Recap on quotients and actions, with a couple of extra-examples (Klein bottle and projective spaces as quotients modulo group actions) then we started talking about connectedness: definition (also for path-connected), the first results and the thm that path connected implies connected, and some examples.
Blackboard photos of Lecture 9.
Coming next:
- The plan for the next lecture (December 12) is: finish with connectedness and start with compactness.
- The exercises for the next werkcollege (December 12) are: 12.5, 12.13, 12.10, 12.15. If you have more time or you want an extra-challenge, look at 12.9.
- The rooms for the next werkcollege (December 12) are:
- Group 1: KBG - 208
- Group 2: KBG - 224
- Group 3: DDW - 1.32
- Group 4: BBG - 165
♣ WEEK 50/Lecture 10 (December 12): Recap on connectedness, with some applications, then discussing connected components and the resulting partition (parts of section 4.1). Then started discussing compactness (definition and first examples- basically the first page of section 4.2). .
Blackboard photos of Lecture 10.
Homework 5..
Coming next:
- The plan for the next lecture (December 17) is: continue with compactness- basic properties (hopefully finish section 4.2, or almost finish it).
- The exercises for the next werkcollege (December 17) are: make sure you did the exercises from the previous werkcollege. Then look at 12.8, then at 12.11 or 12.9, then at 12.48.
- The rooms for the next werkcollege (December 17) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 51/Lecture 11 (December 17): The basic properties of compactness: discussed the main statements from section 4.2 (except for sequential compactness), some corollaries/applications, and then quite a few proofs. See the blackboard fotos.
Blackboard photos of Lecture 11.
Coming next:
- The plan for the next lecture (December 19) is: end the discussion of compactness, sequential compactness, then move to local compactness and 1-point compactifications.
- The exercises for the next werkcollege (December 19) are: 12.19 and 12.21, 12.35 or 12.27(b), 12.40 and 12.41 (maybe leave one for home), 12.32. If you have more time: 12.26.
- The rooms for the next werkcollege (December 19) are:
- Group 1: KBG - 208
- Group 2: KBG - 224
- Group 3: MIN - 0.14
- Group 4: BBG - 165
♣ WEEK 51/Lecture 12 (December 19): parts of 4.2 and 4.3. More precisely, we finished the discussion on compactness and sequential compactness. Then looked at the notions of 1-point compactifications and local compactness, with various examples/remarks, stating the existence and uniqueness theorem- Theorem 4.50- to be proven next time.
Blackboard photos of Lecture 12.
Homework 6 plus a bonus exercise..
And here is a list with some recap exercises (my advise is to try to do the first one, at least, as it takes you through several of the concepts that we have discussed).
Coming next:
- The plan for the next lecture (January 7) is: TBA
- The exercises for the next werkcollege (January 7) are: TBA.
- The rooms for the next werkcollege (January 7) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 2/Lecture 13 (January 7): TBA.
Blackboard photos of Lecture 13 (not available yet).
Coming next:
- The plan for the next lecture (January 9) is: TBA.
- The exercises for the next werkcollege (January 9) are:
- The rooms for the next werkcollege (January 9) are:
- Group 1: KBG - 224
- Group 2: BBG - 165
- Group 3: MIN - 0.14
- Group 4: BBG - 205
♣ WEEK 2/Lecture 14 (January 9): ITBA
Blackboard photos of Lecture 14 (not available yet).
Homework (not available yet).
Coming next:
- The plan for the next lecture (January 14) is: TBA
- The exercises for the next werkcollege (January 14) are: TBA.
- The rooms for the next werkcollege (January 14) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 3/Lecture 15 (January 14): TBA.
Blackboard photos of Lecture 15 (not available yet).
Coming next:
- The plan for the next lecture (January 16) is: TBA.
- The exercises for the next werkcollege (January 16) are: TBA.
- The rooms for the next werkcollege (January 16) are:
- Group 1: KBG - 208
- Group 2: KBG - 224
- Group 3: DDW - 1.36
- Group 4: BBG - 165
♣ WEEK 3/Lecture 16 (January 16): TBA.
Blackboard photos of Lecture 16 (not available yet).
Homework (not available yet).
Coming next:
- The plan for the next lecture (January 21) is: TBA.
- The exercises for the next werkcollege (January 21) are: TBA/.
- The rooms for the next werkcollege (January 21) are:
- Group 1: BBG - 017
- Group 2: BBG - 103
- Group 3: MIN - 0.11
- Group 4: DDW - 0.42
♣ WEEK 4/Lecture 17 (January 21): TBA
Blackboard photos of Lecture 17 (not available yet).
Coming next:
- The plan for the next lecture (January 23) is: TBA
- The exercises for the next werkcollege (January 23) are: I think you have quite a few left from last weeks (including the page with exercises to prepare for the exam).
- The rooms for the next werkcollege (January 23) are:
- Group 1: KBG - 208
- Group 2: KBG - 224
- Group 3: BBG - 115
- Group 4: BBG - 165
We zijn er Bijna ...
♣ WEEK 4/Lecture 18 (January 23):
Blackboard photos of Lecture 18 (not available yet).
♣ EXAM: Thursday, January 30, 2025, 13:30-16:30, EDUC - BETA
♣ RETAKE: April 17, 2024.
IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. I.E., IF YOU MISS A CLASS, PLEASE KEEP IN TOUCH WITH YOUR COLLEAGUES THAT WERE PRESENT, JUST TO MAKE SURE YOU STAY INFORMED.