Topology (Inleiding Topologie)
Blok 2, 2023/2024
Related to the exam:
Here are the exercises of the exam: in English and in Dutch.
Before I add the next link, I want to make one thing clear: I am really not a fan of making "the solutions" of exercises available to you, so that you can "practice the exercises". I think those are the wrong concepts if your aim is to learn. Of course, there is some value in having some material so that, if you are stuck, you can briefly consult to get some extra-idea. Also if you want to see some models of how one could possibly write down/explain a solution. The problem is that usually there are many possible solutions, each solution can be explained (equally well) in different way and, at the end of the day, the way something is explained/the solution one finds, is also rather personal...
... and, all that being said, since many of you have asked for such solutions, I am providing here some work outs of the exam questions. But please be aware: there are statements (in black), solutions (in green) plus comments (in magenta). For some questions I discuss several solutions/ideas/approaches. Also, please be aware that I am writing with a bit more words than one would normally do, with the aim of getting more clarity- so keep that in mind and don't get overwhelmed (one more reason I am not so happy doing this ...). But I hope it is of some help to you.
I am also looking into the idea of having a couple of extra exercise classes for those that are interested in doing the retake. More info will be posted here soon.
This is the web-site for the course "Inleiding Topologie" (Wisb243) given in blok 2, Winter 2023/2024. But, before anything else, I would like to start with the following note:
please reflect about the possibility of adding an educational minor to your bachelor!!!!
Pros:
It takes only half a year extra.
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. And 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).
Cons:
It does take half a year extra (well, I was pointed out that students can do it in their 'profileringsruimte', so it doesn't have to take longer!).
I can't think of any other cons!
If you are interested you can have a look at the page Educatieve minor bèta.
UPDATES:
We will try to keep here all the practical informations about the course, updates/changes that take place during the year, extra-material etc. The most recent additions are:
December 12: Homework 5 (last homework this year).
December 5: Hints for homework 3.
December 5: Quotients in the year 2023. This file is meant as a replacement of pages 25-27 and 52-57 from the current dictaat. (Since we did not spend too much time with the first part of the dictaat on gluings, both the handwaving ones as well as the discussion of equivalence relations and corresponding quotients, this file is better adapted to the way we discuss it in the class this year (and, as a matter of fact, also the next year ...)).
November 16: Pictures of what was written on the blackboards during lecture 2
November 16: Proof of the claim that two metrics on a set $X$ give rise to the same convergent sequences if and only of they induced the same topology.
nd also homework 1.
November 9: the old lecture notes.
November 9: This page was launched!
THE LECTURES:
- Tuesdays: 11:00-12:45. (RUPPERT - PAARS)
- Thursdays: 17:15-19:00 (KBG - PANGEA).
LECTURER: Marius Crainic.
THE WERKCOLLEGES :
- Tuesdays: 09.00 - 10.45; at various locations, depending on the date and on the group (please consult your mytimetable).
- Thursdays: 15:15 - 17:00; at various locations, depending on the date and on the group (please consult your mytimetable).
TEACHING ASSISTANTS:
Group 1: Anna Fokma, Jelle Bloemendaal
Group 2: Max Blans
Group 3: Samira le Grand
Group 4: Anouk Eggink
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.
Group problems: we will try to experiment once or twice with the idea of having "group problems" during the werkcolleges
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).
The marks that you receive for the homeworks will count for 20% of the final mark.
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 (typically 0.25-05 pts). The points that you receive in this way is sum up to a number of points, call it B.
- 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 quizzes will be decisive in how the rounding is done.
- Final mark: is computed using the average of the homework marks HW, the mark E for the final exam and the bonus points B, according to the formula: TO STILL BE DECIDED (min{10, B+ (4 E+ HW)/5}, or keep on using the old formula that has been used for many years now: min{10, B+ max{(7 E+ 3 HW)/10, (17E+ 3H)/20}}?)
- Retake: the same rules apply for the retake.
LECTURE NOTES: We will be using the old lecture notes although I will try to update them as the course proceeds.
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 14, RUPPERT - PAARS): What this course is about in comparison with other courses (main objects, main maps, isomorphisms and key-words), then adopted the temporary (sloppy) definition that "topological spaces are sets together with data that allows us to talk about CONVERGENCE (of sequences) and CONTINUITY (of functions)", and the first attempt (precise and pretty good) to model such "spaces": metric spaces. Then we looked at some examples, rising the red flag whenever metrics seemed not to be the best choice. We focused on circles (the unit one and various other models), cylinder (various models), the Moebius band (including a concrete model), then at the end I mentioned the torus and the Klein bottle as well (and mumbled something about the projective space). These is all concrete, and rather cool, but occasionally not so precise as we would like. Soon we will start doing things in a very precise way (and perhaps ... not so concrete as some would like ... but hopefully still cool).
Blackboard photos of Lecture 1 (partial) ... unfortunately something went wrong with the last fotos and they got lost (more precisely: the blackboard with the cyclinder, the blackboard with the Moebius band, and the blackboard with the torus and the Klein bottle).
Werkcollege rooms for this day:
The exercises for todays's werkcollege were: read page 7 (basic definitions on metric spaces, to refresh your memory),
Exercise 1.6, Exercise 1.7, Exercise 1.10, then read/understand the explicit formulas for the torus (6.1) and the Moebius band (5.1) and then do 1.14 and 1.15.
For the next werkcollege (Thursday) the exercises are: make sure you did the previous ones, then have some fun with 1.4, then move on to 1.8, 1.13, then 1.15 on a picture or 1.16 (or both), then move on and try to do as much as possible from 1.36 and 1.37. Those werkolleges are in the following roms:
♣ WEEK 46/Lecture 2 (November 16, KBG - PANGEA): Definition of topologies and topological spaces and then lots of examples:
Example 1-4 were the discrete, trivial, cofinite and cocountable topologies (defined on any set), then metric topologies, then we looked on the real line
where we discussed the Euclidean topology (induced by the Euclidean metric) and noticed that, in the various descriptions, the open intervals could be replaced
by intervals of type [a, b), or (a, b] (or [a, b], but that was less interesting). Using those of type one obtains a new topology, called the lower limit topology on \R (please be aware that this topology appear in the notes in an exercise at the end of the chapter). And we noticed the curious fact that the lower limit topology contains the Euclidean one. Then I mentioned the Euclidean topology on subsets $A$ of Euclidean spaces $\R^n$. Finally, the notion of induced topology, and we introduced the notion of ``B open in A'', and ``B closed in A''.
In other words, we mainly did pages 32 and 33 of the lecture notes. We stopped at Exercise 2.3. Here are the blackboard pictures of lecture 2.
On the other hand, our motivation was coming from the attempt of using metrics and the two red flags that we marked during the first lecture. The first one was that two different looking metric can give rise to the same notion of convergence. And I stated, without proof, that that happens if and only of the two metrics induce the same topology. For the interested students (which also has the time for it), you can see here a proof that I wrote in the blackboard after the lecture.
Here is the Homework 1.
The due date for this assignment is 23/11/2023 at midnight. Please submit it via blackboard, in a single pdf file (do not submit multiple scans!).
The plan for the next week:
- The plan for the next lecture (November 21) is to discuss convergence and continuity (sections 2 and 3 of this chapter, and hopefully start with section 4 as well).
- The exercises for the next werkcollege (November 21) are: Exercise 2.2, 2.3, 2.14, 2.18, 2.20, 2,27.
- The rooms for the next werkcollege (November 21) are:
♣ WEEK 47/Lecture 3 (November 21, RUPPERT - PAARS): Continuity, continuity at a point and sequential continuity (hence also neighborhoods and convergent sequences), basis of neighborhoods, 1st countability axiom and the main theorem saying: that (continuity) <==> (continuity at any point) ==> (sequential continuity at any point) , where the last implication is actually an equivalence if the domain of the function satisfies the 1st countability axiom.
Blackboard photos of Lecture 3.
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 plan for the next week:
- The plan for the next lecture (November 23) is: Recall the notion of basis of neighborhoods of a point, then talk about basis of opens and 2nd countability, Hausdorffness, interior and closure of a subset of a topological space.
- The exercises for the next werkcollege (November 23) are: Prove that if $f: (X, T_X)\to (Y, T_Y)$ is a continuous function, $A\subset X$ and we endow $A$ with the induced topology, call it $T_A$, then the restriction of $f$ to $A$, $f|_{A}: (A, T_A)\to (Y, \T_Y)$, is continuous as well. Then move to 2.30, 2.31, 2.46, 2.6, 2.32. Then do the following exercise: consider $A\subset R^n$, endowed with the induced Euclidean topology. We wonder about the property that, for any sequence $(a_n)_{n\geq 1}$ of elements of $A$, the sequence converges in the space $A$ (to some element of $A$) if and only if it converges in the space ^n$ (to some element of $R^n$). Show that this happens if and only if $A$ is closed in $R^n$. Then, if you still have time, you can have a look 2.13 and 2.36 (these two can also be done in the train/bus, if you are bored with doing sudoko ...) Or you can challenge yourself with 2.48. Anyway, do the first 3-4 and, after that, do as much as you can from the rest but without worrying that you must do them all.
- The rooms for the next werkcollege (November 23) are:
♣ WEEK 47/Lecture 4 (November 23, KBG - PANGEA): Reminder on neighborhoods, bases of neighborhoods, convergence, continuity (in all its versions and the theorem relating them), then made some comments about the usefulness of the notion of basis of neighborhoods and 1st countability: when local properties are usually required to be checked "for all neighborhoods of $x$", very often it suffices to check them for all $V$ in a basis only; also, the slogan that 1st countability is the hypothesis under which sequences can be used in their full power. Then moved to Hausdorfness, proved it is equivalent to the uniqueness of limits (... if the space is 1st countable)and noticed metric topologies are always Hausdorff. Then moved to the notion of "basis of a topological space" and 2nd countability, looked at some examples (Euclidean topology, lower limit topology), related them to basis of neighborhoods, we explained that a subspace of a 2nd countable space is 2nd countable and, therefore, any subset $A\subset \R^n$ endowed with the Euclidean topology is automatically 2nd countable. Then I defined what the interior and the closure of a subset in a topological space is, with some explanations, then stated the lemma which give a recognition criteria for when a point is in the interior or in the closure of $A$, respectively.
Here are the blackboard pictures of lecture 4.
Homework: here is the Homework 2.
Please submit this assignment before 30/11/2023 at 15:00 (i.e. before the werkcollege); for how to submit it, please follow the instructions in the file.
The plan for the next week:
- The plan for the next lecture (November 28) is: we start discussing constructions of topological spaces. We start with products and with topologies constructed via topology bases.
- The exercises for the next werkcollege (November 28) are:
- prove the claims made in Example 2.39,
- then do Exercise 2.49 (page 48),
- then prove one of the claims made in the lecture: that the collection of all open intervals $(p, q)$ with $p$ and $q$ rational numbers forms a (countable) basis for $\R$ endowed with the Euclidean topology (you may also try the other one that I mentioned, that intervals of type $[p, q)]$ with $p, q$ rational do not form a basis of $(\R, \mathcal{T}_l)$ (the real line with the lower limit topology),
- then 2.54 and 2.58.
- The rooms for the next werkcollege (November 28) are:
♣ WEEK 48/Lecture 5 (November 28, RUPPERT - PAARS): After the reminder we moved to Chapter 3, starting with the product topology (section 5 of that chapter) and then, motivated by that and other similar constructions (see the blackboard photos) we discussed the notion of topology basis (section 7) then, in the last part, generated topologies and initial topologies, as explained in section 8 of that chapter.
Blackboard photos of Lecture 5.
The plan for the next time:
- The plan for the next lecture (November 30) is: spaces of functions, then start with quotient topologies.
- The exercises for the next werkcollege (November 30) are: please do as many as possible from the following, in this order:
- make sure you did Exercise 2.58 and exercise 2.54 where, for the last one, please make sure you use Lemma 2.37.
- Exercise: 3.30 using Lemma 3.18
- Exercise: 3.4 (page 58)
- Exercise 2.55 (using Lemma 2.37) together with 3.32.
- Exercise 3.31,
- Understand Example 3.13 and do Exercise 3.35.
- Exercise: 3.41.
- A cute one is exercise 3.29, as it shows that the 3-sphere can be broken into two solid tori ...
- Then you can look at one (or more) of the exam exercises with topologies induced by a topology basis: 3.42, 3.44, 3.46, etc.
- ... and a more theoretical one: 3.34.
- The rooms for the next werkcollege (November 30) are:
♣ WEEK 48/Lecture 6 (November 30, KBG - PANGEA): Spaces of functions (pages 66 and 6), then started looking at quotient topologies (from page 52, mainly Theorem 3.1, and the example with the projective space from page 56).
Blackboard photos of Lecture 6.
Homework: here is the Homework 3.
Please submit this assignment before 07/12/2023 at 15:00 (i.e. before the werkcollege). Your solution may be handwritten (please write clearly) or latexed. Handwritten
solutions should be handed in physically, latexed solutions via blackbloard.
Update on the homework: the hand in deadline is now December 12, before the werkcollege, and here are some hints for homework 3.
The plan for the next week:
- The plan for the next lecture (December 5) is: continue with the sections related to quotient topologies (equivalence relations, examples, group actions).
- The exercises for the next werkcollege (December 5) are:
- for the various notions of convergences have a look at Exercise 3.50 and 3.51.
- Exercise 3.54.
- last time we discussed (see the blackboard picture) how starting with a set $Y$ and a surjective map $\pi: X\to Y$, where $(X, T)$ is a topological space
one induces a topology $\pi_*(T)$ on $Y$. When $Y$ was the projective space $P^n$, we used that discussion to put a topology on $P^n$: we choose $X$ to be $R^{n+1}- \{0\}$ and took $\pi$ to be the map that send a point $x$ in the Euclidean space (other than the origin) to the line though the origin and $x$. One of you said that we could also use $X= S^n$, and another one that we could use for $X$ to be the upper hemisphere. Please implement that (what is $\pi$ in each case?). In that way you get three topologies on $P^n$ (one from the lecture, and the other two from your colleagues). Try to prove that they are actually all the same.
- Exercise: define a surjective map from the square $[0, 1]\times [0, 1]$ to a concrete models of the 2-torus, sitting inside the Euclidean space $R^3$ . Use that map to induce a topology on the 2-torus. Try to prove that that topology coincides with the Euclidean topology.
- If you have more time, return to the list of exercises from the previous werkcollege and try to finish those.
- The rooms for the next werkcollege (December 5) are:
♣ WEEK 49/Lecture 7 (December 5, RUPPERT - PAARS): Topological quotient maps/induced quotient topologies. With the example of projective spaces and Moebius bands. Then, with the aim of making gluing more precise: equivalence relations, quotients and topological quotients, the abstract quotient and the abstract topological quotients, then the proposition saying the all quotients are isomorphic to the abstract one, with consequences on the Moebius band. And I encouraged you to look at Proposition 0.24/0.25 (same proposition really).
NEW MATERIAL : Here is the discussion of quotients that is better adapted to the way we discuss it in the class this year (and hopefully also next years):
Quotients in the year 2023. This file is meant as a replacement of pages 25-27 and 52-57 from the current dictaat.
Blackboard photos of Lecture 7.
And here are some hints for homework 3
The plan for the next week:
- The plan for the next lecture (December 7) is: reminder on quotients and Proposition 0.24/0.25, then more special classes of quotients: collapsing subspaces, quotients modulo group actions.
- The exercises for the next werkcollege (December 7) are:
- from the "Quotients in the year 2023": look at the Examples 0.12, 0.14, 0.22 and check what is stated there.
- from the dictaat: Exercise 3.10, 3.13, 3.17.
- By a solid torus is a full donut (you hold not a circle but a disk and rotate 360 degree holding it in your hand). It is homeomorphic to $S^1\times D^2 $ (the circle product the disk), and it is usually pictured inside the 3-dimensional Euclidean space. Its boundary is precisely the torus. Cool thing: if you take two solid tori there is a way to glue them along their boundary tori so that ... the outcome is the 3-dimensional sphere! If you want to see that, do exercise 3.29!
- The rooms for the next werkcollege (December 7) are:
♣ WEEK 49/Lecture 8 (December 7, KBG - PANGEA): Recap on quotients. Then special quotients: collapsing a subspaces, cyclinders, cones and suspensions, with various examples, then moved to group actions and the quotient modulo a group action.
Blackboard photos of Lecture 8..
Homework (to be handed in by December 14, before the werkcollege): Homework 4.
The plan for the next week:
- The plan for the next lecture (December 12) is: Finish with quotients, then start with connectedness.
- The exercises for the next werkcollege (December 12) are: 3.11, 3.12, Example 0.29, Exercise 3.23, 3.15, 3.21, 3.13.
- The rooms for the next werkcollege (December 12) are:
♣ WEEK 50/Lecture 9 (December 12, RUPPERT - PAARS): We finished with quotients modulo group actions and then we talked about connectedness (pages 78 and 79, and also added the example of $R$ with the lower limit topology).
Blackboard photos of Lecture 9.
The plan for the next time:
- The plan for the next lecture (December 14) is: discuss connected components and some more applications of connectedness (using also the "removal one point trick") then move on and start discussing compactness.
- The exercises for the next werkcollege (December 14) are: TBA.
- If you did not do 3.13 last time, try it now.
- do 3.24 or start thinking about 3.25 (and continue at home, so that you do not spend the entire werkcollege "just" with one exercise).
- 4.19, 4.13
- make sure you also look at 4.24
- 4.21
- if you have more time do 3.22 (which I did mention in the class) and 4.14
- The rooms for the next werkcollege (December 14) are:
Homework 5 (last homework this year) (to be handed in by January 8th, before the werkcollege).
♣ WEEK 50/Lecture 10 (December 14, KBG - PANGEA): reminder on connected/simply connected spaces with some of the applications (recalled also the 1-point removal trick), the example which is connected and not path connected, then connected components. The we moved on to compactness: definition and first examples/remarks.
Blackboard photos of Lecture 10.
Homework 5 (last homework this year) (to be handed in by January 8th, before the werkcollege).
The plan for the next week:
- The plan for the next lecture (December 19) is: compactness (as much as possible).
- The exercises for the next werkcollege (December 19) are:
- 4.14
- 4.12
- 4.11
- 4.4
- 4.21
- 4.22
- The rooms for the next werkcollege (December 19) are:
♣ WEEK 51/Lecture 11 (December 19, RUPPERT - PAARS): Compactness: the main properties of compactness: all the statements and proofs of most of them (left over: compactness of products and relationship with sequential compactness- see the blackboard fotos).
Blackboard photos of Lecture 11.
Werkcollege exercises for December 21:
The exercises for the next werkcollege (December 21) are:
- use compactness to do Exercise 4.9
- Exercise 4.31
- Show that the projective space $P^2$ can be embedded in the four-dimensional Euclidean spaces $R^4$.
(hint: see Exercise 1.26 in the first chapter).
- Exercise 4.45
- Exercise 4.47
- Exercise 4.30
- Exercise 4.29
- Exercise 4.41
The plan for the next week:
- The plan for the next lecture (December 21) is: finish discussion on compactness (compactness of products and relationship with sequential compactness) and then start with local compactness and the 1-point compactification.
- The rooms for the next werkcollege (December 21) are:
♣ WEEK 51/Lecture 12 (December 21, KBG - PANGEA): Discussed the last parts on compactness, with proofs: the tube lemma, the product of two compact spaces is compact, then def of local sequential compactness and then things done a bit more differently than in the lecture notes: we stated a theorem whose first part is the fact that, for 1st countable spaces, compactness implies sequential compactness (which appears in the lecture notes, but we now gave another proof) and a second part which is not in the notes, saying that for metric spaces compactness is equivalent to sequential compactness. Then introduced the notion of one-point compactifications and looked at some examples.
Blackboard photos of Lecture 12. On the very last page, in red, you can also see the main topics that will be coming next year.
The plan for the next time:
- The plan for the next lecture (January 9) is: discuss local compactness and the main properties of one-point compactifications. If the time allows we will start making a ling with the course on rings, via "the algebra of observables".
- The exercises for the next werkcollege (January 9) are: make sure you have done the previous exercises on compactness. Take your time. Then, during the last 30 min (or at most 45 min) start looking at examples of 1-point compactifications such as:
- 4.57
- 4.62
- 4.63(ii)
- 4.69(ii)
- The rooms for the next werkcollege (January 9) are:
♣ WEEK 2/Lecture 13 (January 9, RUPPERT - PAARS): One point compactifications and local compactness (but skipped the discussion on exhaustions).
Blackboard photos of Lecture 13.
The plan for the next week:
- The plan for the next lecture (January 11) is: start discussing the algebra $C(X)$ of observables and then the Stone-Weierstrass theorem.
- The exercises for the next werkcollege (January 11) are:
- 4.58 (Hint: stereographic)
- 4.59
- 4.67
- 4.69
- 4.74
- 4.62 (notice that the last question there is very very similar to the exercise I left in the class, with the example of a convex subspace of the plane which is not locally compact)
- 4.73 (Hint: try to combine 1-point compactifictations with versions of the removal one point trick ... applied locally ...)
- If ou have more time you can pick up a few more exercises, such as: 4.62, 4.63.
- The rooms for the next werkcollege (January 11) are:
♣ WEEK 2/Lecture 14 (January 11, KBG - PANGEA): In this lecture we focused on $C(X)$ (consisting of real-valued continuous functions on $X$, also called "observables" or "measurments" on $X$) and its relationship with the point-set topology of $X$. That was the main topic and we did it by collecting several topics spread in several places in the lecture notes. Here are the highlights:
- We looked at the algebraic structure present on $C(X)$ with the conclusion that $C(X)$ is an algebra
- I informed you that there is a Gelfand-Naimark theorem that says that $X$ can be recovered from $C(X)$ and its algebraic structure. The more precise statement
will be discussed later in its more precise form provided by Theorem 8.22 in the notes.
- another illustration of the relevance/usefulness of real-valued continuous functions appears when discussing separability (Definition 2.54 in the notes, and the notion of normal space from Definition 2.55). Then I stated the Urysohn lemma (Theorem 5.21) trying to convey to you how remarkable that result actually is- in particular I mentioned that the lemma can be used to prove a metrizability theorem, known as Urysohn's metrizability theorem, which I stated (in the notes is Theorem 7.1).
- Then I returned to $C(X)$ looking also at more-structure present on it, this time not only algebraic but also topological. The main upshot is that $C(X)$ comes equipped with a norm (the sup-norm) that makes it into a Banach space and induces the sup-distance on $C(X)$. That norm interacts nicely with the entire algebraic structure, with the ultimate conclusion on all the structure on $C(X)$: it is a Banach algebra.
- Then we moved to the Stone-Weierstrass. The sup-distance on $C(X)$ allows us to talk about (i.e. it makes sense to talk about): density inside $C(X)$. The starting point is the more classical result on $X= [0, 1]$ which says that any continuous function on $[0, 1]$ can be uniformly approximated by polynomial functions. This can be restated also as saying that for any $f\in C([0, 1])$ there exists a sequence of polynomial functions $p_n: [0, 1]\to \R$ such that $p_n$ converges uniformly to $f$. Yet another way (a third one) to formulate this is that the subset $C_{pol}([0, 1])$ of polynomial functions on $[0, 1]$ is dense in $C([0, 1])$ (with respect to the metric topology induced by the sup-metric on $C([0, 1])$.
- Then I moved on to general $X$ (but still assumed compact and Hausdorff) and, while "polynomial functions on $X$" no longer makes sense in that generality, the question was: what manageable conditions should one ask on a sub-set $\mathcal{P}\subset C(X)$ that would imply density, generalizing the result on $[0, 1]$. I discussed two conditions (being a sub-algebra, and being point-separating) and then I stated the Stone-Weerstrass theorem. The proof will be given next time.
- One last thing: defined a sequence of polynomials on $[0, 1]$ that converges uniformly to the square-root function, leaving it as an exercise to prove the (uniform) convergence.
- ... i see, this has been quite a lot ... though I think it is quite interesting stuff ...
Blackboard photos of Lecture 14.
The plan for the next week:
- The plan for the next lecture (January 16) is: prove Stone-Weierstrass then start looking at Gelfand-Naimark. This time I will follow the notes linearly.
- The exercises for the next werkcollege (January 16) are:
- Show that any compact Hausdorff space is normal.
- 2.33 and then, using that, prove that any metric space is normal.
- 8.2
- 8.3
- 3.54
- 5.17
- if you have more time, look back at 4.22.
- The rooms for the next werkcollege (January 16) are:
♣ WEEK 3/Lecture 15 (January 16, RUPPERT - PAARS): Finished the proof of Stone-Weierstrass and then started the discussion related to Gelfand-Naimark: introduced ideals, maximal ideal then, for $A= C(X)$ (the algebra of real-valued cotinuous functions on a compact Hausdorff space $X$), I pointed out that any point $x\in X$ gives rise to an ideal $I_x$ (consisting of all $f\in C(X)$ that vanish at $x$). Then I stated the result saying that each $I_x$ is a maximal ideal, and each maximal ideal is of this type, with the big conclusion that $x \mapsto I_x$ defines a bijection between $X$ and the set of maximal ideals in $C(X)$.
Blackboard photos of Lecture 15.
The plan for the next week:
- The plan for the next lecture (January 18) is: the plan for the next lecture is to complete discussion of the Gelfand-Naimark.
- The exercises for the next werkcollege (January 18) are: TBA.
- make sure you did do the first ones assigned to the previous werkcollege: 8.1, 8.2, 8.3
- 8.5 (this will also provide a good opportunity to remind yourself about quotients).
- If A and B are two algebras (over reals), a map F : A → B
is called an algebra homomorphism if it is |R-linear and satisfies F(xy) = F(x)F(y) for all x, y in A.
Show now that any continuous function f: X → Y between two topological spaces gives rise to a map
F: C(Y) → C(X) which is an algebra homomorphism.
- Show that if F : A → B is an algebra homomorphism then the kernel of F (defined as the subset of A consisting of those elements a of A with the property that F(a)= 0) is an ideal of A.
- 8.18 (where "character on A" means any algebra homomorphism from A to |R).
- The rooms for the next werkcollege (January 18) are:
♣ WEEK 3/Lecture 16 (January 18, KBG - PANGEA): Discussed the Gelfand-Naimark theorem, where some of the proofs of the intermediate results are different from the ones in the lecture notes. Here is a file with a brief explanation on how you can get an idea of what the spectrum of an algebra look like/how it can be computed (how to think about it/how to approach it).
Blackboard photos of Lecture 16.
Homework: no more homeworks. Instead, here is a list of recap problems that may be helpful for preparing the exam. You can ask questions about them during the last two werkcolleges.
The plan for the next week:
- The plan for the next lecture (January 23) is: recap the Gelfand-Naimark and look at one example, then move on to new material: finite partitions of unity.
- The exercises for the next werkcollege (January 23) are: 8.6, read the notes added above (3 pages on how to deal with the spectrum of an algebra), then move on to 8.12, 8.14, 8.13, 8.15, 8.19 and if they still have time try to look at 8.22.
- The rooms for the next werkcollege (January 23) are:
An important message sent to me by Thimo-Louis, Commissaris Onderwijs van A–Es :
Volgende week woensdag 24 januari organiseert A–Eskwadraat een Vak Ondersteunende Sessie waarbij jullie onder begeleiding van ouderejaars studenten nog een oud tentamen kunnen maken en al je laatste vragen kunnen stellen.
Het VOSje is op woensdag 24 januari in BBG 2.19 van 13:00 - 15:00.
Er is gratis koffie en thee en er zijn zelfs nog eens lekkere koekjes aanwezig om je die laatste extra motivatie te geven voor je tentamens.
Succes alvast allemaal met jullie tentamens en ik hoop veel van jullie te zien tijdens het VOSje!
♣ WEEK 4/Lecture 17 (January 23, RUPPERT - PAARS): The first hour (and a bit) was a recap on algebras and characters: a few principles and several examples. Then in the second part we looked a bit at finite partitions of unity (main definitions, explanations/a bit of motivation, and the first theorem).
Blackboard photos of Lecture 17.
The plan for the next week:
- The plan for the next lecture (January 25) is: TBA
proof of Urysohn lemma, perhaps followed by a few more comments on partitions of unity.
- The exercises for the next werkcollege (January 25) are: I think you have quite a few left from last week (including the page with exercises to prepare for the exam).
- The rooms for the next werkcollege (January 25) are:
We zijn er Bijna ...
♣ WEEK 4/Lecture 18 (January 25, KBG - PANGEA):
Blackboard photos of Lecture 18.
♣ EXAM: Thursday, 1 February 2024, 13:30-16:30, OLYMPOS - HAL3
♣ RETAKE: ??????
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