"Inleiding Topologie", blok 3, 4, 2012

All the grades.

Also, for the people below: whenever you are in the Uithof and have time, please drop by my office for 2 minutes (room 802, 8th floor).

- Ragnar Groot Koerkamp

- Tom Bannink

- Huibert hed Lam

- Frits Verhagen

- Maxim van Olderbeek

This is the web-site for the course "Inleiding Topologie", 2012. 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:

THE LECTURES: Wednesdays, 13:15- 15:00, in room BBL169.

LECTURER: Marius Crainic.

THE WERKCOLLEGES: Wednesdays, 15:15- 17:00, in rooms BBL 071 and MIN 202.

TEACHING ASSISTANTS: João Nuno Mestre, Maria Amelia Salazar, Roy Wang.

FINAL MARK: there will be

- two exams. The first one will be on week 16 (hence on April 18), from 13:30 to 16:30 in EDU gamma. The second one will be in week 26 (hence on June 27), from 13:30 to 16:30 in EDU beta.

- hand in exercises: there will be one hand in exercise (almost) every week. You will receivce the exercise at the end of the werkcollege. You have to hand it in one week later, at the beginning of the lecture.

THE RULES FOR PASSING THE COURSE: The average of the two marks for the exam will make the exam mark E.Similarly for the homework mark H. The final mark will be (H+ 2E)/3. To pass, the final mark should be greater then 6, and the marks in both exams should be greater than 5.

For the re-examination in August: the homework will not count, but the exam will be more difficult.

Lecture notes: We intend to use the same lecture notes as last year:

- .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. Here is a short description of the chapters:

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 6/Lecture 1 (February 8): Some words about the course (keywords: points, points close to each other, convergent sequences, continuity, "isomorphisms"= homeomorphisms, "the category of topological spaces" in comparison with that of groups, linear spaces, sets). The definition of metric spaces, convergence and continuity for metric spaces, the Euclidean space (with the Euclidean and then with the square metric). Then we moved to examples of subspaces of the Euclidean spaces (e.g. on the "smiling faces" from the notes, the circle- pointing out the standard parametrization, and the fact that it gives an example of a continuous bijection which is not a homeomorphism), concentrating on examples that can be obtained from a square by gluing (some of) its sides: cylinder, Moebius band, torus, Klein bottle, the sphere (pointing out the need of an extra-dimension in order to perform the gluing). The discussion was "intuitive"- without many explicit formulas; for formulas, see the lecture notes (but remember that, in the lecture, I explained the way you can get the formulas, or even find them yourself).

Exercises for the werkcollege: 1.1, 1.3, 1.4, 1.13, 1.17, read the explicit formulas for the torus, then for the Moebius band (lecture notes), 1.18.

Homework (to be handed in): 1.8 (Remark: here I expect "arguments" similar to the ones that I gave during the first lecture- i.e. on the picture, making clear in words what the map is, and explaining (similarly) why it is a homeomorphism. For this exercise it does happen that you can write down the explicit formulas, hence you can give a "proper proof" (and you are welcome to do that!)).

- WEEK 7/Lecture 2 (February 15): reminder; various ways to look at the sphere (e.g. as one-point compactification of the plane- via the stereographic projection), various descriptions of the projective plane; abstract gluing and quotients (with some examples from group theory+ "uniqueness" and "existence").

Exercises for the werkcollege: 1.24, 1.11, 1.29, 1.31, 1.32, 1.33, 1.30.

Homework: Explain how the projective plane P^2 can be realized as a subspace of the 4-dimensional Euclidean space R^4. (If you use Exercise 1.22, then prove it!).

- WEEK 8/Lecture 3 (February 22): Opens in a metric space, the definition of topological spaces with first examples, the metric topology and metrizability, induced topology (on a subset of a topological space) and the related terminology, continuity, homeomorphisms.

Exercises for the werkcollege: First make sure that you finished 1.31, 1.32, 1.33 (from last time). Then do: 2.2, 2.3, 2.14, 2.18, 2.25.

Homework: 2.16.

- WEEK 9/Lecture 4 (February 29): Reminder + various descriptions of the lower limit topology on the real line (the one from Exercise 2.19). Continuity, neighborhoods, convergence, sequential continuity, Hausdorffness (and relevance to the uniqueness of limits), basis of neighborhoods, 1st countability.

Exercises for the werkcollege: 2.19, 2.30, 2.41, 2.37, 2.60, 2.39.

Homework: 2.31.

- WEEK 10/Lecture 5 (March 7): Rest of the chapter.

Exercises for the werkcollege: 2.35, 2.44, 2.48, the last exercise on page 45, 2.52, 2.37, 2.56.

Homework: 2.50.

- WEEK 11 (March 14): No course (Hertentamen).

- WEEK 12/Lecture 6 (March 21): Quotient topologies. Example: the abstract torus and models. Then I described an "algorythm" to be used for shoing the a quotient can be embeeded in some Euclidean space- and this is not in the lecture notes! In particular, I explained (and used) the following proposition: if f: C ---> Y is a continuous injective map, where C is a subset of the Euclidean space which is closed and bounded (i.e. compact) and Y is a Hausdorff topological space, then f is automatically an embedding. Then we moved to quotients modulo group actions.

Exercises for the werkcollege: 3.10, 3.1, 3.2, 3.14, 3.16.

Homework: 3.18.

- WEEK 13/Lecture 7 (March 28): Reminder on quotients and quotients modulo group actions (and the examples with the projective plane), product topology, collasping a subspace to a point, cylinders/cones/suspensions of spaces, topology basis and the criteria for comparing the induced topologies.

Exercises for the werkcollege: 3.23, 3.31, 3.29, 3.32, 3.21, read example 3.15, 3.33.

Homework: Exercise 2.39.

- WEEK 14/Lecture 8 (April 4): finnishing the chapter on constructions and topologies, then started discussing connectedness and path connectedness.

Exercises for the werkcollege (tentative): 3.36, 3.38, 4.17, 4.11, 4.12, 4.18 and then, if you still have time: 4.13, 4.14, 4.20.

Homework: Show that the property of being connected is a topological property.

- WEEK 15/Lecture 9 (April 11): Finnished the proofs of the main results about connectedness (up to the theorem that path connected implies connected) and discussed some more application (not in the lecture notes), and gave an indication of algebraic topology and fundamental groups when one wants to compare Euclidean spaces of higher dimensions. Than we moved to compactness: definition, first examples, and stated the main results about compactness.

Exercises for the werkcollege:

Homework: none.

- WEEK 16 (April 18): First exam, 13:30-16:30, EDU gamma. As mentioned in the class:

- the exam covers everything that was discussed in the class, up to compactness (without it).

- at the exam, you are allowed to use some sheets of papers (that you prepared at home). Up to 5 but, as I mentioned in the last lecture, I very very strongly recommend to prepare just one sheet with the MOST IMPORTANT highlights of what has been discussed.

- WEEK 17/Lecture 10 (April 25): We stated again the main results about compactness:

- [0, 1] is compact.

- closed inside compact is compact

- compact inside Hausdorff is closed.

- product of two compacts is compact.

- subspaces of an Euclidean space are compact iff they are closed and bounded.

- continuous functions send compacts to compacts.

- A continuous bijection from a compact to a Hausdorff space is a homeomorphism.

- A continuous injection from a compact to a Hausdorff space is an embedding.

- any compact manifold can be embedded in some Euclidean space.

and then we discussed the proofs.

Exercises for the werkcollege: 4.25, 4.29, 4.38, 4.39, 4.27, 4.32.

Homework: Exercise 4.28- for which you should write down carefully a complete, proper proof (like the ones from the lectures!!!).

- WEEKS 18- 24: Unfortunately, starting with this lecture, I was not able to update this page. Here is the material that was covered from the lecture notes, during the remianing lectures:

- sequential compactness: pp. 76.

- local compactness: pp. 77.

- one-point compactifications: pp. 78-79.

- partitions of unity: 85-89.

- Urysohn's lemma: pp. 91-92.

- the Urysohn metrization theorem (pp. 104) and consequences (pp. 106).

- the Stone-Weierstrass theorem and Gelfand Naimark: sections 1, 2 and 3 from Chapter 8, except for the small discussions on C^* and *-algebras.

- WEEK 25 (June 20): No course (Studieweek).

- WEEK 26 (June 27): Second exam, 13:30-16:30, EDU beta. The rules for the exam are the same as for the previous one.

Enjoy the sphere (and not only).