"Inleiding Topologie", blok 3, 4, 2013
This is the web-site for the course "Inleiding Topologie", 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 .
When, where, etc:
THE LECTURES: Wednesdays, 13:15- 15:00, Min 208 (except for February 20, when it will be in BBL 001).
LECTURER: Marius Crainic.
THE WERKCOLLEGES: Wednesdays, 15:15- 17:00, in WISK 611AB and MIN 202.
TEACHING ASSISTANTS: Boris Osorno Torres, Koen van Woerden, Frits Verhagen (for the entire semester) and then Valentijn
Karemaker for blok 3 and Sebastian Klein for blok 4.
The students are divided in two groups (check here to check the group that you belong to).
The werkcolleges for Groep 1 will be in room Wisk.611ab and for Groep 2 in MIN 202.
FINAL MARK: there will be
- two exams. The first one will be on week 16 (hence on April 17). The second one will be
in week 26 (hence on June 26).
- 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.
NEW- THE RULES FOR PASSING THE COURSE:
The rule formulated at the start of the course was relaxed to the
following:
There will be two exams hence two marks: Mark-A and Mark-B, refering to the two periods of the course, respectively.
To compute Mark-A, ones uses the mark Exam-A (from the first exam), Home-A (the average of the marks for the homeoworks related
to the first part) and Bonus-A (the points gained by solving bonus exercises related to the A-part; by the way, note that we
only had one such exercise so far). With these,
Mark-A := Max(Exam-A, (Exam-A+ Home-A)/2) + Bonus-A.
Similarly for Mark-B. The final mark is the average of Mark-A and Mark-B. To pass, the final mark should be greater
then 6, and both Mark-A and Mark-B 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 6): the key-words of topology (convergence,
continuity and, a bit later, homeomorphisms); we postponed the (abstract!) definition of (topological) space, concentrating
first on metric spaces; recalled the definition of metric spaces, convergence and continuity for metric spaces, then
I mentioned two main questions of topology: how can one decide when two spaces are homeomorphic or not; when is a given space metrizable?
Then the Euclidean and square metric in R^k, subspaces, some examples of homeomorphisms, various ways to look at the circle,
the spehere obtained from a square by gluing. Then, at the very end, I mentioned very fast how, starting from a square,
and performing some gluing of its sides, one gets other interesting spaces such as: cylinder, Moebius band, torus, Klein bottle
(to be explained in more detail in the next lecture).
Exercises for the werkcollege: 1.2, 1.3, 1.4, 1.6, 1.1, 1.10, 1.13.
Homework (to be handed in): 1.11 .
WEEK 7/Lecture 2 (February 13): We continued with
the examples: spheres, Moebius band, torus, Klein bottle, projective space. Then we talked about abstract gluingsd: the notion of equivalence relations,
quotients, then I mentioned existence and uniqueness (the final part- i.e. the abstract quotient, will be briefly discussed at the beginning of the next lecture).
Exercises for the werkcollege: 1.18, 1.19, 1.24, 1.22, 1.26, 1.27.
Homework: 1.23.
WEEK 8/Lecture 3 (February 20- note the different location
for this week: BBL 001): Finished abstract gluing (equivalence relations and quotients), with various examples (not in the notes),
then discussed opens in a metric space, then passed to the general notion of topological space, metric topologies, other examples,
subspace topology, continuity.
Exercises for the werkcollege: 2.1, (2.13 and 2.37), 2.3, 2.27, (2.19 and 2.31)
Homework: 2.18.
WEEK 9/Lecture 4 (February 27): Neighborhoods, convergent sequences,
continuity, continuity at a point, sequential continuity, Hausdorffness, basis of neighborhoods, 1st countability,
the various propositions relating them:
- continuity is equivalent to continuity at each point
- continuity implies sequential continuity; the converse holds if X is 1st countable (for f: X --> Y).
- Hausdorffness of X implies that every sequence in X has at most one limit. The converse is true if X is 1st countable.
- metric spaces are Hausdorff and 1st countable.
Then I mentioned what the interior, closure and boundary of a subset of a topological space is.
Exercises for the werkcollege: 2.30, 2.38, 2.40, 2.41, 2.33 2.44, 2.48, 2.56, 2.49.
Homework: none.
WEEK 10/Lecture 5 (March 6): We had a rather generous reminder. Then we finnished the discussion on
interior, closure and boundary, with several examples that were not in the notes (looking also at the lower limit topology on the real line);
then the notion of basis for a topology, second countability (with various remarks), explaining that subspaces of Euclidean spaces are 2nd countable;
then the notion of topological manifold.
Exercises for the werkcollege: re-do 2.44, 2.48, 2.56 and 2.49 (try to use the picture, byut explain a bit what you are doing).
Then do 2.47, 2.45, 2.50, 2.58. During the werkcollege, the teaching assistants also presented on the blackboard exercise
2.52, writing down a complete solution.
Homework: During the werkcollege, the teaching assistants also presented on the blackboard exercise
2.52, writing down a complete solution. For the homework, you were asked to formulate of analogue of exercise 2.52 for closures
instead of interiors (with parts 1., 2., 3., as in 2.52, but interchanging intersections with unions if necessary), then solve the exercise, with the
proof explained carefully (as was done for you on the blackboard for 2.52). If you were not present at the werkcollege, it is
probably a good idea to talk to your colleagues which were present.
WEEK 11 (March 13): No course (Hertentamen).
Exercises for the werkcollege:
Homework:
WEEK 12/Lecture 6 (March 20): Quotient topologies,
the torus as
an example, quotients modulo group actions, the projective space.
Exercises for the werkcollege: 3.1, 3.10, 3.11, 3.13, 3.14 , 3.16, 3.18.
Homework: 3.15.
WEEK 13/Lecture 7 (March 27): Reminder on the type of
constructions we have seen so far; then continued with the product topology, collapsing a subspace to a point, cylinder,
cone and suspension of a space, construction of topologies via a topology basis.
Exercises for the werkcollege: finish teh ones from last time, and then do:
3.12, 3.20, 3.26, 3.29, 3.31, 3.24, 3.23.
Homework: 3.33.
WEEK 14/Lecture 8 (April 3):
Exercises for the werkcollege (tentative): 2.39 (page 43 of the notes), 4.16, 4.18, 4.11, 4.12, 4.13, 4.15, 4.17.
Homework: TBA.
WEEK 15/Lecture 9 (April 10): Compactness: definitions
and the main properties, proving the simplest ones.
Exercises for the werkcollege: 4.25, 4.28, 4.36, 4.38, 4.39.
Homework: 4.27.
WEEK 16 (April 17): First exam, 13:30-16:30, EDU gamma.
- the exam covers everything that was discussed in the class.
- at the exam, you are allowed to use some sheets of papers (that you prepare at home). Up to 5 but I would very
very strongly recommend to prepare just one sheet with the MOST IMPORTANT highlights of what has been discussed.
WEEK 17/Lecture 10 (April 24): I recalled the
notion of compactness and the main results, proving most of the more difficult ones.
Exercises for the werkcollege: the same as for last week.
Homework: no-one. But please be aware of the "bonus-exercise" given during the lecture (not compulsory).
WEEK 18/Lecture 9 (May 1): local compactness, one point compactification.
Exercises for the werkcollege: 4.,40, 4.41, 4.42, 4.47, 4.46 (tentative).
Homework: TBA (but will not be announced on this web-page).
WEEK 19/Lecture 9 (May 8): Partitions of unity, paracompactness.
Exercises for the werkcollege: will not be announced on the web-page.
Homework: will not be announced on the web-page.
WEEK 20/Lecture 9 (May 15): Partitions of unity; paracompactness
(the rest of the chapter).
Exercises for the werkcollege:
- finish what was left over from last time (5.4, 5.5, 5.6, 5.7).
- continue with 5.8, 5.9, 5.10.
Homework: see here.
WEEK 21/Lecture 9 (May 22): Chapter 7
(several metrizability theorems). There are no exercises for this chapter, so the exercises for this werkcollege will be about
normal spaces, local finiteness, partitions of unity.
Exercises for the werkcollege: 2.64, part 5 of 2.53. Then, assuming 5.10
(in case you did not do it), do 5.11 and 5.12.
Homework: here.
WEEK 22/Lecture 9 (May 29):
Reminder on metrizability theorems; then we moved to chapter 8, discussing the various structures on C(X) and the Stone-Weierstrass
theorem.
Exercises for the werkcollege:
Homework:
WEEK 23/Lecture 9 (June 5): Gelfand-Naimark.
Exercises for the werkcollege: 8.8- 8.12.
Homework: the second exercise from the Exam B of 2011.
WEEK 24/Lecture 9 (June 12):
Exercises for the werkcollege:
Homework:
WEEK 25 (June 19): No course (Studieweek).
WEEK 26 (June 26): Second exam.
The rules for the exam are the same as for the previous one.
IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. MORE PRECISELY, ALTHOUGH I DO TRY TO KEEP IT UPDATED (E.G. BY SPECIFYING
WHICH IS THE HOMEWORK), SOMETIMES I AM NOT ABLE TO DO IT. IN SUCH CASES, PLEASE CONTACT YOUR COLLEAGUES THAT WERE PRESENT
AT THE LECTURE OR WERKCOLLEGE.
Enjoy the sphere (and not only).