# "Inleiding Topologie", blok 3, 4, 2011

Here is the list with the final marks. Note: for exam B, the marking started from 0.5 instead of 0 (as originally announced).
This is the web-site for the course "Inleiding Topologie", 2011. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc.

When, where, etc:

THE LECTURES Wednesdays, 13:15- 15:00, in room BBL061.

LECTURERS: Marius Crainic and Sergey Igonin.

THE WERKCOLLEGES: Wednesdays, 15:15- 17:00, in room BBL061.

TEACHING ASSISTANTS: R.W. Storm, M.A. de Reus, M.A. Salazar.

EXAMS: there will be

- two exams (WEEK 16 and WEEK 26). The average of the two marks make the exam mark E.

- hand in exercises (not every week). The average of the marks make the exercise mark W.

THE RULE FOR THE HAND IN EXERCISES is the following: you will receive the exercise at the end of a werkcollege, say the one in week n. You can think about it, try to solve it and then discuss it in the next werkollege (the one of week n+1) with the teaching assitants- in case that you do go to the werkcollege. Then you have a few more days to improve your solution (if needed). You have to hand it in at the start of the next hoorcollege (i.e. the one in week n+2). Due to experiences from previous years, no exceptions from these rules will be allowed!

THE RULES FOR PASSING THE COURSE: the final mark is the weighted average (3E+ 2W)/5. To pass, the final mark should be greater then 6, and the exam mark E should be greater then 5.

For those who tried to redo the exam A at home: I was really hoping that you will use this opportunity to learn a lot better the first part, and I receive perfect solutions. This was not the case, and here is my conclusion after looking at your second solutions:

- for 3470989: it was a clear improvement, and I propose that I change the previous mark (3.50) into a 5.50 (by making some kind of average).

- 3117510: I did not see enough improvement to motiate me in changing the original mark. I advise you to spend more time with the course, and to come at the retake at the end of August.

- 349669 and 3375986: there is enough improvement to change the mark (e.g. to 5.5-6), but your solutions are just too similar. It is not at all clear to me what happened, and I would like to talk to you before I take a decision.

Lecture notes: We will provide lectures notes as the teaching of the course proceeds.

- Chapter 1 (and here are two more pages for the end of chapter 1) : short intro to the course + many examples of "spaces". Note that some of the pictures use colours. This chapter will be discussed during the first two lectures.

- Chapter 2-3: Topological spaces; Constructions of topological spaces. Note: this is a new version of the notes for chapters 2 and 3. The orginal version had many typos, so we decided to replace it with the new version. The more substantial changes are on pages 32 and 33 (the present version is closer to the presentation in the class). Yes, I know it is not very convenient to start making new versions; we will do our best not to do that with the coming chapters.

- Chapter 4: Topological properties (connectedness, compactness, local compactness, one-point compactifications).

- Chapter 5: Partitions of unity and paracompactness (we skipped this chapter for now).

- Chapter 6: Metric properties versus topological ones.

- Chapter 7: Metrizability theorems (the Urysohn and the Smirnov metrizability theorems; consequences for compact and locally compact spaces).

- Chapter 8: Spaces of functions (the Stone Weierstrass theorem, the Gelfand Naimark theorem, then general function spaces- equicontinuity, etc , Arzela-Ascoli).

- Chapter 9: Embedding Theorems.

- THE COMPLETE LECTURE NOTES: all the chapters above, plus the table of contents, in one file.

The schedule week by week:

WEEK 6/Lecture 1 (February 9): Introduction to the course; metric spaces; first examples of "spaces".

For the werkcollege: Exercises 1.2, 1.3 and 1.4, 1.6, 1.7, 1.13, 1.18.

WEEK 7/Lecture 2 (February 16): Reminder from the previous lecture; spaces obtained by gluings edges (Moebius band, torus, double torus, Klein bottle, projective space). The abstract gluing (equivalence relations, quotients).

Exercises for the werkcollege: 1.19, 1.21, 1.22, 1.25, 1.28, 1.31, 1.32

Here is the take home exercise given in this week.

WEEK 8/Lecture 3 (February 23): Reminder; the abstract definition of a topological space; examples on an arbitrary set (the trivial, discrete, co-finite, etc); metric topology; metrizability; subspace topology.

Exercises for the werkcollege: 2.27, 1.33, 1.34, 2.1, 2.13, 2.3, 2.16, 2.19, 2.30, 2.31 (try to do as many as possible, if you are done then look at the other exercises from the notes).

WEEK 9/Lecture 4 (March 2): Neighborhoods; continuity (also homeomorphisms, embeddings), continuity at a point, converegence, sequential continuity, basis of neighborhoods, 1st countability (and the various results from the lecture notes). Inside a topological space: interior, closure, boundary.

Exercises for the werkcollege: try to finnish what is left over from the last time. Then: 2.16, 2.18, 2.27, (2.30 and 2.37), 2.40, 2.24, (2.19, 2.31 and 2.41), 2.4, 2.8.

Here is the take home exercise given in this week.

WEEK 10/Lecture 5 (March 9): The rest of the chapter.

Exercises for the werkcollege: 2.44, 2.45, 2.48, 2.49, 2.52, 2.56, 2.58, 2.64, 2.61, 2.62.

WEEK 11 (March 16): No course (Hertentamen).

WEEK 12/Lecture 6 (March 23): Quotient topology, topological quotient maps, characterization of continuous maps from quotient spaces, quotients modulo equivalence relations; the quotient of a Hausdorff space is not always Hausdorff; construction of the torus as a topological quotient; quotients modulo group actions; the circle as the quotient of R modulo Z; the quotient of a Hausdorff space by a finite group is Hausdorff.

The "take home" exercise: Exercise 3.18.

Exercises for the werkcollege: 3.1, 3.9, 3.10, 3.14, 3.16.

WEEK 13/Lecture 7 (March 30): Projective spaces as quotients; product topology; topological groups; collapsing a subspace to a point; cylinders, cones, suspensions; bases for topologies, generated topologies.

Exercises for the werkcollege: 3.4, 3.5, 3.11, 3.12, 3.22, 3.23, 3.26, 3.27

WEEK 14/Lecture 8 (April 6): Generated and initial topologies; the space of functions from an interval to R^n; types of convergence of functions: pointwise, uniform, uniform on compacts; the corresponding topologies on the space of functions; convergence for continuous functions; completeness with respect to the sup metric. Then started with Chapter 4 (definition of connectedness and some examples).

The "take home" exercise: Exercise 3.33.

Exercises for the werkcollege: 3.6, 3.7, 3.8, 3.29, 3.30, 3.31, 3.34, 3.35, 3.36.

WEEK 15/Lecture 9 (April 13): Connected spaces, path connected, connected components+ several examples/applications. Then we stated the main properties of compact spaces (but we will return to these next time, and also talk about their proofs).

For the exam, we agreed that the part on compact spaces will not be required, but the one on connectedness (and path connectedness) will.

WEEK 16 (April 20): No course (Tentamen Week 3).

WEEK 17/Lecture 10 (April 27): Recalled the definition of compactness and the list of the main properties:

- Example 4.16 (4): [0, 1] is compact.

- Prop 4.17: closed inside compact is compact

- Theorem 4.18: compact inside Hausdorff is closed.

- Theorem 4.21: product of compacts is compact.

- Corollary 4.23: subspaces of an Euclidean space are compact iff they are closed and bounded.

- Theorem 4.25: continuous functions send compacts to compacts.

- Theorem 4.26: A continuous bijection from a compact to a Hausdorff space is a homeomorphism.

- Corollary 4.27: A continuous injection from a compact to a Hausdorff space is an embedding.

The rest of the lecture was dedicated to the proofs of these results (including the various lemma's used in the proofs). I also stated and commented on:

- Theorem 4.30: any compact manifold can be embedded in an Euclidean space.

Exercises for the werkcollege: 4.8, 4.9, 4.26, 4.29, 4.38, then "Show that there is no continuous injection of a bouquet of two circles into the circle", 4.27, 4.36, 4.35.

Hand in exercise: 4.39.

WEEK 18/Lecture 11 (May 4): Local compactness: exhaustions and the existence/uniqueness of the one-point compactification. Examples of one-point compactifications. Sequential compactness. Then, from Chapter 6 (which can be downloaded now- see the links above), we concentrated on Theorem 6.6 and 6.7, stating them and explaining their statement (e.g. Cauchy sequences, completeness, totally boundedness). By the way, for now we skip Chapter 5!

Exercises for the werkcollege: 4.42, 4.45, 4.47, 4.46, 4.30, 4.49, 4.10.

WEEK 19/Lecture 12 (May 11): reminder on the notions of completeness and total boundedness, then the proofs of Theorems 6.6 and 6.7. Then we discussed finite partitions of unity (definitions + the existence theorem and its proof). I remind you once again: the importance of partitions of unity comes from the fact that it "allows one to pass from local to global" (an immediate consequence is that any function can be written as a sum of functions "concentrated on small opens"- and this is an indication of the general principle).

Exercises for the werkcollege: 6.3, 6.5, 5.1, 5.2, 6.1, 6.2, and 'then do more exercises about one point compactifications'.

Hand in exercise: 6.4.

WEEK 20/Lecture 13 (May 18): Reminder on separation axioms and normal spaces, and also on finite partitions of unity. Then a sketch of arbitrary partitions of unity: the notion of locally finite family of subsets, the notion of locally finite family of continuous functions, the notion of arbitrary partition of unity. Then the notion of refinement and finally that of paracompactness. I remind you once again: you should think about "paracompactness" as the topological condition that is nedded for the existence of partitions of unity!!! It is good to have in mind that most spaces that we encounter are paracompact. I mentioned that compact spaces are paracompact (obvious) and that all metric spaces are paracompact (I only mentioned it; it is a non-trivial theorem, which appears with a proof in the lecture notes- Theorem 6.11).

We skipped section 4 (from chapter V), and we will not return to it (it is there for the interested reader). Then I concentrated on Urysohn's lemma (Theorem 5.21 in the notes), Urysohn's metrization theorem (Theorem 7.1 in the notes) and its consequences (Theorems 7.3, 7.4, 7.5): we stated them and then discussed the proofs.

Exercises for the werkcollege: 5.4, 5.5, 5.6, 5.7. Then try 5.7+5.8+5.9+5.10 (very similar to each other), but be aware that it is much easier if you use Theorem 5.19.

WEEK 21/Lecture 14 (May 25): The algebra C(X) of continuous functions n a compact Hausdorff space X is a complete metric space; he uniform topology on C(X); Banach algebra and C*-algebra structures on C(X); point-separating" property; the Stone-Weierstrass theorem in the real case.

Exercises for the werkcollege: 8.1, 8.2, 8.3, 8.5, 8.6.

Hand in exercise: 8.4 (to be handed in at the wercollege of June 15).

WEEK 22 (June 1): No course (Hertentamen blok 3).

WEEK 23/Lecture 15 (June 8): Reminder on the various structures on C(X) (algebraic: vector space, algebra, *-algebra; topological: metric, Banach space, Banach algebra, C-star algebra). Then discussed the results from the section on the Gelfand-Naimark theorem.

Note: section 5 of this chapter has been skipped and is not required for the exam.

Exercises for the werkcollege: 8.8, 8.9, 8.10. 8.11, 8.12.

WEEK 24/Lecture 16 (June 15): Chapter 9.

WEEK 25 (June 22): No course (Studieweek).

WEEK 26 (June 29): Exam (tentamen week).

plaats/tijd: woensdag 29 juni van 17.00-20.00 uur in Educatorium Bètazaal

Enjoy the sphere (and not only).