Topologie - 2019

Andrea, 12.08.2019 10:50-11:10

Topics:

• What is the Fundamental group of the Möbius strip?

When two spaces have the same Fundamental Group? Def of Homotopy equivalent spaces and find the homotopy equivalence between the cirle and Mobius strip.

• exercise: you have (xn) sequence with limit x. Prove that the set containing only the sequence and the limit is compact.

I took a long time for the first part and unfortunately after there was no time left.

Sisto is calm and helps you if you make confusion.

Nico, 13.08.2019, 11:30-11:50

Questions:

• Let \(X\) be a compact metric space. Suppose that \( f: X \rightarrow Y \) is continuous and \( f(x) \neq x \) for all \( x \in X \). Prove that there exists \( \epsilon > 0 \) so that \( d(x,f(x)) >\epsilon \) for all \( x \in X \).
• Let \( f , g : X \rightarrow Y \) be homotopic maps, \( X , Y \) topological spaces. Show how the induced map of \( f \) and \( g \) are related.

I needed more than half of the time for the first questions where I couldn't conclude at the end. He tried to help me and sometimes he let me think for a couple of seconds without saying something. We run out of time before I finished everything of the second question.

In front of his office is a couch with some chocolate, one could eat. That is really nice.

Viera, 13.08.2019, 13:40-14:00

Questions:

• Let \(A \subseteq X \) be a discrete subset of a compact space. Is A finite? If not, which additional condition makes it true?
• Give two examples of covering of the circle.

After the 2nd question, he wanted another example of a covering. He moved on to ask me, what happens on the level of fundamental groups. He wanted to know, whether there is a covering, where the fundamental group is mapped to the trivial group (this would be \(\mathbb{R}\)), and then asked if there is also a compact space with this property. However, Luca stopped him and told him, this would be a too hard question, so we stopped at this point.

Tobias, 13.08.2019, 14:20-14:40

Sisto says hi, you give Luca your Legi and you start (really no smalltalk at all). You have a stack of paper and pens ready.

• Let \(X\) compact, \(A \subseteq X\) a discrete subspace. Is \(A \) finite? Which conditions are additionally needed?

I was quite nervous at the beginning and had some problems despite knowing the anserwer that \(A\) should be closed. After some time I came up with the solution.

• Do you remember roughly how we proved that \(X\) compact, \(A \subseteq X\) closed implies \(A\) compact?

I explained the idea of the proof in two sentences without writing something down.

• Let's do something from the second part of the lecture. Sisto lets Luca choose the fundamental group of the Klein Bottle.

I started explaining van Kampen, chose sets A and B as in the lecture but was most of the time stuck trying to show that the fundamental group of the outer set was \( Z*Z \)

• What is the structure where we faced free groups?

I drew the two rings which intersect at one point.

• Are these homeomorphic?

Yes. Then I though again about the question and noticed that they only have the same Fundamental group but are probably not homeomorphic.

• How do you show they have the same fundamental group?

I came with the idea of homotopy equivalence but didn't really know where to start. He gave me some hints how to choose f and g. The time was already over, I explained with one sentence how I get the condition \( abab^{-1} = 1 \).

Sisto waits quite long with giving hints. So take your time, don't get nervous.

Xuwenjia, 13.08.2019 15:40-16:00

Questions

• Describe the construction of the Cantor set with a picture, and state some properties of it.

(With Proves.)After drawing the picture of Cantor set, I wanted to start with totally disconnected and no isolated points, but he stopped me as I was trying to prove totally disconnected, said it's getting too far and wanted me to state some other properties of it, but I didn't get his words at first. What he wanted to hear is the compactness.

• Define the limit of a sequence. Is the limit of a sequence always unique?

(want you to prove: that the limit of a sequence is unique if X is Hausdorff.) In between, he asked me what exactly the open set which contains the limit x is. I answered it is for all open sets which contain x.

• In a compact metric space, does a sequence always have a limit?

(no, but sequentially compact tells the subsequence converges)

If you didn't get an idea, he won't help you at the very beginning but give you a chance to think a bit. Sometimes I didn't get exactly what I should do.

Ole, 13.08.2019 16:20-16:40

There was a 15 minute delay, but luckily Kirill was after me and came early, so it was chill.

• Are subspaces of Hausdorff spaces Hausdorff? Products? Closures?

• What's the fundamental group of (D2 with the equivalence relation from q63).

- There's another way of showing this (I show it again using van Kampen).

- There's another way still. This space is homeomorphic to the projective plane. Show this.

(after a second of what the fucking internally I start trying to intuitively explain this, saying things like "Do you have anything sphere-like? I'll show you.")

- "Alright, give me the concrete homeomorphism."

I did not get anywhere, after like 5 mins he was like ok time's up.

The atmosphere was pretty good

Kirill, 13.08.2019 16:40-17:00

As I came in we started immediately. Sisto took a review table and sad that there is a lot of stuff there to ask, so after 2 or 3 min he asked me thу first question. Overall, I think, 5 out of 20 min Sisto thinks what to ask you simultaneously consulting with Luca.

Questions from the table:

-Are spaces equipped with trivial/discrete topology compact/path-connected? In the case of compact he also asked about countable and finite space. (Because I assumed at the beginning that we talk about uncountable spaces as it is in review tables)

-Are spaces in cofinite topology first-countable?

He asked me to proof my answers. Then we moved to the questions list. He asked about (21).

-Give an example of continuous bijection which is not a homeomorphism. Can you give a general criterion for showing that a continuous bijection is a homeomorphism? I gave an example with id from (X, discrete) to (X, trivial) and sad that it is sufficient for a domain to be compact and for codomain to be Hausdorff. Then he asked whether I remembered the proof, so I started to bring up some useful facts and then he stopped me on one of them and asked me to prove it.

-Why if X is compact and Y is Hausdorf then every continuous function from X to Y is a closed map? After we moved to (106).

-Let p: X−→Y and q: Y −→Z be two covering maps. Assume moreover that all the fibers of q are finite. Describe a strategy to prove that q ◦ p is a covering map.

He said he needed only the set up because we didn't have much time left. So I described the construction of a cover of z in X and he said that that was enough.

Overall the mood was very chill. Although they were in a hurry because of the delay there was no rush during the exam.

Emanuele, 14.08.2019 10:30-10:50

Questions:

• Prove that \(\mathbb{Q}\) is totally disconnected. He asked me to prove also other related facts that I used in the proof (for example that a subset of \(\mathbb{R}\) is totally disconnected iff it does not contain any non empty interval.)
• Prove that \(\mathbb{R}^2\) is not homeomorphic to \(\mathbb{R}^3\)
• Explain why there is an isomorphism between the fundamental group of homotopic equivalent spaces. (He just asked for the idea, not a rigorous proof.)

Silvio, 14.08.2019, 11:30-11:50

Questions:

• Question 20: Let \(f : X → Y\) be continuous, and let Y be Hausdorff. Show that

\(\{(x, f (x)\} \subset X × Y\) (the graph of the function) is closed.

• Question 105: Give two examples of covering of the circle

After the second he wanted the following

- Another connected cover

- Describe what the induced map of the covering does

- Does it exists a cover of the circle by the circle such that the image of the fundamental group under the induced map by the covering map is trivial?

Lukas, 14.08.2019 11:50-1210

Questions:

• (26) State and prove the characterisation of continuity in terms of limits for first-countable spaces.
• (106) Show that for p from X to Y and q from Y to Z two covering maps, and q has in addition a finite fibre. Show that the composition is again a cover.
• (not on the List) State the fundamental group of the Two-Torus without interior.

The atmosphere was relatively nice. Proof Sisto did point out mistakes, and when he sees that you got the idea, he jumps to the next question.

Wayne, 14.08.2019, 14:00-14:20

Questions:

• (1) Prove that f is continuous in the topology sense and the analysis sense are equivalent.
• (107) Show that the induced map f star between the respective fundamental groups is well defined and a homomorphism.

I took an unnecessary amount of time trying to prove (1), expecting it to be easy. I had relatively little time to prove the second question. Sisto makes you think for a worryingly long time before giving you a hint.

Fabian, 14.08.2019, 15:20-15:40

The whole exam was about the line with double zero, which we will call X. First, Sisto recalled the definition of the space X.

• S: Show that X is path-connected.
• Me: Distinguishing three cases, constructing paths for each case
• S: (15) Prove that there is no injective path connecting the two zeros.

I presented my prepared solution, but there were two mistakes in it. After I corrected them, half of the time had passed. Then Sisto asked Luca: "Shall we go for the crazy question?", to which Luca replied: "Yes, I think you can ask the surface question." - "No, I actually meant the other question." - "Well, it's actually the same question." Then, they discussed some time which question they should ask, and finally I said: "I would like to take the crazy question." Sisto seemed very happy.

• S: So, I can tell you that the fundamental group of X is non-trivial. Can you guess some loop which is not null-homotopic?
• After some time I found one.
• S: So, how can you prove that the fundamental group is non-trivial?

I tried to prove that the claimed loop is not null-homotopic, but he recommended me to choose another approach. So I tried to find some cover. After we found out, that the two copies of \(\mathbb{R}\) don't form a cover of X, he asked me to find the universal cover of X. I couldn't find it right away, so I said it would be helpful if I knew the fundamental group, so I would know the degree of the cover. He told me it is \(\mathbb{Z}\), and so I came up with the construction of the universal cover. The time was nearly up, but he already let me go.

Jingi, 14.08.2019, 16:20-16:40

• - S: Show that X is compact and Hausdorff implies X is normal
• - S: What happened if we let the compact out?
• - I: Then it is not normal?!?
• - Then he started to think out loud and said something about A,B, compact etc......(Me: ???)
• - S: OK, Can you give me an example where X is not Hausdorff and A in X is compact but not closed?
• - I: OK, we need a space which is not Hausdorff- e.g. line with double zero!
• - S: (to Luca) Can we work with that?
• - L: yeah sure! We can do it.
• - I: OK.. Or we can work with particular point topology on R.
• - S: (laughed, already writing down the definition of the line with double zero...)
• with some hints and fails I came to the conclusion that {0+} is compact but not closed where we choose trivial topology for {0,1} on {0,1)xR......(???)
• - L: We only have 5 minutes left.
• - S: OK. Then tell me why R2 is not homeomorphic to R3
• I didn't manage to explain rigorously why the fundamental group of R3\{p} is trivial but that was the end of the exam...

15.08.2019, 9:50-11:10

He first started going trough the "subspace"-column of the table, and asked me about compactness, local compactness, then he asked me about normal, I said if it is closed, and he said "okay, and what if it is not closed, try to prove it then", I tried to prove it, and got very confused, after a long time struggling with the proof he he said "okay, maybe it's a little too confusing" and then he changed the topic and asked me number (108) from the list of questions, but we only had enough time for the (\ \Leftarrow \) direction.

The ambiance is relaxed, Sisto does not stress you for answers, and helps you when you get stuck in the proofs.

Viola, 15.08.2019, 16:00-16:20

The whole exam was about the induced homomorphism.

He started by asking for the well-definedness (Question 107).

Then he wanted to know when the induced homomorphism is an isomorphism (he wanted to know homotopy equivalences). It took me some time to actually get the answer because his way of asking questions can be very confusing at times..

Next (Question 59): I stated the Lemma that for f,g: X—> Y homotopic maps there exists a path gamma so that g* = beta f* He then asked me if Y would have to be path-connected. After I came to the conclusion ‘probably not’, he claimed ‘no’ and however gamma would always exist. He then wanted me to prove the existence of gamma by explicitly constructing it using the homotopy property. I struggled a bit but eventually came to the right conclusion at which point there was only 1 minute left and he said “There’s no more time now” and the exam was over.

Samuel, 21.08.2019, 11:10-11:30

• Q1: Let X be compact and Hausdorff. Show X is normal. He also wanted to know why a closed subset in a compact space is compact.
• Q2: Show that the fundamental group of the unit circle are the whole numbers Z.

Sisto waits some time before giving hints and i didn't quite understand when something is trivial enough to not prove it and when he wanted a complete proof. Overall a very nice atmosphere.

Joel, 21.08.2019, 15:20-15:40

I come in and Luca welcomes me very friendly, afterwards I greet the Professor.

• S: Let's start with the Cantor Set. How is it defined?
• I: (I draw a picture and describe how the \( C_k \) are defined.) C is the intersection of the \( C_k\).
• S: Right, can you state some properties of the Cantor Set?
• I: Totally disconnected, compact and no isolated points.
• S: Okay, why is it compact?
• I: Because [0,1] is compact and Hausdorff, a closed subset is also compact. C is an intersection of closed sets and therefore also closed and so compact.
• S: Correct, can you show it is totally disconnected?
• I: Draw a picture and describe how to show it.
• S: Alright, now can you formalize your proof?
• I: I struggle to show that there can't be any intervall in C.
• S: (Not really satisfied, he completes my proof.)
• S: Now lets talk about Contractible Spaces. Can you define, when a space is contractible?
• I: (Write down the definition and called the contraction point p.)
• S: Okay, now lets take another point q in our Space. It it contractible to this point as well?
• I: Yes, because there is a path from q to p via the contraction, we can first contract to p and then take the path back to q.
• S: Correct, can you formalize your proof?
• I: (Again struggeling) Yes we concatinate the two. (Write down the concatination).
• S: Okay, there was a Lemma in the Lecture, stating a property, when such a map is continuous. What was it?
• I: (No idea at all) Eeehm ,if the two parts coincide at t=1/2?
• S: That means that the function is well defined.
• S: (After realizing that I have no Idea) The sets have to be closed. (Note: This is exactely the last point on the Email, Prof. Sisto sent on 23.08.)
• S: The time is over.

Overall pretty calm, but he seemed pretty bored and looked at his watch every time I stopped talking. (I was the last Student that day, I think).