Difference between revisions of "Topologie - 2019"

Revision as of 15:18, 14 August 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)

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

(no, but sequentially compact tells the subsequence converges)

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

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.

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...