Topologie - 2018

Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like me kicking you with my fists.

Tomás, 24.08.2018 11:30-11:50

Topics:

• First countable compact implies sequentially compact
• Exercise: For family \(C_i\) of connected sets, \(C_i \cap C_{i+1} \neq \emptyset\), show connected.

Sisto was super nice and friendly, yet it took me a long time to solve the exercise, and the time was over before we could do anything else. Wish I could have done much more.

Michael, 27.08.2018 11:30-11:50

Topics:

• First X,Y first countable, f continiuos (def open set) iff every sequence (xn) with limit x, f(xn) converges to f(x)
• f:X->X,f(x)=/x for all x in X, continuos, metric space, X comp., -> Exists Epsilon>0 s.t d(f(x),x)>=Epsiolon.

Sisto has a list with statements and exercises and chooses randomly.

Kaye, 27.08.2018 11:10-11:30

Topics:

• exercise: you have (xn) sequence with limit x. Prove that the set containing (I think he meant only, because the solution wouldn’t make sense, but he didn’t really say it.) the sequence and the limit is compact.
• f:R—>R continuous function in R. Proof of one direction of the equivalence between the definition of continuity with open sets and the epsilon, delta definition from analysis.
• the fundamental group: what is the “meaning” of x0? He wanted to know that if there exists a path gamma from x0 to y0, then we can find a group isomorphism between the two fundamental groups. (Proposition seen in class).

Then the time was already over! Sisto is really nice, and gives you time to think about the answer and gives useful hints if necessary.

Emanuel, 27.08 13:10-13:30

Topics:

• Prove that compact sets in Hausdorff spaces are closed.

• XxY, Y cpt, p:XxY -> X projection map, prove that p is a closed map. This was actually rather hard, but he was nice enough to say that beforehand. He had to help me a lot but we managed to do it in the end.

The atmosphere is super relaxed. He has a nice couch infront of his room and gives you chocolates :D

Skander, 27.08 13:50-14:10

Topics:

• Let X be a Hausdorff space, C a compact subset of X, show that C is closed.

• Exercise: For family \(C_i\) of connected sets, \(C_i \cap C_{i+1} \neq \emptyset\), show that the union of the connected sets is connected. (The luck for this one is absolutely unreal, I know)

Schöne Ferien euch allen! ;)))

Laurena, 27.08.18 14:10-14:30

Topics:

• X in Rn is compact iff closed and bounded, proof of one implication
• X first countable, C closed iff limit of converging sequence in C is in C, proof of one implication

Exercises:

• Same exercise as Skander and Tomás
• X topological space, topology containing empty set and complement of finite sets, prove that it is not 1st countable

Gideon, 27.08 15:40-16:00

Topics:

• Heine Borel with proof in one direction of choose
• Closed subset of a compact space is compact.
• Prove that the union of a sequence and its limit is compact.

Viviane, 28.8.2018 9:50-10:10

• 3 equivalences for compactness in metric spaces and Proof of compactness in a 1st-countable space implies sequential compactness
• Proof of path connected implies connected
• Proof that the Rational Numbers are totally disconnected

Berno Binkert, 28.8.2018 10:10-10:30

Statements:

• 3 equivalences for compactness in metric spaces
• Proof of compactness in a 1st-countable space implies sequential compactness

Exercises:

• Show that the standart topology in a finite metric space is discrete
• Show that a sequence (in X a topological space, united with its limit is compact in X

Paul, 28.8.2018 10:50-11:10

• State the 3 equivalences of a cpt metric space
• Prove that a cpt first countable space is seq cpt
• Give an example of a continuous map F : X -> Y such that the induced group morphism from the fundamental group of X to the fundamental group of Y is not injective and not surjective. I think he wanted me to find a F that is surjective as well since if we don't require F to be surjective we can simply pick F to map every point of S1 to a unique point of S1 .

Raphael, 28.8.2018 11:10-11:30

• 3 equivalences cpt in metric space
• Proof 1st countable cpt implies sequentially cpt
• Example of two top. Spaces, which are homotopic but not homeomorphic (S1 and R2\0)
• Proof pathconnected implies connected

Joël, 28.8.2018 13:30-13:50

• Prove that \(\mathbb{Q}\) is totally disconnected.
• Under what condition is the limit of a sequence (if it exists) unique? Answer is Hausdorff, then I had to prove it.
• Example of two topological spaces that are not homeomorphic but that are homotopy equivalent. I also chose \(\mathbb{R}^n \setminus \{0\}\) and \(S^{n-1}\), then I had to give the outline of the proof that \(\mathbb{R}^2\) is not homeomorphic to \(\mathbb{R}^n\) for \(n \geq 3\).

Andrin 29.8.2018 11:50-12:10

• finde a continuous bijection which is not a homeomorphism (Id on a space with more then one point where definition space is equipped with discrete topology and image space equipped with trivial topology)
• prove f: X -> Y continuous, X cat, Y HD implies f closed.

- He also wanted the proof of Y Hd, A \sub Y cpt implies A closed and subsequently Y Hd, A \sub Y cpt, y \in Y\setminus A implies exists U,V open with A \sub U and y \in V and U\cap V=\emptyset.

• Then he wanted me to guess the bijection between {f: S1 -> X continuous } modulo homotopic and \(\pi_1(X,x0)\) modulo conjugation. The bijection is \(f \mapsto B_\gamma(f)=\gamma *f *\gamma ^{-1}\) where gamma is a path from x0 to f(0). We identify the circle with the interval [0,1] where 0=1. He wanted me to proof well defindness so showing that the path gamma is replacable and surjectivit.

Johannes, 29.8.2018 16:00-16:20

• finde a continuous bijection which is not a homeomorphism
• proof f: X -> Y continuous, X cat, Y HD implies f closed.
• proof X Hausdorff subset C compact implies C closed.
• proof X Hausdorff implies X Normal
• proof X metric space implies X normal

These where the proofs i had to give we then talked about Hausdorff spaces in general I had to give sometimes small proofs but most of the time a picture was enough.

Emie, 30.8.2018 13:50-14:10

• definition (path)-connected component
• proof connected component closed
• proof A connected, closure of A connected
• are path-connected components closed (no but open when locally path-connected)
• when path-connected component = connected component

Exercise: X a Hausdorff, compact space. We have a inclusion "getting smaller" sequence of compact, non-empty, connected spaces Ci (meaning Ci+1 is in Ci). C is the intersection of all Ci.

• C is closed (use X Hausdorff, compact -> closed)
• C is non-empty (use X compact)
• C is connected (use X normal and part 2)

Idk how to use Latex here so hope you get what I mean.

Manuel, 30.8.2018 16:00-16:20

• Def P(x)
• Def C(x)
• why is P(x) disjoint to P(y)?
  • Proof concatenation continuous
    • proof Lemma which states continuous on closed decomposition is continuous too
• C(x) closed
• Proof of all the used lemmas:
  • Closure connected if A connected
  • Closed in Subspace top of closed iff closed in general topology
• Show P(x) disjoint decomp of X
• Start proof C(x) disjoint decomp of X

He gives little to no feedback and talks really "kryptic" (didnt find out what he wanted straight away), probs my fault. Thats it

Florian, 30.8.2018 17:40-18:00

- What are path components? - What are connected components? - What properties do connected components have? Are they open, closed? (Closed, since closure of conn set is connected) Examples: - If two sets are disconn, are their closures also disconn? (No, for ex [0,1) and (1,2) or two open 1-balls in R^2 at (-1,0), (1,0) - If a set is conn, is its interior conn as well? (At first I can't think of a counter ex, so I start writing down a proof to see where it fails, he advises against and reminds me of the prev example with the two balls. I realise: one closed union one open ball is conn, but the interior is disconn.)

- Connected components of the set X=(1/n)_n>0 union {0} (Conn comp of 1/n are singlets (clopen) for all n). - Are they open? (I say yes, since intersecting X with (1/n-eps,1/n+eps)=1/n for eps small enough) - Is the conn comp of {0} open in X? (No, since for every eps>0: [0,eps) intersects X)

- I should prove the following statement: (X,T), (Y,S) top spaces, p:X->Y covering map, s.t. for all y in Y: p^{-1}(y) is finite and Y compact. Then X is compact. (I start writing an open cover of X and apply the map p to it. I should show that p is an open map. I write down definition of covering and try to write p(U_i) as union of open sets, but don't manage. He gives some hints and draws a sketch to help me, but I don't see how. Then time is up.)

Wünsch allen schöne Ferien! :D

Carlo, 30.8.2018 18:00-18:20

• Def P(x)
• Def C(x)
• why is C(x) disjoint to C(y)?
• why are C(x) closed?
• example: subset oft R consisting oft 1/n for any natural n united with its limit 0. Which C(x) are open?
• Other topic:

Let p:X->Y a cover, Y cpt. Assume preimage of points is finite. Show that X is cpt

• first he wanted me to find an example to illustrate this. I was struggling, but finally found an example Which was not so interesting. His idea would have been the cover of RP2 by S2
• Then I tried to prove the statement by considering an open cover Oi of X. He told me that p(Oi) is not the open cover of Y we were looking for. I did not See immediately that the p(Oi) are open, so I had to prove it, and was struggling again. I was not able to finish the proof of the initial statement

Zheng Chen, 31.8.2018 10:50-11:10

• State the three equivalences of compactness in metric spaces
• Prove totally bounded and complete implies sequentially compact

Exercise: Let p:X->Y a cover, Y sequentially compact. Assume preimage of points is finite. Show that X is sequentially compact

• Let (x_n) sequence in X, yn = f(x_n)
• (y_n) has a converging subsequence (y_nk) -> y
• Consider the subsequence (x_nk) where f(x_nk)=y_nk
• Claim: (x_nk) converges, proof by contradiction:
• Let {x'_1,...,x'_m}=f^-1(y), U open in Y with y in U, U_i such that f^-1(U) is the disjoint union of the U_i
• Then there is {U_1,...,U_m} such that x'_i is in U_i
• Conclude by showing that (x_nk) converges in the intersection of the U_i

Sisto saw me waiting outside and asked if I wanted to start the exam earlier, which I did.

Beat, 31.08.2018 11:50-12:10

• State the three equivalences for compactness in a metric space and prove that sequentially compact implies totally bounded and complete.

• Suppose f:X->Y continous and Y Hausdorff. Show that the graph of f is closed.
• Show that the Mobius strip and the cylinder are not homeomorphic. I was actually able to do something with hints from him but was not able to prove it. I asked him afterwards how the proof works and it is very hard. He never expected me to be able to solve the problem but wanted to see how I approach the problem.

Joel, 31.08.2018 13:30-13:50

• Every open cover of a compact space has a lebesque number. (with proof that d(x,A) is continuous)
• X simply connected, are two paths homotopic?
• X top space, when are two paths homotopic? (with respect to the fundamendal group.)
• X top space, what are the homotopy classes of paths starting at x_0 and ending in y_0 in X. (they're in bijection with the fundamental group. he was really vague here, so I didn't know what to do.)