Recent content by Korybut

The original statement is false, even though written in a textbook. Counter example is well known actually Consider the map ##(0,2\pi)\rightarrow \mathbb{R}^2## such that ##t\rightarrow (\sin 2t, \sin t)##. Image of this map is figure-eight, which is closed in ##\mathbb{R}^2##, map itself is...

As it happens usually once you post you are full of ideas I have the following sketch of the proof. Since ##f(N)=M## is locally closed for any point ##x\in M## there is open ##U\subset S## such that ##M\cap U## is closed in ##U##. By assumption ##f:N\rightarrow S## is continuous. Therefore I...

Hello! There is a simple line in the textbook. If ##S## is a manifold, an injectively immersed submanifold ##M## of ##S## is embedded if and only if ##M## is locally closed in ##S##. Recall the definition. M is locally closed if for each point ##x\in M## there open ##U\subset S## such that...

Actually everything is finished or at least perfectly clear to me now. These sets ##\lbrace W_i^j \rbrace## are indeed locally finite refinement of arbitrary cover ##\lbrace U_\alpha \rbrace_{\alpha\in A}## (Not the best notation IMO since ##\lbrace U_n \rbrace## is basis for topology).

Problem occur due to my terrible misunderstanding of what is locally finite space. In my notes I have "Topological space where each point is contained in finitely many open sets", don't know where I found this one. With correct definition everything is obviously fine with the proof.

Thanks! Your example exhibits the obstruction I was missing. How about the last step of the proof? I know it is technical, sorry for that

I believe this should insist that ##\lbrace W_i^j \rbrace## is locally finite however I am confused. If ##|j-k| <4## then this intersection can be non empty so locally finite refinement is not presented. Or I miss something... Need help I get why within 4 steps intersection is trivial...

Not necessary. I can not the proof that was presented. I can pick any ##U_n## from the basis ##\lbrace U_n \rbrace## and choose any point say ##y## within ##U_n##. Due to local compactness there are open ##U## and compact ##C## such that ## x\in U \subseteq C##. However ##U_n## might not be a...

It is clear now and it is very easy. Let closure of ##V_1## is covered by finite amount of open sets which are not necessary basic elements. Let some of this open sets are generated by infinite union of basic set. This is still a cover and due to compactness there is finite subcover of basic...

Hello! I have some troubles diving in the proof of this lemma Lemma. Let ##S## be locally compact, Hausdorff and second countable. Then every open cover ##\lbrace U_\alpha \rbrace## of ##S## has a countable, locally finite refinement consisting of open sets with compact closures. Proof...

I am very thankful to everyone for clarifying the subject to me. Proof in the Lee's book is indeed consistent and there are no questions left

Sorry, Indeed book is available online https://www3.nd.edu/~lnicolae/Lectures.pdf My screenshots are dealing with example 1.2.22 on page 15

Hello! I would like to be sure about my understanding of the definition provided in screenshot below 1. What is this ##\mathcal{E}(G,N)##? I know that not all extension are isomorphic so I wonder What are the elements of ##\mathcal{E}## groups? Or maybe all Es diffeomorphic to each other...

Hello! There is a proof that Grassmannian is indeed a smooth manifold provided in Nicolaescu textbook on differential geometry. Screenshots are below There are some troubles with signs in the formulas please ignore them they are not relevant. My questions are the following: 1. After (1.2.5)...