Manifolds 2 | Interior, Exterior, Boundary, Closure
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- Опубліковано 21 гру 2024
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This is my video series about Manifolds where we start with topology, talk about differential forms and integration on manifolds, and end with the famous Stoke's theorem. I hope that it will help everyone who wants to learn about it.
x
#Manifolds
#Mathematics
#Differential
#LearnMath
#Stokes
#calculus
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
Manifolds makes me happy!
Me too. I love topology
What amazing times we live in that such complex topics can be made so accessible. Thank you ☺
Yes! As an undergraduate physics student who tries to self study extra math courses on the side, this and the functional analysis series are perfect!
Thank you :)
Same is the case with me
They’re both awesome courses fr dude
Man, I wait videos from this series like I wait for new episodes of my favourite shows. Thanks for your amazing work !
That is what learning should be :) Thank you very much!
Glad I'm not the only one who feels this way! :D
Same here!
I love how mind-bending it can be to work with "different" topologies, so fun!
I would like to thank you once again for delivering this amazing content, it is such a joy to watch your videos! It feels great to refresh on some subjects or learn new ones with every new video, and in such an enjoyable way.
There is a lot of effort put into these videos and it shows!
Thanks again!
I like your method of defining the various types of points one after the other, with the associated Venn diagrams; it makes the definitions very easy to compare (and easier to remember than from the typical topology text, IMHO)
It took me some time to understand the boundary of S, but it really deepens my understanding after it!
Also one little question: can I say that the boundary of S is every thing in its left hand side "in this topological space (X,T)"?
You mean for the example here?
Yes
@@brightsideofmaths
Finally a great video on the topic ! Shurkan 🙏🏻🙏🏻
Glad you liked it!
Thanks
Thank you :)
0:00 Intro
0:25 Quick recap: topology definition
1:27 Important points (interior, boundary etc.)
5:36 Important sets made of points (including closure)
8:05 Example. Non standard topology on R
That's amazing lesson. I want to cry in front of such pure form for teaching. Congrats !!!
Thanks :)
Thanks for this amazing series
Glad you enjoy it!
Thank you so much!!
A great effort ♥️
Good stuff, good channel
Much appreciated
Very good example at the end
I like this manifold series.
The following seems fit the given axioms of topology (@ 0:48): X = [0,3] and T = {ø, X, [1,2]}, but I thought closed interval [1,2] couldn't be an open set (except in edge cases). I'm probably missing something but can't see how this violates the axioms.
Or is it that that *is* a topology, and an arbitrary closed interval can be an open set, and it's more that it's not of interest and, in particular, the collection of *all* closed sets gives the discrete topology so isn't of interest?
If you say "closed" interval, you always mean "closed with respect to the standard topology of R". This is interval might not be closed with respect to another topology.
Hi! Sorry it's a silly point on notation/aesthetics, but I notice you sometimes use the ":" (or :\iff in latex) when, in contrast to the assignment/definition symbol for variables (e.g. a := 69) you'd use ":" to define properties rather than variables. Is that a correct reading of it? If so then it's a really nice shorthand for writing down "we call x something if and only if P(x) is true".
Yes, the colon in ":" is just a reminder that this is actually the definition of the thing on the left.
Dang, your examples make my brain melt, but I find them fascinating and I learn so much! Thanks again for taking the time to make and share everything.
Why does the definition of a topology require finite intersections but does not put any restrictions on unions? Would it work if it was the other way around or are there problems with such a definition?
We want to conserve the essence of open sets in metric spaces. Intersections could shrink the set to a non-open set. Unions don't make a problem.
Consider for example in the euclidean topology of R the intersection of all open sets of the form (-1/n, 1/n) for all natural numbers n, this is {0}, which isn't an open set
One example: imagine the open intervalls ]-a, 1+a[, a>0 being a real number. The intersection of all these sets is [0,1], which is a closed set
@@brightsideofmaths Thank you, and everyone who replied. I understand it better now.
@@TheSandkastenverbot And what does it being closed tell me about it not being open? 😄 (Nevermind I did the same mistake before, shhh)
For the boundary point p of S, let's say it's not included in S (a little shifted to the left), but it belongs to U (an element of the topology) whose intersections with both S and complement of S are not empty. Can we still call point p the boundary point of S?
Why not?
@@brightsideofmaths In my intuition, any points that belong to the line of S is the boundary points. But in this case, although U contains p and intersection of U and S (and complement of S) are not empty, but p doesn't lie in the line of S (but still in U). Can you please explain more about it?
@@angtrinh6495 For such discussions, the community forum is better :)
@@brightsideofmaths Thank you! I'll take a look there!
Hmmm it seems to me that being an accumulation point of S is equivalent to being either an interior or boundary point of S, but I can't manage to prove it only with these definitions... I guess that depends on the choice of topology ?
@Luca Zz What the heck, you just destroyed my brain. Can't wait for the rest of this series :)
@@StratosFair Think about the interior and boundary points of the set {1}u{2} as a subset of the real line with the usual topology.
All your lectures are just awesome!!!
On the important names:
What about points of S that are not in any open set.
Can you check the definition of boundary points. Are all p belonging to U the way it is defined boundary points?
Thanks a lot! I don't get your questions exactly. Do you have a problem with the names?
After studying the example thoroughly at the end of the video, my doubts are cleared. Thank you for the amazing explanations.
Hi, what texts would you recommend for someone who wants to study this in a bit greater depth?
Do you mean topology or manifolds?
Hi, thank you so much for your videos! I have a question. On the example the exterior of S does not include the interval (-inf,0] because it is a closed set right? Then why is the boundary point of S (-inf,1] ? Isn't it supposed to be an open set? Why is (-inf,0] closed while (-inf,1] is open?
Thanks for the questions! Who says that (-inf, 1] is open?
The exterior is always an open set.
@@brightsideofmaths oooh okay, then if I understood correctly that means that the boundary of S does not need to be an open set? Btw thanks again for answering! I'm learning this in uni but I don't understand the professor 😭
No, the boundary is always closed.@@xxoloveitt
@@brightsideofmaths got it, thanks!
Hello folks,
I don't get why (0,1) is told to be not open. Any single point grabbed in this interval has an interval around it included in (0,1).
I missed something ?
You miss the important part of topology: the sets we call open are fixed in T.
Thanks for replying and the good work you do day in day out.
Does it mean that (0,1) needs to enumerated in the collection of the open sets ? I'm confused because if we set a=0, (0,1) belongs to (a,+inf). So the statement : " there is no such interval in (a, + inf) " doesn't hold anymore.
Edit : i get myself wrong. By definition, we decided that only (a; +inf) intervals are open in our topology. So, (0,1) is not open in our topology.
For the record : the answer to my question is YES. The open sets has to be explicitly said to be open.
@@medounendiaye3123 “More generally, one defines open sets as the members of a given collection of subsets of a given set” on wikipedia it says :-)
Does this mean it would be impossible for a subset of a space with a discrete topology to have an accumulation point?
Indeed!
shouldn't a circle be S(superscript)1?
That would be the notation we use later for the circle as a manifold. Here we are on an abstract level :)
Is it true that if T = P(X) then there is no boundary point and accumulation point, since {p} is contained in T?
This is the discrete topology, where every singleton is an open set. So you can show that a set cannot have boundary points.
Thanks manyfolds!
Helped me alot, Thanks!
yayyy more manifolds :D
Thank you.
You're welcome!
gracias.
so, always it can be said that ∂S = X \ (S° ∪ Ext(S)) or is this a result just for the case of this example?
Yes, this follows directly from the definition.
@@brightsideofmathsthanks! this is my first contact with this subject formally and your videos have been of great help
You state that a boundary point of S should be neither in the interior nor in the exterior of S, but formally define such points by an open set whose intersection with both S and its complement is not the empty set. However, X is in the topology of X, hence an open set, so what if I choose U = X ? In the definition you do not require U to be a subset of X....
The point is that the property of the intersection holds for *all* open sets U.
@@brightsideofmaths Got it ! Thanks ! I very much enjoy these videos.
Hi just thinking, isn't (0,1) an open set? you just said it isn't and I was wondering how prof. Thanks
"open" is always with respect to a given topology.
In example, S(0,1) is an openset.
I got the accumulation points of S to be { x | x
Nice work!
Can u please tell me why you used (a,inf) as topology and said it’s important, why not [a,inf)?
It will still satisfy the 3 conditions and will be called open sets. Where am i wrong?
If you have [a,inf), you could write the union of [a+eps, inf) for eps > 0 and get (a,inf).
So accumulation points are the union of interior points and boundary points.
In the example, the derived set S' should be the same as the closure of S.
No, not all boundary points are accumulation points.
@@shakesbeer00 In this example, it's correct.
@@brightsideofmaths You are right. It seems that S' contains such points of S that each has a sequence in S converging to it. Thanks for the prompt clarification!
interesting example
I hope the next one is about Charts, Atlases, Compatibility, Differentiable Manifolds etc. ;)
First, we finish topologies and then we start with differentiable manifolds :)
Brilliant lecture. May I know what software (app) you are using to write on the screen with your (stylus) pen?
Xournal :)
Hi, I am 14 and I’d really like to learn advanced physics, but I can’t find high quality videos like yours. Can you please start a Classical or Quantum physics series? (or Relativity). Thx
You may enjoy teaching yourself from textbooks. For Classical Mechanics I like David Morin's Intro to Classical Mechanics which you can prob find online. That should be a good start.
may I suggest using the search function to look up the International Winter School on Gravity and Light 2015
ua-cam.com/video/pyX8kQ-JzHI/v-deo.html
nice video！
Thank you! Cheers!
Where is Quiz for video part-2?
Oh, you can find it here: tbsom.de/s/mf
Does anyone know what note-taking/recording software this is?
I know it!
@@brightsideofmaths haha, I'd sure hope so! Do you mind spilling your secrets?
P.S. Thank you for making all these great videos!
@@jacobadamczyk3353 Of course. I read your question like you just asked about existence.
I use Xournal :)
@@brightsideofmaths I should've expected that much 🤣 thank you!
I don’t get how S can be a subset of X but not necessarily in T, given X itself is in T???
You have to be careful when talking about elements and subsets relation. We have S ⊆ X and S ∈ P(X) but also T ⊆ P(X). However since T is not equal to P(X), it can happen that we have S ∉ T. In short: a set can be an element or a subset of another set. These two notions are different.
@@brightsideofmaths Thanks for clarifying, I see what you mean!
so (0,1) is NOT in the topology, right?
It is not an element of T. :)
What if a=0 -> (0, infinity) in TAu. (0, 1, 2, 3... infinity). The elements 0,1( and the infinity elements between, when it comes to the elements of the set of reals) in this interval
is not already contained in tau ?
I can choose arbitrary u then any p can be shown as boundary interior exterior rip :’(
Definitions were simple; the example, not so much...
I really like that he makes his videos that way. They start by laying a foundation, and then they take it in a whole new direction!
@@PunmasterSTP definitely true.
💛🙏
Hola como esta?
hablaré esta vez en español. Surge una importante pregunta de clases peculiares de variedad de curvas Mg en algún 3-pliegue de CY. La pregunta en cuestión es probar que para n=3 de muchos pliegues todo CY\times{1}= CY\{1\}, que prueba para tal parte unitaria como es de proyectiva una variedad Mg, en Mg\times{} CY\{1\}= Mg\times{} \{-\prime}, esto pues la parte primitiva de una variedad Calabi-Yau con muchos pliegues, es capas de generar "puntos" de curvas muy parecido al espacio Q-racional. Las preguntas de investigación que yo con otros investigadores (basado en la obra de Dirchilf y Joyce) es entender cómo para este espacio Q-rational sustituimos el unitario CY\{1\}, por un space-modulo de Hodge que son todas las bases de un diagonal D^{\prime{} - 1}= D\times{} C. Aquí se construye un módulo D(1) para incrustar los 3-pliegues de un CY en su única y constante parte primitiva. De hay por ejemplo se podría entender muy bien con las curvas conjeturas por Dirchilf-Joyce son muy altas en una superficie-Enrique, que sea capas de producir curvas semi-estables no necesariamente degenerada (calculo "estable" para pliegues de el invariante DT), que puede ser escrito también como un grado de la curva, general al módulo D(1)-Hodge incrustado. Esto pues todo reflejo de la superficie-Enrique es cuadrática y a semeja cualquier curva a un esparce alto G-global muy próximo.
You do a miserable job with examples. Your examples suck, and you don't help with intuition. You just explain rules
What is wrong with explaining rules?
Very good example at the end
Thank you very much :)