Manifolds 2 | Interior, Exterior, Boundary, Closure

[@malawigw]

Manifolds makes me happy!

[@gdash6925]

Me too. I love topology

[@chris1324_]

What amazing times we live in that such complex topics can be made so accessible. Thank you ☺

[@lucaug10]

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!

[@brightsideofmaths]

Thanks again!

[@StratosFair]

Man, I wait videos from this series like I wait for new episodes of my favourite shows. Thanks for your amazing work !

[@brightsideofmaths]

That is what learning should be :) Thank you very much!

[@lucaug10]

Glad I'm not the only one who feels this way! :D

[@zathrasyes1287]

Same here!

[@Duskull666]

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 :)

[@obaidurrehman2464]

Same is the case with me

[@arisioz]

They’re both awesome courses fr dude

[@Syrian.Coffee]

Finally a great video on the topic ! Shurkan 🙏🏻🙏🏻

[@brightsideofmaths]

Glad you liked it!

[@scollyer.tuition]

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)

[@shirleymoon9934]

It took me some time to understand the boundary of S, but it really deepens my understanding after it!

[@shirleymoon9934]

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)"?

[@brightsideofmaths]

You mean for the example here?

[@shirleymoon9934]

Yes @@brightsideofmaths

[@ahmetfurkanemrehan]

That's amazing lesson. I want to cry in front of such pure form for teaching. Congrats !!!

[@brightsideofmaths]

Thanks :)

[@NewDeal1917]

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

[@arpit_prajapati11]

Thanks for this amazing series

[@brightsideofmaths]

Glad you enjoy it!

[@PunmasterSTP]

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.

[@miguelsantiagoradasanchez9623]

Thanks

[@brightsideofmaths]

Thank you :)

[@Dede-qq8qv]

Thank you so much!! A great effort ♥️

[@natepolidoro4565]

Good stuff, good channel

[@brightsideofmaths]

Much appreciated

[@carl3260]

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?

[@brightsideofmaths]

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.

[@tensorfeld295]

I like this manifold series.

[@christianharriviktorreibol5521]

Very good example at the end

[@arrowmancy]

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".

[@brightsideofmaths]

Yes, the colon in ":" is just a reminder that this is actually the definition of the thing on the left.

[@zazinjozaza6193]

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?

[@brightsideofmaths]

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.

[@mgostIH]

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

[@TheSandkastenverbot]

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

[@zazinjozaza6193]

@@brightsideofmaths Thank you, and everyone who replied. I understand it better now.

[@mgostIH]

@@TheSandkastenverbot And what does it being closed tell me about it not being open? 😄 (Nevermind I did the same mistake before, shhh)

[@sidharthd4400]

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?

[@brightsideofmaths]

Thanks a lot! I don't get your questions exactly. Do you have a problem with the names?

[@sidharthd4400]

After studying the example thoroughly at the end of the video, my doubts are cleared. Thank you for the amazing explanations.

[@dqrksun]

Helped me alot, Thanks!

[@medounendiaye3123]

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 ?

[@brightsideofmaths]

You miss the important part of topology: the sets we call open are fixed in T.

[@medounendiaye3123]

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.

[@TBXu]

@@medounendiaye3123 “More generally, one defines open sets as the members of a given collection of subsets of a given set” on wikipedia it says :-)

[@mastershooter64]

yayyy more manifolds :D

[@StratosFair]

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 ?

[@StratosFair]

@Luca Zz What the heck, you just destroyed my brain. Can't wait for the rest of this series :)

[@heimrath007]

@@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.

[@zaccandels6695]

Hi, what texts would you recommend for someone who wants to study this in a bit greater depth?

[@brightsideofmaths]

Do you mean topology or manifolds?

[@angtrinh6495]

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?

[@brightsideofmaths]

Why not?

[@angtrinh6495]

@@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?

[@brightsideofmaths]

@@angtrinh6495 For such discussions, the community forum is better :)

[@angtrinh6495]

@@brightsideofmaths Thank you! I'll take a look there!

[@xxoloveitt]

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?

[@brightsideofmaths]

Thanks for the questions! Who says that (-inf, 1] is open? The exterior is always an open set.

[@xxoloveitt]

@@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 😭

[@brightsideofmaths]

No, the boundary is always closed.@@xxoloveitt

[@xxoloveitt]

@@brightsideofmaths got it, thanks!

[@kingshukdutta2064]

shouldn't a circle be S(superscript)1?

[@brightsideofmaths]

That would be the notation we use later for the circle as a manifold. Here we are on an abstract level :)

[@ihavezerofriends]

Thanks manyfolds!

[@c.m.139]

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

[@santiagoarce5672]

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.

[@sossupummi]

may I suggest using the search function to look up the International Winter School on Gravity and Light 2015

[@deansmith7163]

kzbin.info/www/bejne/pqq7aZ6HYq-tfqs

[@alijoueizadeh2896]

Thank you.

[@brightsideofmaths]

You're welcome!

[@smolfish3470]

Is it true that if T = P(X) then there is no boundary point and accumulation point, since {p} is contained in T?

[@brightsideofmaths]

This is the discrete topology, where every singleton is an open set. So you can show that a set cannot have boundary points.

[@guigomartins]

so, always it can be said that ∂S = X \ (S° ∪ Ext(S)) or is this a result just for the case of this example?

[@brightsideofmaths]

Yes, this follows directly from the definition.

[@guigomartins]

​@@brightsideofmathsthanks! this is my first contact with this subject formally and your videos have been of great help

[@wenanyaugustine3311]

Hi just thinking, isn't (0,1) an open set? you just said it isn't and I was wondering how prof. Thanks

[@brightsideofmaths]

"open" is always with respect to a given topology.

[@HeirOfCuba]

Does this mean it would be impossible for a subset of a space with a discrete topology to have an accumulation point?

[@brightsideofmaths]

Indeed!

[@shobhitkhajuria7464]

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?

[@brightsideofmaths]

If you have [a,inf), you could write the union of [a+eps, inf) for eps > 0 and get (a,inf).

[@dangthanhtuanit]

In example, S(0,1) is an openset.

[@bobbybannerjee5156]

Brilliant lecture. May I know what software (app) you are using to write on the screen with your (stylus) pen?

[@brightsideofmaths]

Xournal :)

[@roozbehr93]

I hope the next one is about Charts, Atlases, Compatibility, Differentiable Manifolds etc. ;)

[@brightsideofmaths]

First, we finish topologies and then we start with differentiable manifolds :)

[@养兔大户]

nice video！

[@brightsideofmaths]

Thank you! Cheers!

[@trondsaue7860]

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....

[@brightsideofmaths]

The point is that the property of the intersection holds for *all* open sets U.

[@trondsaue7860]

@@brightsideofmaths Got it ! Thanks ! I very much enjoy these videos.

[@SamSarwat90]

I got the accumulation points of S to be { x | x

[@brightsideofmaths]

Nice work!

[@shakesbeer00]

So accumulation points are the union of interior points and boundary points.

[@shakesbeer00]

In the example, the derived set S' should be the same as the closure of S.

[@brightsideofmaths]

No, not all boundary points are accumulation points.

[@brightsideofmaths]

@@shakesbeer00 In this example, it's correct.

[@shakesbeer00]

@@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!

[@fabiangn8022]

gracias.

[@loden5677]

I don’t get how S can be a subset of X but not necessarily in T, given X itself is in T???

[@brightsideofmaths]

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.

[@loden5677]

@@brightsideofmaths Thanks for clarifying, I see what you mean!

[@samlai5715]

interesting example

[@jacobadamczyk3353]

Does anyone know what note-taking/recording software this is?

[@brightsideofmaths]

I know it!

[@jacobadamczyk3353]

@@brightsideofmaths haha, I'd sure hope so! Do you mind spilling your secrets? P.S. Thank you for making all these great videos!

[@brightsideofmaths]

@@jacobadamczyk3353 Of course. I read your question like you just asked about existence.

[@brightsideofmaths]

I use Xournal :)

[@jacobadamczyk3353]

@@brightsideofmaths I should've expected that much 🤣 thank you!

[@nazmulhasanfahim731]

Where is Quiz for video part-2?

[@brightsideofmaths]

Oh, you can find it here: tbsom.de/s/mf

[@michuosas]

so (0,1) is NOT in the topology, right?

[@brightsideofmaths]

It is not an element of T. :)

[@marcelo4303]

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 ?

[@keeperofthelight9681]

I can choose arbitrary u then any p can be shown as boundary interior exterior rip :’(

[@ativjoshi1049]

Definitions were simple; the example, not so much...

[@PunmasterSTP]

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!

[@ativjoshi1049]

@@PunmasterSTP definitely true.

[@cristhiangalindo4800]

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.

[@antonellomascarello4698]

💛🙏

[@zSliz1]

You do a miserable job with examples. Your examples suck, and you don't help with intuition. You just explain rules

[@brightsideofmaths]

What is wrong with explaining rules?

[@christianharriviktorreibol5521]

Very good example at the end

[@brightsideofmaths]

Thank you very much :)