One day when I'm working and have money, you're the first TH-camr I'll support on Patreon. I've learned so much from your "What is a Tensor?" series of lectures. Channels like these make me wish I went to college and studied pure and applied mathematics. Super excited for Manifolds! I've been having logical gaps from Tensors that I really hope these videos will go some way towards addressing. Keep doing what you do, your project is changing lives (and learning math has brought some meaning and confidence back to my life). Best, -Float Circuit.
@17:50 The number of elements actually comes out to be 2^x, where x is the number of elements in the set, And also, thanks for the awesome vids!! Really helped me quite a lot!
Bro, I appreciate your treatment of these subjects. I'm a self taught math/science enthusiast. Much of my knowledge has it's roots in your lessons. Thank you. Plus, I must add. When I have my balls I can make anything as well! The truth of mathematical abstraction strikes again ❤🎉😂
Hello thank you for making these videos, 5 years ago in high-school I started self teaching myself topology with these fantastic videos as a guide. I eventually went on to reading some textbooks after a few weeks of watching on youtube and found the subject very rich and fascinating, specifically algebraic topology which I spent a year or two learning a lot about in my free time! Now I have stumbled upon your channel once again to try to grasp general relativity and tensor calculus on manifolds after taking a 2-year long hiatus from mathematics because I was studying other things in university that was taking up a lot of free time but now I have time to get back to it and am learning about general relativity and tensor calculus which I never spent much time reading about except for differential forms. Thanks again =)
Hi, I loved the lecture! At 7:16, you say that the union of any two open sets is open, and that this immediately implies the arbitrary union of open sets is open. I don't follow this. I can see why it would follow that any finite union of open sets is open, but I don't see how it follows that an infinite union is open. Are you using induction? Even with induction, it would seem that would only mean a countably finite union of open sets is open. How would it imply that a uncountably infinite union is open? Thanks!
The assumption is that any union (finite, countable, uncountable) of open sets is open. The assumption that the union of any two sets is open would only get you that any finite union is open. Note the intersection of any two open sets is open only implies that any finite intersection is open. There are easy examples where even a countable union of open sets is not open (think of open intervals around a common point that shrink in the limit to that point).
Just a sincere request, I believe the new generation of physicists can use a good course on group theory and maybe a short review of complex algebra. Since you explain these concepts amazingly, I thought, it's not the worst idea to ask about such courses. Thank you very much for your great videos.
This is a good idea. Complex analysis is a very important topic, along with special functions. I may work it all into the Lie Group lecture series. Thanks for watching!
Thank you for a fantastic set of lectures. What can sound daunting you make simple by focusing on the underlying concepts and explaining clearly the different ideas. Maybe you could create a play-list putting together the different chapters. That would help in ensuring that things are followed in the proper sequence. Also you have some older lectures posted. How do they dove-tail with your current set. Again thank you very much for a very enjoyable lectures. George (from Versailles in France)
When you said that the union of any two open sets is an open set, doesn't that only imply that finite unions are open? Because induction only holds up to the finite case. Surely the general theorem should be that the union of any subset of the topology is a member of the topology?
Sorry I know it's 2019, but for the line topology described from 20:21 onwards, how does one get the null set since the smallest open set is (0 - 0.5) when N=2? I supposed X is the open set (0 - 1) when 1-1/infinity? Cheers.
I have a little question, the vedio say the open ball can form any arbitrary shape in plane, but I still don't understaned how to use "FINITE" intersections and unions of circles to form rectangle. It doesn't make sense to me, can somebody explain it to me clearly?
Unions need not be finite; so take unions. By making the balls arbitrarily small you can get arbitrarily close to the shape of the rectangle, so that in the limit the difference between what you get and the rectangle is null. Imagine it as covering a rectangle with as small and as many circles as tou need.
Sir, I have a question. When you defined a topology on the interval (0,1) as the set of all intervals of the form (0, 1 - 1/n), where n belongs to {2,3,4...}, then which element in the topology is the null set(phi) ? For that matter, what is the definition of a null set(phi) ?
I have just watched the first lecture in the lecture series on manifolds. Really liking it so far! Are there any recommended, or otherwise relevant, exercises to go along with the series on manifolds?
For the people interested, two other very common topologies are the box and product topologies. Although they are same for finite products, but they are different for infinite products. As for the real line there are the lower and upper limit topologies except the standard topology.
15:35 You state that the union of disjoint sets is part of the topology. Is that necessarily true if we define the topology as all open intervals on the line? I.e. if we took the union of {1,2,3} and {5,6,7} the set {1,2,3,5,6,7} wouldn’t be in the topology right? (but the set {1,2,3,4,5,6,7} would be)
The rule is that the union of any two open sets must also be an open set. I don’t understand your example, but if {1,2,3} is open and if {567} is open then {1234567} must be open too.
Thanks for the fast reply! I am looking at the 3 rules for a formal definition of a topology. From the Wikipedia definition I have "2. Any union of elements of τ is an element of τ." So in my example, the union of the disjoint sets would be {1,2,3,5,6,7}, excluding 4 and this set wouldn’t necessarily exist with the definition of the basis used for the standard topology right? It seems you are stating the requirement to define the topology is that the union of 2 sets has to be an open set but in your lecture (and elsewhere) I am seeing that the union of 2 sets has to be AN ELEMENT of the topology. I am wondering if these 2 things are equivalent and if so how would the set {1,2,3,5,6,7} end up in the topology given the basis used to define the standard topology. Sorry if I am mixing up some terminology here, I’m a little bit new to the subject.
Ah ok, along with your explanation at 23:39 and an external reference, I realized I was misunderstanding the definition. The basis is only part of the definition of the standard topology, the union of all sets of the basis is also part of the topology! Taken from the external reference: A topology on the real line is given by the collection of intervals of the form (a, b) along with arbitrary unions of such intervals. Let I = {(a, b) | a, b ∈ R}. Then the sets X=R and T ={∪αIα |Iα ∈I} is a topological space. This is R under the “usual topology.”
Hello, I would really appriciate some clarification on the subject of topology consisting of open sets. If that is the case then, if I understand it correctly, there could be no set X in topology, because set X contains all the points in plane, and therefore boundary too, so it is not an open set. Is that thought correct, or am I missing something?
I’ve always thought it most helpful to say a Topology just is a subset of the powerset of whatever set you’re interested in (rather than leaving it at open sets).
In physics the metric is determined by physical laws. General Relativity can be thought of as the theory of the spacetime metric. The Einstein Equation is the mathematical statement of the physical law that gives us the metric of spacetime.
I am following your lectures for some time although I am not a physic or math student, I am recently studying geometric deep learning and I find your lecture series really helpful for that. I hope you continue your great effort to make those recondite topics accessible to all. Also can you please suggest to me which books are following in this playlist?
At 21:08 you wrote points from equation 1-1/n for n=2,3,4,... and you wrote 1/2, 3/4, 4/5... but between 1/2 and 3/4 there should be 2/3. Nevermind, you can drop this one from open sets collection.
just to be sure as this is a crucial point: at 10:4 you define what an open ball is in the plane using |X-P|< r where r belongs to the real numbers. But r must be strictly positive. Thus the point P itself does not belong to the open ball. Is this correct ?
My excuse. I was confused and my last sentence should be skipped. As r must be strictly positive (this should be corrected in the video), the point itself belong to the open ball. Only a set that only contains the point P is not an open ball.
Jacques Smeets If r=0 and P was in an open ball all by itself that would be the discrete topology! But the presentation is the “standard topology” where, as you say, r > 0.
Thanks a lot for the extremely helpful video! One question: Why is it that open balls and open rectangles are two different bases, but the discussed two alternatives for the open set on the line are not two different bases, but two different topologies?
If one base can be constructed from the other, then they are bases of the same topology. If you can not construct one base from the other, then they are bases of different topologies.
I thought you were a physicist who uses and respects math as a tool. Surveying your patreon page I see that you have a love for the beauty of the mathematics itself. Homeboy Poincare would certainly not disagree. Subscribed.
Would it be correct to say that the "null set" rule is a corollary of the intersection rule? For example, if the intersection of any 2 open sets is an empty set, then I can find two such sets whose "intersection" is the null set, which is therefore an element of the topology. Similarly for the union rule and X being an element..?
I thought a bit more about it. The one problem that occurs to me is if we took the entire set as open and started just with that. Then your method will not produce any other open sets, including the null set. I think we need to stipulate the null set and the entire set must be in every topology.
Let X be an infinite set. If you take the set of subsets of X whose complements are finite as a topology for X, then intersections will not include the empty set. This is because these subsets will disagree only on finitely many points and so the intersections will always be infinite subsets of X.
Not with this material actually, I cover a broad range of topics. You need a good introduction you book on standard topology (point-set topology, not algebraic topology) mostly. There are many books and articles about manifolds out there. I think I used “Differential Geometric Structures” by Walter A. Poor (Dover). That is certainly good.
Montavon Jean-Baptiste Yes, it is possible. In the discrete topology every set is topologically open, including all "closed" sets. In fact, there are topologies that have as a basis half-open sets of various types. And, topologically speaking, the entire set X is a member of the topology and it is BOTH topologically open and topologically closed (its complement is the null set which is open and the complement of an open set it by definition closed). Such a set is sometimes called "clopen".
It is so by definition (easy answer) and the null set is a subset of every set. It is the complement of the entire set, so it is both open AND closed! It is "clopen" :)
Do you think that you would be able to make a video or a series of videos on the math of String theory in the current 11 spatial dimensions with respect to M-theory.
This is awesome! Thanks so much. Thorough in a perfect way, for people (like me) who just needs to understand it and get the intuitive meaning of it, without all the rigorous definitions and proofs. Yay!
Jonas Kyhnæb I'm glad you like it. I do suggest that you find an elementary point set topology text and read it after you finish these lectures. It will be much easier and more interesting. GOod Luck1
@@XylyXylyX Same here, I'm a dumbass who has hope that they can relearn maths and understand topics like this thanks in part to great explanation. I really hate when I'm trying to learn something and it's just 'ok, this is how we solve this class of problems!'. Much prefer the explanation of whats going on. Subscribed, hope ye haven't quit youtube :)
JT Coriolis This is a HUGE observation! In fact if we define a topology in terms of "closed sets" (an equivalent way to do it) then the rules regarding unions and intersections are reversed! THink this way: if we allow infinite intersections then we can create an infinite list of open sets whose infinite intersection is *only one point*. Therefore every point would be a member of the topology and then *every topology* would be the discrete topology. How uninteresting! By the way, understand these rules were settled on after many many years of struggle to understand what were the most important concepts required to understand continuity and limits. THey are NOT intuitive at all. To me at least....:)
@@XylyXylyX Why are discrete topologies uninteresting? They would seem to give you best detail or most accurate representation of some metric space and perhaps easier to identify isopmorphisms between different discrete topological spaces.
Miguel Amaral The problem is that every mapping is continuous in the discrete topology and therefore no function is really distinguishable in that sense. Also, topological spaces don’t usually have “isomorphisms” but they do have “homeomorphisms.” If all the spaces involved use the discrete topoloy then finding homeomorphisms would indeed be easier because all the mappings and the inverse mappings would be open mappings. The only requirement would then be that the mappings were 1-1 and onto.
You're absolutely right about physics students taking Complex Analysis. I did that years ago and it completely cleared up the mess of that subject when it was taught by physicists. Another physics major mentioned the same thing. "Advanced Calculus" also recommended, for the epsilon-delta definition of limit. That blew all the cobwebs away from that concept. Why don't they teach that straight away instead of all this rubbish about "getting closer and closer to ... but never reaching it"!! And the most lucid book of any kind I've read is the one on Topology ... can't remember the author at the moment. Update: it'sTopology a First Course, by Munkres. And the above is no slight against the physics profs who taught me a lot of wonderful physics; it's just that most of them were not equipped to teach mathematics, and the texts that they used in class were no help.
It is all about time, usually. I learned the most after graduation. We all need to make sure that when we complete our coursework that we have learned how to teach ourselves!
I have been telling people to find a book they like themselves, because there are so many. I can’t believe that these lectures leave topology books behind....you must mean the manifold material isn’t in a typical topology book? Regardless, do not rely on my lectures to learn this stuff. Find a good book too! Most published books are good, in my opinion. Just be sure they are aboutPoint Set Topology, not something else.
"open sets" are just the name we give to the sets in a topology. "closed sets" are just the name we give to the complement of open sets. The point is the logic we derive from the concept of a topological space, not the name we give to things.
@@atiurrahman7907 Buuuut, I should point out that all of the theory of topology can also be derived by defining a "closed set" a inverting all of the basic principles defining a topological space. So you do bring up more than just a matter of semantics...the notion of topology can be approached several different ways. In fact, one way involving the definition of "neighborhoods", if I recall correctly, doesn't even need a topology at all!
Who would have thought I'd make it through lecture one. Thank you, your explanations were crystal clear! Just curious, may you clarify the concept of a topological base, I got lost in your discussion of it towards the end. Alternatively, do you have any online references in mind that explain topological bases well enough for someone that just stumbled on this math by chance (not a rigorous math person)?. Keep on being awesome! -Float Circuit.
A base of a topology is a collection of specific open sets that can be used to create all possible open sets by arbitrary union and finite intersection. It is actually quite a simple concept in principle. Topology is not something to casually dabble with, in my opinion. On the other hand, elementary point-set topology isn't very difficult as long as you can allow yourself to think abstractly. ANY introductory topology text will do. None are all that much better than any other, in my opinion. try: archive.org/details/introductiontoto00game I liked that book a lot.
No, it is not redundant. X is a set. The pairing (X, T) is a list of two different sets. It is true that T contains X, but there are many different Ts that could be a topology on X.
I LOVE learning from you. I am an engineer, but I am loving this stuff not only because it is leading me to manifolds, but I love it in and of itself. THANK YOU.... Moving on.... Just at about 10 minutes, you declared that the set X with a topology is what we were after -- a MINIMAL group (not in the algebraic sense of the word) of attributes. But you ALSO said that your set X was equipped with a notion of "distance."I cannot see how you can have a notion of distance for a set X when the only attributes are the notions of open sets: union is in the set, finite intersection in the set, 0 in the set, and X in the set.I cannot see how you can acuquire a sense of distance from those minimum attributes. Now at 11 minutes you define the open ball. So are you suggesting that that notion of an open ball is all you need? ALSO (second issue) WHY is a single point NOT an open set? It seems to me that you should have supplemented your definition of open set with the fact that it must contain at least two points.
JT Coriolis Lets take these one at a time. First: the notion of "distance" is, as you have stated yourself, an extra concept that is applied to a set. An open ball can not be defined if we cant define "r". Ultimately this means that we must endow our set X with a metric. In this case it is the standard Euclidean metric. However, the SAME topology can be constructed from rectangles and rectangles do not need an "r": they are just defined by their coordinates. You are correct: the topology does not define "distance" in any geometric sense. For that we must assert a rule that measures distance and in this case we use standard Euclidean distance. However a topology does provide a sense of "nearness" in that ultimately it is the topology that determines what points are the limit points of various subsets of S. 2) Regarding single points as open sets: It is entirely our choice what counts as an open set as long as those rules work. We can choose the "discrete topology" and make every point an open set if we wish. However, as you will see later, if you choose the discrete topology then every function is continuous and no set or series has a limit point. So, we choose topologies that are "weaker" than the discrete topology that provide interesting and useful results regarding functions.
You should add this axiom in the first place "The arbitrary union of open sets is open. If you just put the axiom that the union of two open sets is open, it will give you that the finite union of open sets is open.
Very good video, but slight nitpick at 26:53 you refer to the empty set as phi, but it is actually not the greek letter phi that denotes out but the Danish-Norwegian letter Ø. Pronounced like the u in up.
And a second question, it seems that a point set without a metric, or 'structure' as you put it, could still have topology, as the discrete and trivial topology always hold. So am I right in saying that topology is built on the structure of such set, but does not specify the structure? And in this case the metric is what structure really means.
高远 IT is not a "problem" really, but if you did you would simply have the discrete topology because the intersection of two balls touching at one point would also be an open set and therefore every point would be its own open set.
高远 if I understand you correctly, yes. A topology can exist for ANY set and a metric is not required. Even the usual topology of the plane with a metric is the same as the "open rectangle topology" of the plane without a metric, for example. THis is probably because the plane with the open rectangle topology is "metrizable" regardless of whether or not a metric has been established.
Well I think being metrizable is equivalent to having metric, or there exists the notion of distance. The specific form of metric does not matter. What I mean by not having a metric is there is no notion of distance, the set is just collection of points. And in such case rectangle would make no sense either. But even with a set that is just collection of points, we could still establish the trivial and discrete topology, right? It seems 'topology' is not interchangeable with 'structure' of the set, as you developed the logic.
May I ask a more general question, that can we study manifold and coordinate without engaging geometry? Or at least we engage only what is necessary in a step by step manner? Just from simplest set of points, then requiring it being a metric space(maybe?), and then by somehow it is justifiable to map the elements of the set to a coordinate so that we can study the set quantitatively.
One definition of an open set is any set that contains a neighborhood of all of its points. If a set contains its boundary then the points on the boundary do not have a neighborhood that lies inside the set. It is good to link up all of the different ways of defining open sets to see how they are all equivalent.
Yes, except no one ever does that. Profs teach a neighborhood Heine-Borel definition and then ask you to prove sequence-type Bolzano Weierstrass statements on the test...or vice versa...
Sandy Grungerson True. Usually a course in topology picks a single definition of open sets and derives everything from there. I can't complain too much, it is probably less confusing for a first pass through the subject. Topology is incredibly deep.
Hi xylyxlyx. I am a Patreon since 4 months ( 5€/months). I stil have not received any copy ( “way does it move”). One month ago I tried to contact you via the patron arena, but No reply. Therefore I repeat my request here: when/ how can I expert the copies ? Note, I am not the only one who is complaining about this. Would highly appreciatie to receive your quick reply. Thanks.
As a mathematician I found your youtube site looking for some general information on topology. My primary interests are math logic, set theory, group theory, info theory and analysis topics roughly in that order. As I watched your intro and following lectures in this series, I was struck by the lack of any proofs given. At first, this was a minor annoyance, but as I watched I realized your target audience is quite possibly not mathematicians. Then I watched this initial video and it became clear you are directing your lessons to applied math viewers. Now, I'm actually relieved you don't go thru lengthy proof schemes for every statement you make in these lectures, it certainly speeds up the instruction. One comment regarding continuous homeomorphisms between a topological space and R2. You seem to imply it has to be unique. I can envision a continuous map onto that is not unique. As long as the set and its topology are mapping unique elements to R2, we should be able to map to the same R2 space. The Axiom of Choice should guarantee this. Which btw I was surprised you didn't mention in the lecture either. Well all in all, I'm finding your lectures very informative and enjoyable.
Robleh100 This set of lectures is meant to be a sidetrack for the “What is a Tensor?” Lectures. The idea is to give physicists a working vocabulary of topology so they can have a deeper appreciation of the concept of manifolds which are used in general relativity to model space time. A real topology course would be, as you say, a continuous stream of proofs. However, note from time to time I do drop in a proof but in no way is this lecture series suitable for mathematics students. The axiom of choice is a level deeper than I want to go. I do mention it in the series about the foundations of quantum mechanics when I discuss the Hamel basis of a vector space, I am not sure where I may have implied that homeomorphisms should be unique, I certainly know better! :) Thank you for you comments.
one last comment before I try to refrain from becoming a frequent commenter, your youtube... how shall I say it...screen name... I guess... XylyXylyX, it has begun to intrigue me and I'm probably reading too much into it, but it seems to be maybe a way to list the order of a group? I thought hmmm ....3,4,2 for the letters.... lemme see could be a group number under some modulus, then it struck me... ohhh palindromic yes, yes, it's a palindrome. This seemed interesting too since I've become interested in recurrent palindromes on the 12-hour clock, but again I could be making something outta nothing...so just what is that name all about? I didn't look at your about area.....
I have searched TH-cam for a good explanation for this subject, and this is by far the best. Liked and subscribed.
@@kingassassin7953 Thank you for your kind comment. Good luck with your studies!
One day when I'm working and have money, you're the first TH-camr I'll support on Patreon. I've learned so much from your "What is a Tensor?" series of lectures. Channels like these make me wish I went to college and studied pure and applied mathematics. Super excited for Manifolds! I've been having logical gaps from Tensors that I really hope these videos will go some way towards addressing.
Keep doing what you do, your project is changing lives (and learning math has brought some meaning and confidence back to my life).
Best,
-Float Circuit.
Float Circuit Thank you for your kind words :) Good luck with your studies!
Same. I don't even have money to buy 1 kg rice.
You're SO much better at explaining things than my topology professor it's unbelievable
You just taught half the topology course I took in undegrad in a half hour video. Well done.
this is just half of the chapter 2 of Munkres....
@@henrydavidh.3515 Yeah what kind of course was it lmao, we did this in half a lesson of calculus 2
I really appreciate such videos since they are the source of deep learning 🎉
Starting this series today. Thank you so much for all the work you've put into these videos.
Finally someone who could tell me about the applications of the intervals in Topology, and the meaning of the theorems. Thank you
@17:50 The number of elements actually comes out to be 2^x, where x is the number of elements in the set, And also, thanks for the awesome vids!! Really helped me quite a lot!
Bro, I appreciate your treatment of these subjects. I'm a self taught math/science enthusiast. Much of my knowledge has it's roots in your lessons. Thank you.
Plus, I must add. When I have my balls I can make anything as well! The truth of mathematical abstraction strikes again ❤🎉😂
Thank you for your kind comment
Hello thank you for making these videos, 5 years ago in high-school I started self teaching myself topology with these fantastic videos as a guide. I eventually went on to reading some textbooks after a few weeks of watching on youtube and found the subject very rich and fascinating, specifically algebraic topology which I spent a year or two learning a lot about in my free time! Now I have stumbled upon your channel once again to try to grasp general relativity and tensor calculus on manifolds after taking a 2-year long hiatus from mathematics because I was studying other things in university that was taking up a lot of free time but now I have time to get back to it and am learning about general relativity and tensor calculus which I never spent much time reading about except for differential forms. Thanks again =)
Hi,
I loved the lecture! At 7:16, you say that the union of any two open sets is open, and that this immediately implies the arbitrary union of open sets is open. I don't follow this.
I can see why it would follow that any finite union of open sets is open, but I don't see how it follows that an infinite union is open. Are you using induction? Even with induction, it would seem that would only mean a countably finite union of open sets is open. How would it imply that a uncountably infinite union is open?
Thanks!
The assumption is that any union (finite, countable, uncountable) of open sets is open. The assumption that the union of any two sets is open would only get you that any finite union is open. Note the intersection of any two open sets is open only implies that any finite intersection is open. There are easy examples where even a countable union of open sets is not open (think of open intervals around a common point that shrink in the limit to that point).
At 13:25 he just said if I start with my balls I can make anything 😂😂😂😂😂😂😂😂
I studied topology in my first semester.. I just didn't completely understood it. But after I studied from you.. I learnt a lot.. Thank you sir...
Just a sincere request, I believe the new generation of physicists can use a good course on group theory and maybe a short review of complex algebra. Since you explain these concepts amazingly, I thought, it's not the worst idea to ask about such courses. Thank you very much for your great videos.
This is a good idea. Complex analysis is a very important topic, along with special functions. I may work it all into the Lie Group lecture series. Thanks for watching!
@@XylyXylyX keep goin', you rock
Thank you for a fantastic set of lectures. What can sound daunting you make simple by focusing on the underlying concepts and explaining clearly the different ideas.
Maybe you could create a play-list putting together the different chapters. That would help in ensuring that things are followed in the proper sequence.
Also you have some older lectures posted. How do they dove-tail with your current set.
Again thank you very much for a very enjoyable lectures.
George (from Versailles in France)
A playlist is a good idea. Especially since I am diverting for a short while into this idea of manifolds. Ok. I'll do it.
When you said that the union of any two open sets is an open set, doesn't that only imply that finite unions are open? Because induction only holds up to the finite case. Surely the general theorem should be that the union of any subset of the topology is a member of the topology?
You are right. Otherwise, we would have also infinite intersections, which we do not have.
Love the background airplanes taking off.
Interesting and inspiring for those who love mathematics
You are amazing, Thank you so much for explaining such complicated concepts in an easy way
At 10:26 does open ball consist of the only set of points at the edge of the ball? Or it consist of all points inside the boundary?
Sorry I know it's 2019, but for the line topology described from 20:21 onwards, how does one get the null set since the smallest open set is (0 - 0.5) when N=2?
I supposed X is the open set (0 - 1) when 1-1/infinity? Cheers.
We just include it! The null set is simply declared to be part of the topology.
it is very motivating lecture
Very good presentation! Thank you.
Ralph Dratman You are welcome. I hope you enjoy the other lessons in the series.
I have a little question, the vedio say the open ball can form any arbitrary shape in plane, but I still don't understaned how to use "FINITE" intersections and unions of circles to form rectangle. It doesn't make sense to me, can somebody explain it to me clearly?
Unions need not be finite; so take unions. By making the balls arbitrarily small you can get arbitrarily close to the shape of the rectangle, so that in the limit the difference between what you get and the rectangle is null. Imagine it as covering a rectangle with as small and as many circles as tou need.
Sir, I have a question. When you defined a topology on the interval (0,1) as the set of all intervals of the form (0, 1 - 1/n), where n belongs to {2,3,4...}, then which element in the topology is the null set(phi) ? For that matter, what is the definition of a null set(phi) ?
I have just watched the first lecture in the lecture series on manifolds. Really liking it so far!
Are there any recommended, or otherwise relevant, exercises to go along with the series on manifolds?
For the people interested, two other very common topologies are the box and product topologies. Although they are same for finite products, but they are different for infinite products. As for the real line there are the lower and upper limit topologies except the standard topology.
15:35 You state that the union of disjoint sets is part of the topology. Is that necessarily true if we define the topology as all open intervals on the line? I.e. if we took the union of {1,2,3} and {5,6,7} the set {1,2,3,5,6,7} wouldn’t be in the topology right? (but the set {1,2,3,4,5,6,7} would be)
The rule is that the union of any two open sets must also be an open set. I don’t understand your example, but if {1,2,3} is open and if {567} is open then {1234567} must be open too.
Thanks for the fast reply! I am looking at the 3 rules for a formal definition of a topology. From the Wikipedia definition I have "2. Any union of elements of τ is an element of τ." So in my example, the union of the disjoint sets would be {1,2,3,5,6,7}, excluding 4 and this set wouldn’t necessarily exist with the definition of the basis used for the standard topology right?
It seems you are stating the requirement to define the topology is that the union of 2 sets has to be an open set but in your lecture (and elsewhere) I am seeing that the union of 2 sets has to be AN ELEMENT of the topology. I am wondering if these 2 things are equivalent and if so how would the set {1,2,3,5,6,7} end up in the topology given the basis used to define the standard topology.
Sorry if I am mixing up some terminology here, I’m a little bit new to the subject.
Ah ok, along with your explanation at 23:39 and an external reference, I realized I was misunderstanding the definition. The basis is only part of the definition of the standard topology, the union of all sets of the basis is also part of the topology!
Taken from the external reference: A topology on the real line is given by the collection of intervals of the form (a, b) along with arbitrary unions of such intervals. Let I = {(a, b) | a, b ∈ R}. Then the sets X=R and T ={∪αIα |Iα ∈I} is a topological space. This is R under the “usual topology.”
Shardul Upadhyay YEs, it sounds like you got it clearly now. A basis is just a subset of the full topology.
Thanks XylyXylyX! Looks like a great set of lectures, looking forward to the rest of them
Hello, I would really appriciate some clarification on the subject of topology consisting of open sets. If that is the case then, if I understand it correctly, there could be no set X in topology, because set X contains all the points in plane, and therefore boundary too, so it is not an open set. Is that thought correct, or am I missing something?
If the base of the topology of open sets on the line (0,1) is uncountable is it even possible to say that the whole topology is bigger than the base?
The reasoning is that the base is a subset of the topology but not the other way around. However, they both have the same cardinality.
Hey you have an amazing channel and I kinda wish it was pronounceable so I could recommend it more easily to people!
Why is the word open so critical to use it in context open set? What is behind to use the word open?
I’ve always thought it most helpful to say a Topology just is a subset of the powerset of whatever set you’re interested in (rather than leaving it at open sets).
Just kidding, you get to that. Good job hahah
Great lesson, awesome teacher!
In general points can have multiple metrics. How does this fit into mathematics and physical norms?
In physics the metric is determined by physical laws. General Relativity can be thought of as the theory of the spacetime metric. The Einstein Equation is the mathematical statement of the physical law that gives us the metric of spacetime.
I am following your lectures for some time although I am not a physic or math student, I am recently studying geometric deep learning and I find your lecture series really helpful for that. I hope you continue your great effort to make those recondite topics accessible to all. Also can you please suggest to me which books are following in this playlist?
You forgot about 2/3 point in example of strange topology on (0,1) interval (near end of video). Thank you for your work!
I am not sure what you mean?
At 21:08 you wrote points from equation 1-1/n for n=2,3,4,... and you wrote 1/2, 3/4, 4/5... but between 1/2 and 3/4 there should be 2/3. Nevermind, you can drop this one from open sets collection.
just to be sure as this is a crucial point: at 10:4 you define what an open ball is in the plane using |X-P|< r where r belongs to the real numbers. But r must be strictly positive. Thus the point P itself does not belong to the open ball. Is this correct ?
My excuse. I was confused and my last sentence should be skipped. As r must be strictly positive (this should be corrected in the video), the point itself belong to the open ball. Only a set that only contains the point P is not an open ball.
Jacques Smeets If r=0 and P was in an open ball all by itself that would be the discrete topology! But the presentation is the “standard topology” where, as you say, r > 0.
Thanks a lot for the extremely helpful video! One question: Why is it that open balls and open rectangles are two different bases, but the discussed two alternatives for the open set on the line are not two different bases, but two different topologies?
If one base can be constructed from the other, then they are bases of the same topology. If you can not construct one base from the other, then they are bases of different topologies.
25:39 Am I following correctly? Is this a circular definition???
I thought you were a physicist who uses and respects math as a tool. Surveying your patreon page I see that you have a love for the beauty of the mathematics itself. Homeboy Poincare would certainly not disagree. Subscribed.
Pairadeau Very Kind of you. I have been slow to produce this month but will be back at it very soon!
Would it be correct to say that the "null set" rule is a corollary of the intersection rule? For example, if the intersection of any 2 open sets is an empty set, then I can find two such sets whose "intersection" is the null set, which is therefore an element of the topology.
Similarly for the union rule and X being an element..?
Marcas O'Duinn Sure. I never thought about it that way. Sounds Ok.
I thought a bit more about it. The one problem that occurs to me is if we took the entire set as open and started just with that. Then your method will not produce any other open sets, including the null set. I think we need to stipulate the null set and the entire set must be in every topology.
Let X be an infinite set. If you take the set of subsets of X whose complements are finite as a topology for X, then intersections will not include the empty set.
This is because these subsets will disagree only on finitely many points and so the intersections will always be infinite subsets of X.
at 10:48 it seems like you mistakenly put an equality in that inequality of |x-p|
Yaman Sanghavi Indeed! THanks.
I think he was just underlining the < symbol...
Do you have a textbook recommendation to follow along to this series with?
Not with this material actually, I cover a broad range of topics. You need a good introduction you book on standard topology (point-set topology, not algebraic topology) mostly. There are many books and articles about manifolds out there. I think I used “Differential Geometric Structures” by Walter A. Poor (Dover). That is certainly good.
How could you make a square out of finite open balls? Then how can a square be a part of that topology?
Alan Zhu Unions are permitted to be *infinte* not finite. Intersections are only allowed to be finite.
It is then possible that some element of one topology are closed intervals, and however called open in the sens of topology ?
Montavon Jean-Baptiste Yes, it is possible. In the discrete topology every set is topologically open, including all "closed" sets. In fact, there are topologies that have as a basis half-open sets of various types. And, topologically speaking, the entire set X is a member of the topology and it is BOTH topologically open and topologically closed (its complement is the null set which is open and the complement of an open set it by definition closed). Such a set is sometimes called "clopen".
how can a null set be a member of the topology of X? im sorry im dumb..
It is so by definition (easy answer) and the null set is a subset of every set. It is the complement of the entire set, so it is both open AND closed! It is "clopen" :)
your teaching style is very awesome pleas provide me your lecture slides that would be very beneficial
Do you think that you would be able to make a video or a series of videos on the math of String theory in the current 11 spatial dimensions with respect to M-theory.
Someday maybe....but it won't be anytime soon. I'd do QFT first anyway.
This is awesome! Thanks so much. Thorough in a perfect way, for people (like me) who just needs to understand it and get the intuitive meaning of it, without all the rigorous definitions and proofs. Yay!
Jonas Kyhnæb I'm glad you like it. I do suggest that you find an elementary point set topology text and read it after you finish these lectures. It will be much easier and more interesting. GOod Luck1
@@XylyXylyX Same here, I'm a dumbass who has hope that they can relearn maths and understand topics like this thanks in part to great explanation.
I really hate when I'm trying to learn something and it's just 'ok, this is how we solve this class of problems!'. Much prefer the explanation of whats going on.
Subscribed, hope ye haven't quit youtube :)
callmedeno Nope, still here. I have some more work coming out this weekend. Thank you for watching!
Why do you restrict the intersections to preclude an infinite number, while you allow an infinite number of sets in the union?
JT Coriolis This is a HUGE observation! In fact if we define a topology in terms of "closed sets" (an equivalent way to do it) then the rules regarding unions and intersections are reversed! THink this way: if we allow infinite intersections then we can create an infinite list of open sets whose infinite intersection is *only one point*. Therefore every point would be a member of the topology and then *every topology* would be the discrete topology. How uninteresting!
By the way, understand these rules were settled on after many many years of struggle to understand what were the most important concepts required to understand continuity and limits. THey are NOT intuitive at all. To me at least....:)
@@XylyXylyX Why are discrete topologies uninteresting? They would seem to give you best detail or most accurate representation of some metric space and perhaps easier to identify isopmorphisms between different discrete topological spaces.
Miguel Amaral The problem is that every mapping is continuous in the discrete topology and therefore no function is really distinguishable in that sense. Also, topological spaces don’t usually have “isomorphisms” but they do have “homeomorphisms.” If all the spaces involved use the discrete topoloy then finding homeomorphisms would indeed be easier because all the mappings and the inverse mappings would be open mappings. The only requirement would then be that the mappings were 1-1 and onto.
An inspiring lecture. Awesome.
Thanks, but note acousticwaves' comment below. I have to make a change in the end of this video, but the next tow example lessons are fine.
You're absolutely right about physics students taking Complex Analysis. I did that years ago and it completely cleared up the mess of that subject when it was taught by physicists. Another physics major mentioned the same thing.
"Advanced Calculus" also recommended, for the epsilon-delta definition of limit. That blew all the cobwebs away from that concept. Why don't they teach that straight away instead of all this rubbish about "getting closer and closer to ... but never reaching it"!! And the most lucid book of any kind I've read is the one on Topology ... can't remember the author at the moment.
Update: it'sTopology a First Course, by Munkres.
And the above is no slight against the physics profs who taught me a lot of wonderful physics; it's just that most of them were not equipped to teach mathematics, and the texts that they used in class were no help.
It is all about time, usually. I learned the most after graduation. We all need to make sure that when we complete our coursework that we have learned how to teach ourselves!
What are the pre requisites for this lecture series?
Topology and manifolds sort of stand alone, but I would prioritize differential equations, linear algebra, and real analysis.
@@XylyXylyX thanks for the reply..btw your doing a great job making these lecture videos.
Do you have a book that goes with these lectures? I have a couple of topology books but you leave them behind after the first five lessons.
I have been telling people to find a book they like themselves, because there are so many. I can’t believe that these lectures leave topology books behind....you must mean the manifold material isn’t in a typical topology book? Regardless, do not rely on my lectures to learn this stuff. Find a good book too! Most published books are good, in my opinion. Just be sure they are aboutPoint Set Topology, not something else.
great lecture
Why open sets sir; why not closed sets. What purpose does it serve when we take open sets. Please clarify sir.
"open sets" are just the name we give to the sets in a topology. "closed sets" are just the name we give to the complement of open sets. The point is the logic we derive from the concept of a topological space, not the name we give to things.
@@XylyXylyX
Thank you sir. Your lectures are unique and totally different from others. Happy 2021.Thank you again.
@@atiurrahman7907 Buuuut, I should point out that all of the theory of topology can also be derived by defining a "closed set" a inverting all of the basic principles defining a topological space. So you do bring up more than just a matter of semantics...the notion of topology can be approached several different ways. In fact, one way involving the definition of "neighborhoods", if I recall correctly, doesn't even need a topology at all!
8:50
it doesn't follof that X is the biggest set. Is it another axiom? That open sets are also members of X?
I'm not exactly sure what your question is. Can you rephrase
it please?
Who would have thought I'd make it through lecture one. Thank you, your explanations were crystal clear! Just curious, may you clarify the concept of a topological base, I got lost in your discussion of it towards the end. Alternatively, do you have any online references in mind that explain topological bases well enough for someone that just stumbled on this math by chance (not a rigorous math person)?. Keep on being awesome!
-Float Circuit.
A base of a topology is a collection of specific open sets that can be used to create all possible open sets by arbitrary union and finite intersection. It is actually quite a simple concept in principle.
Topology is not something to casually dabble with, in my opinion. On the other hand, elementary point-set topology isn't very difficult as long as you can allow yourself to think abstractly. ANY introductory topology text will do. None are all that much better than any other, in my opinion. try:
archive.org/details/introductiontoto00game
I liked that book a lot.
Love you..thank you
Can you please make a video on embedding in topology too??
Did anyone hear 👂He actually made himself yawn at about 4.45 - 4.48! but just kept teaching right through the yawn. Top notch ! 👍
If X is a member of the topology then do we really need to call out the combination of X and the topology on X? Isn't this redundant?
No, it is not redundant. X is a set. The pairing (X, T) is a list of two different sets. It is true that T contains X, but there are many different Ts that could be a topology on X.
Thanks, the day after I made this comment it struck me that there is a good reason and now you have provided it.
I think for the definition of open balls, r must be strictly greater than 0 right?
I LOVE learning from you. I am an engineer, but I am loving this stuff not only because it is leading me to manifolds, but I love it in and of itself. THANK YOU.... Moving on.... Just at about 10 minutes, you declared that the set X with a topology is what we were after -- a MINIMAL group (not in the algebraic sense of the word) of attributes. But you ALSO said that your set X was equipped with a notion of "distance."I cannot see how you can have a notion of distance for a set X when the only attributes are the notions of open sets: union is in the set, finite intersection in the set, 0 in the set, and X in the set.I cannot see how you can acuquire a sense of distance from those minimum attributes. Now at 11 minutes you define the open ball. So are you suggesting that that notion of an open ball is all you need? ALSO (second issue) WHY is a single point NOT an open set? It seems to me that you should have supplemented your definition of open set with the fact that it must contain at least two points.
JT Coriolis Lets take these one at a time. First: the notion of "distance" is, as you have stated yourself, an extra concept that is applied to a set. An open ball can not be defined if we cant define "r". Ultimately this means that we must endow our set X with a metric. In this case it is the standard Euclidean metric. However, the SAME topology can be constructed from rectangles and rectangles do not need an "r": they are just defined by their coordinates.
You are correct: the topology does not define "distance" in any geometric sense. For that we must assert a rule that measures distance and in this case we use standard Euclidean distance. However a topology does provide a sense of "nearness" in that ultimately it is the topology that determines what points are the limit points of various subsets of S.
2) Regarding single points as open sets: It is entirely our choice what counts as an open set as long as those rules work. We can choose the "discrete topology" and make every point an open set if we wish. However, as you will see later, if you choose the discrete topology then every function is continuous and no set or series has a limit point. So, we choose topologies that are "weaker" than the discrete topology that provide interesting and useful results regarding functions.
It’s always easier to explain the infinite line interval by normalizing it to infinity, i.e 0-1
Be careful since you cannot deduce from "The union of two open sets is open" that "The arbitrary union of open sets is open" 07:13
You should add this axiom in the first place "The arbitrary union of open sets is open. If you just put the axiom that the union of two open sets is open, it will give you that the finite union of open sets is open.
Very good video, but slight nitpick at 26:53 you refer to the empty set as phi, but it is actually not the greek letter phi that denotes out but the Danish-Norwegian letter Ø. Pronounced like the u in up.
What would be the problem if we declare closed balls as open sets?
And a second question, it seems that a point set without a metric, or 'structure' as you put it, could still have topology, as the discrete and trivial topology always hold.
So am I right in saying that topology is built on the structure of such set, but does not specify the structure? And in this case the metric is what structure really means.
高远 IT is not a "problem" really, but if you did you would simply have the discrete topology because the intersection of two balls touching at one point would also be an open set and therefore every point would be its own open set.
高远 if I understand you correctly, yes. A topology can exist for ANY set and a metric is not required. Even the usual topology of the plane with a metric is the same as the "open rectangle topology" of the plane without a metric, for example. THis is probably because the plane with the open rectangle topology is "metrizable" regardless of whether or not a metric has been established.
Well I think being metrizable is equivalent to having metric, or there exists the notion of distance. The specific form of metric does not matter. What I mean by not having a metric is there is no notion of distance, the set is just collection of points. And in such case rectangle would make no sense either.
But even with a set that is just collection of points, we could still establish the trivial and discrete topology, right? It seems 'topology' is not interchangeable with 'structure' of the set, as you developed the logic.
May I ask a more general question, that can we study manifold and coordinate without engaging geometry? Or at least we engage only what is necessary in a step by step manner? Just from simplest set of points, then requiring it being a metric space(maybe?), and then by somehow it is justifiable to map the elements of the set to a coordinate so that we can study the set quantitatively.
why do you sometimes use open to indicate that a set lies in the topology and sometimes to indicate that is doesn't contain it's boundary?
One definition of an open set is any set that contains a neighborhood of all of its points. If a set contains its boundary then the points on the boundary do not have a neighborhood that lies inside the set. It is good to link up all of the different ways of defining open sets to see how they are all equivalent.
Yes, except no one ever does that. Profs teach a neighborhood Heine-Borel definition and then ask you to prove sequence-type Bolzano Weierstrass statements on the test...or vice versa...
Sandy Grungerson True. Usually a course in topology picks a single definition of open sets and derives everything from there. I can't complain too much, it is probably less confusing for a first pass through the subject. Topology is incredibly deep.
"So this is less than" (underscores it and makes it a less or equal than...
10:51 you say notice that this is a less than sign and underline it to point it out... But then it's a less than or equal to sign
Oops! Rookie mistake indeed :)
Hi xylyxlyx. I am a Patreon since 4 months ( 5€/months). I stil have not received any copy ( “way does it move”). One month ago I tried to contact you via the patron arena, but No reply. Therefore I repeat my request here: when/ how can I expert the copies ? Note, I am not the only one who is complaining about this. Would highly appreciatie to receive your quick reply. Thanks.
I am very sorry about this! I will get you a link to the books today.
I have posted the links in the Patreon area! So sorry about the delay.
This is my first day of learning manifolds.
it's been 10 months how's your understanding of manifolds now?
Great,, thank you !
Thanks😍
thank you very match
3:53 ;At 8:30
Now that we know that the balls constitute the base, how do we find the base of the balls?
Another ball with r greater than 0 and smaller than the r of the base
Any one help me first I saw the lesson with english subtitle now english subtitle is missed
13:18 I think he laughed at this just like i did.
Thomas rad I didn't until you pointed it out.
I think your definition of open ball should be |x-p| & =
Can you send me a timestamp? It has been a LONG time since I reviewed this lesson…but what you wrote is certainly correct.
@@XylyXylyX at 10:51, you say less than, but then write less thanx and equal
You sound mildly similar to Edward Snowden :)
Q: Why does Jeff Bezos hate topology? A: He disapproves of unions.
Topology is actually defined as the boundary of all open sets within the topology bruh!
I am not sure what you mean.
4:43 i went sleep here
👁🔺
He yawns at 4:50. Supposedly bored by himself.
As a mathematician I found your youtube site looking for some general information on topology. My primary interests are math logic, set theory, group theory, info theory and analysis topics roughly in that order. As I watched your intro and following lectures in this series, I was struck by the lack of any proofs given. At first, this was a minor annoyance, but as I watched I realized your target audience is quite possibly not mathematicians. Then I watched this initial video and it became clear you are directing your lessons to applied math viewers. Now, I'm actually relieved you don't go thru lengthy proof schemes for every statement you make in these lectures, it certainly speeds up the instruction.
One comment regarding continuous homeomorphisms between a topological space and R2. You seem to imply it has to be unique. I can envision a continuous map onto that is not unique. As long as the set and its topology are mapping unique elements to R2, we should be able to map to the same R2 space. The Axiom of Choice should guarantee this. Which btw I was surprised you didn't mention in the lecture either. Well all in all, I'm finding your lectures very informative and enjoyable.
Robleh100 This set of lectures is meant to be a sidetrack for the “What is a Tensor?” Lectures. The idea is to give physicists a working vocabulary of topology so they can have a deeper appreciation of the concept of manifolds which are used in general relativity to model space time. A real topology course would be, as you say, a continuous stream of proofs. However, note from time to time I do drop in a proof but in no way is this lecture series suitable for mathematics students.
The axiom of choice is a level deeper than I want to go. I do mention it in the series about the foundations of quantum mechanics when I discuss the Hamel basis of a vector space, I am not sure where I may have implied that homeomorphisms should be unique, I certainly know better! :) Thank you for you comments.
one last comment before I try to refrain from becoming a frequent commenter, your youtube... how shall I say it...screen name... I guess... XylyXylyX, it has begun to intrigue me and I'm probably reading too much into it, but it seems to be maybe a way to list the order of a group? I thought hmmm ....3,4,2 for the letters.... lemme see could be a group number under some modulus, then it struck me... ohhh palindromic yes, yes, it's a palindrome. This seemed interesting too since I've become interested in recurrent palindromes on the 12-hour clock, but again I could be making something outta nothing...so just what is that name all about? I didn't look at your about area.....
Could go at x2 rate. Excessive number of examples. But a great lecture otherwise.
IMO there is as much examples as needed. Not too much. :)