I guess Im asking randomly but does someone know a way to log back into an Instagram account?? I was stupid forgot my login password. I appreciate any help you can give me!
I believe he strategically placed a small slip of paper with those numbers on the chalk stand beneath the spot on the board where he was writing. Cleverly planned and well executed lecture!
A little mistake at 49:46: the fact that M\M=∅ is open implies that M is closed (by definition). Similarly M\∅=M being open implies that ∅ is closed. By the way, thanks Prof. Schuller for the awesome lectures
Why would that imply M is closed, if i take any interval open, or closed, and remove any number of the elements of that group I am left with another set with no implications on its openness or closure... Please elaborate and show me why you would say otherwise
34:33 This definition of the induced topology, though not necessarily incorrect, technically overcounts identical elements. There could be many U in O such that U ∩ N gives the same result. Also, the notation in this definition gives the immediate false impression that we are using universal comprehension (notice there is no epsilon-relation on the left). These slight issues (or non-issues, depending on how you look at it) can be resolved by defining the subspace topology more formally as {V in P(N) | there exists U in O : V = U ∩ N}. With that, restricted comprehension is manifest. Edit: He actually introduces this definition later at 38:33.
sets dont count their elements! and we dont have an issue with PRC, because it exists by the axiom where a set can be an image of a functional relation.
Overcounting elements doesn't really matter if you want to build a set. It'll just fix itself up by the axiom of set theory. There is no problem at all with universal comprehension here since U and V already a set and the intersection of two sets is a set. It's just a matter of own taste, whether to use (not necessary) more formal definitions in the whole lectures since he already explained all of these issues in his previous lectures on set theory.
I was wondering what kind of Kindergarden he went to that teaches set theory :D Sounds like a place i would like to send my child to if i ever have one...
Hehe, it's part of a movement called "new math" or "modern math", which was a new way to teach math, in a very abstract way to children, starting very very early. It occured mainly in France and the US (that's part of the reason why those 2 are math superpowers, with respectively 12 and 13 Field medals), and to a lesser extent in the UK and Germany (who had lost many brains after WW2), after the Sputnik launch and was meant to boost the science and math curriculum in order to catch up with the soviets who are a great math nation (France already was a math nation but the arrival of new math is due to a member of the Bourbaki group, André Lichnerowicz, becoming head of education ministry.) Problem is, it was so abstract that neither parents nor teachers knew it and could teach it effectively, so, despite highlighting the genius of some students, most children couldn't follow and ended up with a learned helplessness, like a trauma when it comes to math. It thus was abandonned at the end of the 60's.
After the discussion about the nontransitive dice I wonder: does one need a topology for any type of measurement? What about comparing probabilities or densities on various (event-) spaces? Do they always have to have a known topology that supports this?
Wonderful lectures, congratulations, but it would be really nice and even better if the problem sheets could be available. @Frederic Schuller: do you think you could help your you-tube students by providing the problem sheets in some way?
7:49, 18:14, 21:12, 24:56, 27:29 (B_r is defined on M itself not the embedded space), 34:21, 43:36, 48:01, 49:28, 53:20, 1:01:57, 1:04:24, 1:09:34, 1:15:41, 1:17:35, 1:20:30 (rock paper scissor for dice), 1:22:18 (wisdom), 1:28:13, 1:33:12, 1:37:32
I haven't been able to find any, but Wikipedia happens to list many properties of topologies and things like that. I find it useful to prove each of those one by one. Edit: Of course, it is annoying at times not knowing whether the stuff from a particular lecture is enough to prove something.
@@neelmodi5791 @Neel Modi if you are looking for a topology book with problems to solve, I highly recommend the Elementary Topology Problem Textbook by O.Ya. Viro et al.
What he means when he says" you learned in kindergarten', is set theory was called the new math in the 70's and students were expected to learn basic set theory in earlier grades. But, that fell apart.
In the definition of continuity, should the map be surjective?, If not surjective, is the topology in the target space the topology of the entire space or the induced topology of the codomain of the function? For instance, if f(x)= sin(x) then the codomain is [-1,1] but f in itself is also a function from R->R. Clearly, the set [-1,1] has different properties than R. Does it make a difference?
I think it’s fine either way. For the sin function, if you take any open set in the target R that is not in [-1,1], the preimage will be the empty set. For example the pre image sin^(-1) ( (2,3) ) = empty set which is in the topology of the domain
at 30: 21 when he proved the intersection of two set and pick a point in it, I think min(r,v) radius is not enough because assume the point lie in the middle of the intersection there are still ossiblity the radius is too large and lie outside of the intersection.
No, his proof is already complete. If the open ball with min. radius should always lie in the intersection. If not, there will be part of the corresponding open ball that lies outside the defined set.
Excellent lecture. One point though: In your example b) at around 1h28, I think it’s not accurate to say any map phi from M to N where N is equipped with chaotic topology is continuous. The reason is the preimage of N (as an open set in the topology) is a subset of M but not necessarily an open set in M.
@@denisodivertido Thanks a lot for the reply, my bad. One question (if you have time) Phi^{-1}(N) might not be well-defined as under Phi, two different points in M may hit the same point in N. Am I missing some thing here ?
@@hoareg2, phi^{-1}(N) is, by definition, the set of points of M whose image under phi is in N. But, for all x in M, phi(x) belongs to N. Therefore, phi^{-1}(M) = N. Note that phi does not have to be one to one for this definition to make sense. If phi is one to one, however, then, for every subset B of N, phi^{-1}(B) is equal to the range of B under phi^{-1}.
1:15:45 why does Dr say the sequence 1 +1/n that converges against 1using the normal topology not converge using the discrete topology? Is there not a set in P(R) that contains all off the points on the journey to 1?
we say that a sequence converges to x if for each NEIGHBORHOOD U of x, there is an index after which all the elements in the sequence are in that neighborhood. {1} is a neighborhood of 1, and absolutely no element of 1 + 1/n is an element of the set {1}, so the sequence does not converge. this is analogous to not being able to find a delta for a given epsilon in a metric space.
The condition to check was that for all points in the empty subset, there exists a ball of radius r such that the ball contains all such points. However, since the empty set contains no such points, whatever condition was the subsequent of the “for all points in the empty set” predicate would trivially hold Ex-Falso Quodlibet is a Latin term that describes the fact that a false predicate can imply anything.
I don't have any background computing the number of topologies, but from what I've heard it is really hard to do. I believe the number of topologies on a finite set is computed through brute force on a computer. I'm not sure but I think that there isn't a general formula for the number of topologies on a set of size n. If one imposes a condition on the topology, that it be T_0 (meaning no two distinct points are members of the same open sets) then counting the number of T_0 topologies on a finite set becomes equivalent to counting the number of partial orders on the set (which is understood better but I believe still hard).
You can Look at all the subsets of the powerset and consider the topology that is generated by a given subset. Then u need to sort of classify subsets that generate the same topology and the number of those classes is the number of topologys. This is the naive approach. One might be able to derive a formula for finite sets this way
Pllease, help me out. A map frpm R^d to R^f, both sets equipped with the standard topology doesn't seem to recover the usual notion of continuity. Take d = f = 1 and consider a constant map e.g. forall x in R, phi(x) = 1
Hello @Frederic Schuller, I notice that you defer some problems/ideas to problem sheets. I did find a few interesting and would like to check my approach is acceptable. Would you be able to provide a link to those?
Plain old combinatorics. If you have a set of n elements, the powerset must contain: { {}, {1}, {2}, {1,2}, ... {1,2,...,n} {2,3,...,n} {1,3,...,n} ... } So it is simply the number of possible combinations of elements one can construct without regard to the number of elements in the set or the order of elements in the set (given that no elements repeat). I believe the exact equation is: /Sigma_{i=0}^{n} i! Someone feel free to suggest another equation if I am wrong I am usually bad with pattern representation in maths 🙃
I don't have any background computing the number of topologies, but from what I've heard it is really hard to do. I believe the number of topologies on a finite set is computed through brute force on a computer. I'm not sure but I think that there isn't a general formula for the number of topologies on a set of size n. If one imposes a condition on the topology, that it be T_0 (meaning no two distinct points are members of the same open sets) then counting the number of T_0 topologies on a finite set becomes equivalent to counting the number of partial orders on the set (which is understood better but I believe still hard).
I think in axiom II of topology you should add finitely many intersections. If an infinite intersection were allowed one could end up with a ball with radius zero in the standard topology.
@@CrusaderTube your point fits to my intuition nicely as soon as we are in R. But I have a problem. Consider that I have no idea of what open set is. Then the definition suggest that every collection O (curly) is an open set. This is where I was confused. I still think one should add finitely many intersection to avoid the problem that I mentioned. I also take a look at Alan Hatcher notes on point set topology (available on his webpage). He wrote "The intersection of any finite collection of sets in O is in O"
@@pedidep the second axiom says that if U and V are in O for two sets then U intersection V must also be in O. It is possible to prove by induction that this means intersection of any finite collection of sets in O is in O. The axiom presented by Hatcher you mentioned and the axiom (ii) are then equivalent, since the implication clearly holds the other way too (as {U,V} is a finite collection).
@@CrusaderTube I can assume O with infinite subsets then their intersection is again in O regarding Prof. Schuler's definition. All I can understand is that it contradicts Hacher's definition. Can you explain to me how do you avoid infinite intersection in present axiom?
@@pedidep The definition says “intersection of any TWO subsets (of M) that are in O must be in O”, NOT “intersection of any collection of subsets in O must be in O”. I can’t explain it more clearly than this. Read my previous responses and think for yourself. There is no problem with the definition in the lecture.
Did he define what an open set is? From his use it seems to mean any element in the equipped topology of the set in consideration. Is there a more precise definition than this.
No, as that is exactly the definition of an "open set": a set in the topology. The choice of open sets (obeying the three rules) for a space is a topology.
I have always found it funny that in topological set theory technically anything can be a topology so long as the three axioms hold, but that's kinda the point, it's based on a notion of free logic meaning you want the theory to be as free from assumption as possible except where necessitated by the function of the theory. Unfortunate for us that means an open set gets defined when the topology gets chosen for a given space of sets! Basically it comes down to a similar idea: define an element, or set, as lovely as possible... A set is a collection of elements which satisfy that all elements are members of the set... It's crazy but it's the free-est logic we can use so that sets can be used in a wide range of applications while still achieving what we built them for in the first place 🙂
@Notachannel: this is the way, i understand it: "Open sets" and their choice in a topological space define (!), what it means for elements to be "close" together in a space, that actually has no notion of "distance". In a metric space, where you do have a notion of distance, the open sets (at least, when the topology is induced by the metric) are those, that constitute a neighbourhood for all its elements, so that for any element of that open set the others literally surround it (you have an epsilon-ball around each element fitting completely into the open set). Not a single element lies on its boundary. By choosing a topology in a purely topological space, and thus choosing a specific open set, you are also saying: these elements (shall!) constitute a neighbourhood for each other (although you couldn´t have said that without defining it, because there is no notion of distance). If you have an even smaller open subset of that open set, you are saying: these elements of the subset are even closer together - like the elements of the smaller subset are like the inner layer of the neighbourhood, while those, that stay outside are the outer layer. Hope, that´s helpful, but as i said: my personal interpretation and not completely sure, it is correct. I had a hard time myself, trying to make sense of it.
The empty set 0 (as you wrote it) is not an element of every set, rather, it is a subset of every set. What he is doing is subtracting the elements in the empty set, which contains no elements. Therefore M = M\0. Hope this clears up your confusion.
The elements of ∅ are indeed in every set, but that's what's special about nothingness that you can subtract it as many times as you like and it will still be there
@ ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-1wyOoLUjUeI.html "In Kindergarten you wrote this U intersect V" Kindergarten me ate glue :| not did set mathematics.
Watch the earlier lectures. There are objects which are too big to be sets, and we can't take their union and expect to get a set. I don't think he mentions the name of these bigger animals, but they are _classes_.
1:27:56. 1. What if there is no preimage of some element in N? N is not being hit entirely. Then the map is not continious? 2. What if all N is being hit, but we choose a topology, which contains an element, or a set, which contains an element than doesn't lie in N? Is the map not continous?
1. The map is still continous. "Having no preimage" is equivalent to "The preimage is the empty set" or "No elements from M maps to some "a" belonging to N, hence a set of them is empty" and we know that the empty set is on any topology in M 2. A topology on a set N can't contain sets with elements outside N because the topology, by definition, MUST be a subset of P(N), the set of all subsets of N. If some "a" doesn't belong to N, it doesn't belong to any subset of N and hence doesn't belong to P(N), so it can't be in the topology.
phi just needs to be a function from M to N, nothing more. As an example the map phi(x) = x^2 from the reals to the reals (with the standard topology both times) is continuous, and the preimage of the open interval (-1, 4) is the open interval (-2, 2). This is because some real number x has x^2 in the interval (-1, 4) exactly when x is in the interval (-2, 2). One can check that the preimage under phi of any open set in R is again open.
He did that in one of the previous lectures. A set S is then defined open, if for each element x in S, you can find a (positive) r, such that B_r(x) is a subset of S.
Help if you can? What is the difference between a subset symbol with one line under it and a subset symbol with two lines under it? (Google doesn't know)
@@kockarthur7976 two lines most certainly dont mean proper subset! those two lines are an equal sign and mean that the subset might be equal to the full set. to make it proper, those two lines need to be crossed, like an unequal sign. the subset sign with one line is a variant where the bottom of the sign above serves as the top line of the equal sign. also the sign without any lines usually means the same. so 0,1,2 lines, all the same. to make the subset proper, one or two lines but crossed.
Following on from p0gr's reply: at 33:28 Schuller erases the bottom line on the subset symbol ({\subseteq}) to denote "proper subset" ({\subset}), explicitly saying so.
When you define S^1 whit the equivalence relation. how do you get the topology on S^? because you cant induce it from R^1 since now S^1 consist of equivalence classes not elements of R^1 and you cant use your definition of the standard topology since it requires the norm which requires square root and multiplication. Also you said that any norm generates the same standard topology. Is there a more general norm which can be used to define the standard topology for for example equivalence classes? or more simply what operations are necessary for a norm since square roots aren't closed under all sets whit multiplication? Anybody's answer is appreciated :)
This is a bit late but there is a way to inherit a topology through an equivalence relation. As the professor mentioned this is called the quotient topology. If one has an equivalence relation, like x ~ y whenever y = x + 2*pi*k for some integer k (the equivalence relation I believe the professor meant to write) then one can induce a topology on the set of equivalence classes R/~. One way to understand this topology is by considering the "quotient map" q:R to R/~ which sends a real number x to its equivalence class say [x]. Then one induces the topology by saying that a subset V of R/~ is open if and only if the preimage q^{-1}(V) is open in R (this is also called the final topology on R/~ induced by the map q). Essentially given a subset V of R/~, meaning a collection of equivalence classes, one is looking at the points in R that are a member of some equivalence class in V (one could say the union of the equivalence classes in V). Then the induced topology is just saying that a collection V of equivalence classes is open exactly when this union is open in R. Hope that helps and let me know if anything is unclear :)
Lecture 1 Propositional Logic 28:30 "ex falso quodlibet" written on the board. (You sometimes see "ex falso sequitur quodlibet" in texts - "from false follows anything you like".)
Most taught (as opposed to research + thesis) MSc courses will provide a selection of courses from which you have usually some mandatory and several optional subjects.
Utility (quality) function is usually required to be convex. That's wrt economy. For measuring better scientific theories, here's a tip of an iceberg www.cyberneum.de/fileadmin/user_upload/files/publications/TR_145_[0].pdf
I find your comment intriguing. Do you know of some keywords that would allow me to search for more information on what you are saying about the requirement of the utility function to be convex?
@9:23 Frederic Schuller alters the space-time continuum of his own youtube video. In fact, I argue that he formed a discontinuity in the continuity of this space time. LOL
But why can't we equip the space of all theories with chaotic topology? This will make every sequence of theories convergent. But yeah, they will converge to a complete nonsense as well. Why do we define almost constant as constant everywhere, except finite amount of points? Shouldn't it be countable amount?
No, I was wrong. There can be countably but infinitely many element in a set. But I think his point is that if there are finitely many exceptions, then we will eventually skip through all exceptions for some large N, which would converge the sequence. This we cannot do for countably many, as we can have infinitely many elements in the exception set.
Great lecture, some funny jokes, a few too many hence, and sometimes funny pronunciation: e.g. ep-Cylon - is that one of those from Battleship Galactica?
Because it is the most general definitions which turn out to give a good intuitive picture of a space and continuous maps, in many cases. This is something you only understand after seeing examples. This is not to say all examples will be intuitive. But some very important examples will be, like the standard topology on R^d. Also continuous mappings (according to the topology definition) from R^m to R^n in the standard topology corresponds with the usual definition of continuity that you will have seen in multivariable calculus. If you wish, you may post your questions in the Physics Forums, since it is easier to type mathematical notations in there. I will try to find your post and if I don't, someone else will probably have a good answer for you.
This lecture desperately needs to explain the motivations for these definitions. WHY define topologies via open sets? WHY are the 3 requirements what they are? WHY not define closed as not open?
I really love this guy's style (no homeo-ba dum tsh), sad to see that gender studies are very probably even affecting the math and physics departments with their bs.
41:00 - I'm the right age to have had set theory in lower school. I saw it several times, actually. I never saw the point, and I honestly still don't. Not because it's "not relevant," but because it is *never put to use* in a normal "average person training program*. It's just time spent that never gets called upon again. I think it should just be left to be introduced when a student embarks on a program of learning that actually makes relevant use of it. Also, naive set theory should NEVER be taught - there's just no point in teaching something that isn't fully correct and that modern math is NOT based on. Maybe the reasoning was "it will be easier for youngsters to understand," but that's pointless - just postpone it until it's for sure the student will need it, and then teach it RIGHT.
