Topology vs "a" Topology | Infinite Series 

Hi everyone, Thanks for the comments. As Bruno Fagherazzi noted below, I do indeed value your feedback, and I am listening. We at Infinite Series are committed to providing you with challenging, high-quality content, and we take this commitment to heart. In line with that commitment, my goal was to clarify a challenging topic by sharing an analogy which was, admittedly, imperfect. This was the reason for the disclaimer at 0:55. But the analogy has raised more questions than it answered! Mathematics is a dynamic, exciting, and stimulating field of study, and communicating complex mathematical ideas in a fun-yet-rigorous way can be a challenge. It is, however, a challenge that I am fully committed to. It’s also a work in progress! So rest assured, I’ve heard your feedback from the past few episodes, and - with that feedback in mind - I am working on new content for 2018. It is vibrant mathematics of the utmost quality. I can’t wait to share it with you. Happy holidays, Tai-Danae

As a mathematician, I like that the series is taking a turn to explaining formal points of mathematics rather than just presenting cute problems. That said, I doubt there is any good way to motivate axioms of topological space except seeing so many examples that you are convinced that this the right way to go. Standard way to motivate it to math students is through analysis, open sets are good for differentiation and closed sets for integration. Also, explaining limits and limit points of a sequence is a great way to introduce open sets. Looks much less terrifying to the non-initiated than epsilon-delta approach. On the other note, since Gabe is also on the team, it would be great to see some Lie theory in the future videos - you could combine an elegant math theory with a physicist point of view.

I normally don’t make negative comments on videos but I really feel like the analogy used in this video actually made the ideal of topology more confusing than the axioms themselves.

How would you know if the verifier machine is running forever or just taking a long time to terminate? It seems to me that the verification simply reduces to the undecidable halting problem

My goodness. That was a remarkably good explanation that made everything click. I haven't been able to find any other material that gives a solid intuition for what topology is really all about, and this made it click. Thank you a million times over.

I studied a lot of topology things myself (thanks wikipedia) as my hobby and find this theme quite interesting, and I think I grasped at least some basic understanding of its concepts, but this video just confused me. It felt like some simple and intuitive concepts are described in some confusing, complicated way, while it should be exactly opposite case. That's so uncharacteristic for general content of this channel. Presentation is as good as always, but the content itself is a miss.

hey in the LP metric @2:15 you forgot to take the P'th root of the quantity inside the brackets

I love that you quoted Munkres' book

Confusing analogy.

This machine analogy is completely unnecessary and actually counterproductive. As someone who has taken classes with heavy topology, I can say that this is NOT how anyone thinks about open sets. There is not even any (nontrivial) relevance between open sets and indicator functions, the closest mathematical version of this indicator machine stuff. "Verifiable set" isn't even a standard term in topology. On top of being unhelpful, imo it was just a sloppy explanation of what was trying to be conveyed.

please do more video's on topology

One topic at a time please Infinite Series!! By the time the next Gabe video comes along we'll have forgotten all about his last one and the same for Tai-Dinae. Let each one wrap up their topic with a number of videos before the next one comes along.

If the machine gets an input that's not in its range, then how long one will have to wait to believe that "yes it is not in its range."? Maybe, after we leave the machine believing it doesn't belong to the set, it terminates then we will conclude wrong things. Why are we using such a machine? Why can't we use a 0 or 1 binary output machine for getting a yes or no?

I had never heard of finite topologies before. It's certainly not often that a channel like this shows me, not just a new proof, but actually a totally new branch of maths. I love what you and Gabe have done with the place, you're carrying the torch for Lindsay perfectly!

Thanks very much for slowing down a bit! Much more understandable :)

Last video lots of people asked her to talk slowly, and so she did it. Great to see how they are concerned with the public. Great video!

I prefer the book definition to the machine interpretation.

The machine analogy really doesn't work well. I think it is much less intuitive than the geometric meaning of open-ness. And that isn't even very hard to understand: "In a metric space M, we call a set 'open' if for every point x in M, M also contains a sphere around x. Sets like this are called open because they do not have any 'borders': Any sphere around a point on the border would stick outside the set! Now if we have two open sets U and V, it is easy to see that for any point x in the union set, both of the spheres around x that are in U and in V are also in the union. Also, the smaller of the two will also be in the intersection - since it is contained in the larger one, and therefore in both U and V. There are two easy examples for open sets: The whole space is open since every sphere is contained in it, and the empty set is open because it has no element, so anything is true for 'every element' of it. We would like to make this more general, so we don't have to restrict ourselves to sets with measurable distances. So what we'll do is use the three facts we just found out about open sets in metric spaces, but forget the metric part! That is why we define a topology on a set X to be any collection of subset that contains X, the empty set, and that contains all unions and finite intersections of its elements."

This was a fantastic introduction. I am trying to apply it in a specific way and looking for clues.

The part with the array of machines reminds me of how some ADCs (Analog to Digital Converters) work. Basically they use an array of comparators to find which interval the input voltage belongs and then emit a digital number representing it.

I have taken all this stuff in my real analysis class, and i see how this can be helpful and intuitive. But I still do think that example-counterexample is a better way of getting the ideas of sets. To make those more intuitive one could "zoom in" on the space to see how closed and open behave. But great video!!

I really like the rationalization of why you "need" open balls. It's definitely not beguinner friendly and, it would be confusing if you just started topology, but it is a creative analogy between certainty and open-ness in the metric topology sense.

I think the problem most people have with Topology is that they try and relate it to the real world/the real line too often which leads to confusion. Even the terminology of "open" and "closed" sets is flawed as it implies they are opposites. I think some things in maths are better explained by the book definition. However, I appreciate the attempt of analogy and helping abstract maths reach a broader audience. Much respect x

Really good video. Have not had much occasion to really think about topology for many years. I like the machine analogy. The relation on the set that determines if it is open in the space or not. I have carried around a few spaces (Hausdorf 'cuz they are easier to wrap my mind around) that I like - "1" and French Railroad Space. Thanks for the video...

For anyone wondering (as I was) why the infinite intersection of verifiable sets isn't verifiable, consider: take an infinite set of machines labelled 1, 2, 3, ... where the ith machine waits i seconds before terminating. Each individual machine will terminate in a finite amount of time, but there will never be a finite time where all the machines have terminated, because even after i seconds, you're still waiting for at least the i+1th machine to terminate. I'm still not sure what these "verifier machines" have to do with shapes, though.

I personally think the best way to understand the axiomatic approach to topology is studying the properties of convergence in metric or even normed spaces. One quickly discovers that these properties can be derived only from the concept of open balls, and sets that are unions of open balls, even the continuity of a function. Thus, one might ask what really are the properties that define these sets- the open sets. One may think of generalizing them by their 3 key properties, which are exactly the axioms of a topology, and quickly gets a certain "visualization" of the space as being constructed of arbitrary unions of ball like objects. I think explaining topology, even not rigorously, takes a series of videos, not a single one

Thank you so much for doing an easy to understand video about topology! Next time someone who's not a mathematician asks me what is topology, I may just send this video!

Overall, this was excellent. I’d be interested in another video about the boundary of the set.

It is worth noting that your way of constructing a topology is only one of the ways in which we can INTRODUCE a topology to a space - which is that through open sets. Equivalently, you can introduce a topology through neighborhoods, closed sets, closure operation, interior operation, etc, not just open sets. I like to think about the definition of a topology in terms of how it was introduced rather than the collection of open sets that are deduced! Also, regarding the finite intersection analogy. An infinite union of open sets is always open. The infinite intersection of open sets is not always open. For example, consider the intersection of open balls of radius r centered around 0 for r

What and absolutely brilliant exposition. I didn't know about Tai-Dinae Bradley until now, I will be buying up her books on all things math!

Hey Tai-Danae, thank you so much for slowing a bit down, really helps, especially for non-english native speakers! :D That said... not really sure about this episode, may have to re-watch it, not for understanding it (I'm quite familiar with this topic) but for the analogy... Either way, keep at it! Happy Holidays! ^^

I liked the analogy ☺️

A lot of comments criticizing the video, but the like-dislike ratio is 10-2. I guess for the people that found this video helpful nodded happily and continued with their lives.

The "verifiable sets" and the machines to recognize them sounded like a reference to computability theory. Is there a connection between open sets and computable indicator functions?

Unfortunately I still did not understand the relationship between shapes like sphere, torus , and so on, and the definition of a topology on a set

Why is the empty set verifiable? It seems exactly the opposite

Okay, am curious: Given the indicator machine method you used to describe topology... do you in general tend to come at math from an Intuitionist perspective? Just curious. Also, how will those that win t-shirts specify the size they need, given that size is not a topological property? ;)

Confused? You should read the blog post in the description...

I feel like the dictation improved, thank you, and great job

I wonder what the audience for this channel is made up from, maybe it would be abit better to go deeper into the rabbit hole . Explaining theroems and less on definitions .

The definition at the beginning seemed clear enough.

Thank you for this video! I've been thinking a lot about this exact idea and am surprised to find a PBS video on it. It seems that some of the comments have missed the point so I hope this comment might "back you up" a bit. See this MSE post by Qiaochu for other references to this idea: math.stackexchange.com/questions/31859/what-concept-does-an-open-set-axiomatise In my opinion some of the value of this idea is this. The abstract notion of a "topological space" is nice for a while until you open your eyes enough to question if it's "too general". Is there philosophical value in the framework or is it merely a "successful language"? In mathematics we end up working with "real" type spaces like manifolds, Hilbert spaces, etc... and the reals are apparently much more natural than Analysis 1 would have you believe. When you work through Munkres however you get tantalized by the elegance of the language of open sets and wonder what the open set concept is "really" talking about, because it's obviously more general than "space". Why finite intersections and arbitrary unions? So as it states, the video is not addressing what is "topology" (which is generally thought of as the qualitative properties of "space"-ey things pertaining to connectedness and closeness) but what is "a topology". (For the record Tai-Danae, I thought you explained this perfectly and I only hope to emphasize exactly what you have said.) Thank you for the reference to the S. Vickers book, I'd been hoping to find a reference developing this idea. One further thing I'd like to note, famous mathematicians have publicly "admitted" to not completely understanding "what is a topology": mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets

It's a shame that this channel stopped posting videos

The analogy presented seems very confusing. Explaining it in more detail and giving more examples of topologies would certainly help.

9:04 I don't understand axiom #3. It is about the empty set and the embedding set X itself, therefore how can it be a requirement for building a specific collection of subsets? How can it be not verified?

I miss this channel

Congratulations, between this vid and my consumption of wine this evening, this is the first RU-vid vid of which I will confess I have no understanding. Not complaining, just observing.

Why is the empty set verifiable? This is all so confusing: "Suppose we have an indicator machine for the entire set of real numbers. Then the whole set is verifiable". Does this tell me any truth? If we suppose a set is verifiable that only tells me we suppose it's verifiable, but it does not make it necessarily true. I don't even know what to do with that.

Could topology be a new way of looking at Turing machine's halting sets? Does anyone know the topology this induces on infinite sequences of bits?

All these videos start very simple and nice and slowly get more complicated. The problem is that I always lose concentration in the middle

Is this 'open' business and the 'closed' set problem stuff, the reason why we get the idea of 'locally' the same ?

Analysts really ought to tell physicists that entropy is all they need.

Now we got a halting problem haha

A lot of people here are mentioning the same thing, that the machine analogy brings to mind the concept of Turing machines - especially when you mention that the machine may either halt or run indefinitely. As a computer scientist who knows nothing about topology, now I'm dying to know if there's more to that analogy? Is there some important correspondence between *decidable languages* and *open sets?* Is there some niche alternate version of topology defined in terms of computability, kind of like computable analysis?

From 7:15 - 7:35, what do you mean by the machine takes infinite time?? We aren’t actually concerned about the time of termination whether it is a second or an year or a light year. We want if the machine can terminate or not….right? And if there is something fishy behind that infinity, cant it be applied to union of verified sets?

Oh shit this video actually worked for me

why is an empty set open? In the second example in the blog post, why does it fail the third axiom? Why is {b} not open? It looks open to me. The axiom doesn't say all intersections subsets have to be included in X. Just that they have to be open. What if we include {B} originally in X and {b} is the intersection of {a,b} and {b,c}?

More videos on topology via logic please!

First things first, awesome video! Secondly, I'm bothered by a tiny detail that was sort of kind of discussed in an earlier comment thread, but it didn't really get into what I wanted. But anyway, I'm kind of bothered by the idea that we exclude the end points from these sets because no ruler is infinitely precise. However, aren't we mixing real world concepts with abstract ideas here? Surely, (within Pure Mathematics at least) we can allow ourselves to have infinite precision?

Thank You for another fantastic Video!! Could you do a episode on Turing machines? I think that would be fascinating and useful to many people!

I see a close correspondence between the group theory and topology based of this definition!

Why is it necessary for the machine to not halt? Why can't it output a boolean T/F?

Pretty sure you can't verify anything if your verification machine isn't guaranteed to terminate in a finite amount of time. Say a machine terminates at at some finite time T or not at all. In order to verify the result, you have to wait for time G such that G > T for all possible values of T. Since no finite value is greater than all other finite values, you can't verify anything in finite time.

A topological space is a strictly more generalized concept than a metric space in the sense that there exists a space whose topology is not metrizable. Take the reals *R* with lower limit topology, or the space of all functions defined on *R* into *R* with topology of pointwise convergence, or any set S with cardinality 2 or greater, where the topology consists merely of S and the empty set.

Observation 2 is weird, because the number could be the limit of one of the subset, and that particular would not end?

You can't just suppose there are an infinite number of machines and then proceed to list them and arrange them in an array. That only works for a countably infinite set. There are certainly uncountable topologies.

Kelsey, hurry back. The videos are much better when you're doing them. You have a great way of explaining the topics which make them much better.

Loved this video! :) Its really awesome when you explain things from textbook in a beautiful way. Hope to see more such videos in future.

I think I got lost at around 09:03. I still don't really grasp "a topology on X is a set of subsets which satisfy these three axioms". I don't feel I grasp the point about the null set being 'verifiabile'. And finally, I'm not really sure what the 'indicator machine' is, and what open sets are, in general. Thinking about the ruler earlier in the video, is the 'indicator machine' function analogous to the question: 'can a human with a finitely divided ruler or meter stick broken off from that space, and perfect measurement repeatability and reliability, measure the specific point in finite time'? The language, pace and animation seemed so clear and well-presented, which frustrates me that I feel I don't I get it. Don't get me wrong, I really enjoyed this, and think it was so well-made. I'm just frustrated at my own ignorance I suppose.

Very clean vid

Very unsatisfied with 5:05 onwards to 5:20. A very bad if not incorrect explanation for why the endpoints are not included. To follow that logic, we'd have to exclude all points.

Damn..ty for the video. I just wished this kind of videos are around when I took Topology in 2016. I failed Lol.

Great video!

07:10 why would it take infinite amount of time ? If we have infinite machines we can run them in parallel, if all terminate then x is in the set of intersections, if one does not terminate it is not ?

A number that's not in you... Graham's Number is certainly not in you, because if you try to remember in, your hand will collapse into a black hole.

This analogy feels wrong. You can run an infinite number of indicator machines in parallel and the maximum time for all the machines to finish is bounded.

So, what's something that's not a topological space? Is a set, S, not automatically a topological space of any set which contains S?

Wait, are you leading towards homotopy type theory by way of Turing machines? This is a topic I want to learn about but haven't yet, so if so I'm quite excited.

Stops on an input in the range and runs forever if not? I'm hearing echoes of the Halting Problem here...

This doesn't show the motivation for why we use topology. The starting point is the question "what is it about the set [0,1] that makes every continuous function achieve a maximum and minimum?" The brute force solution is that it contains all its limit points, but the elegant and insightful reason is that every collection of open sets that covers [0,1] has a finite subcollection that also covers [0,1]. If you don't start there, then what you're saying may be technically true, but it misses the point.

thankfully real world space is not infinitely divisible

I will to this day never understand the need, other than as a mere interesting curiosity, of putting a topology onto uncountably infinite groups, such as Lie Groups.

I don't see the value of introducing the links between topology and computability at this introductory level, which tends merely to confuse the issue, and introduce yet more concepts. While what is presented can be followed and can stand alone, it's pretty useless by itself, and the student still needs to follow a more standard path if they want to proceed past the trivial. So it feels more like showing off than anything else.

uhhh I'm gonna have to go back and watch a bunch of previous videos

is a torus a closed surface? I am asking from a physics perspective. For example, can gauss's law apply to it?

If a number is in the intersection of a verifiable set then all the machines will eventually terminate, but the video claims that we'd have to wait infinitely long for that to happen? Why would we have to wait any longer than the machine that takes the longest to verify the number?

While not a fantastic explanation of why we needed to explore this idea of "verifiable machines" to understand open sets, I think it is in no part the fault of the presenter. She's taken note of our comments from last video and taken the speed of her speech down a notch here. I think it's fair to say we all understood what she was trying to explain, regardless of whether or not we thought it was helpful. I'm just saying, I don't think there's a glaring problem with the new host this episode; it seems like if there were any gripes anyone should have it would be with the writing staff.

7:30 But wait. If you put x into three machines, but one of them is still running, you can't say it is in the intersection, because it's still running, but you can't say it isn't in the intersection, because you can't know if the machine will terminate. I'm confused. How can you know x is in the intersection of finite intersection of intervals? And the empty too. You would have to wait an infinite amaount of time to verify if x is in the empty set.

It was great to see some connections with topology which is like language theory. Deciding things halt/running infinite is kind of a hobby of a computer science student.

Phrases: Open, Empty, and finding things inside U ? Is math always this sexy?

Could you do a video on p vs np and the complexity theory?

The argument that an infinite union of sets is verifiable is incorrect/incomplete in subtle but significant way (imo). By stating that you can run the machines in parallel on a given input you are suggesting that you can run more than one machine at a time. If this were the case I believe you would have a much more powerful logic that could in fact verify the intersection of infinitely many sets. Instead you want to run the machines concurrently. In fact it is not obvious that you could successfully run all of the machines to completion. For example suppose that you numbered the machines such that they map to the naturals (i.e. 0, 1, 2 ...) if you were to run each machine in order for exactly one second, and it took every machine at least two seconds two terminate, then we would never compute that an element is in the union of all of the sets. Instead we must come up with a clever way to run the machines. Suppose that we instead dovetail the machines in the following way. run M0 for 1 second, then run M0 and M1 for 1 second each. Repeat this process indefinitely such that on the nth iteration you run M0-M(n-1) each for 1 second. Once any single machine terminates we can terminate the entire program, since we know that the set is in the union. Notice that as n->infty every machine will have run for infinity seconds. e.g M0 M0 M1 M0 M1 M2 ... where each instance of Mn refers to running a machine for a single second. You will notice that this dovetailing of our machines looks a lot like the enumeration of the rationals and cantors pairing function. If we attempt the same dovetailing trick on the intersection of an infinite set of sets we will run into the problem that we need every machine to terminate. To give a simple but incomplete argument as to why we could not concurrently run all of the machines to termination, suppose that every machine would terminate after one second. It would take an infinite amount of time for all of the machines to terminate regardless of how we dovetail, since one times infinity is infinity. If however we could run the machines in parallel, then it would take us only a second for all of the machines to run to completion.

How do you input a real number to a finite machine in a finite amount of time?

I don't like the idea of using terminating machines for these purposes.

I feel the explanation for why the intersection of an infinite set of machines doesn't terminate is wrong: We can't feed all the inputs to all machines at once - it's just the machine which is the intersection would always have another machine which it needs to feed it's input into, and never be sure if all the machines would answer positively, in contrast to an infinite union, in which the first positive answer out of an infinite collection of machines would suffice and thus terminate in finite time.

Tai-Danae I am fascinated by the way your mind works. You have a genius for taking a subject that makes my brain hurt and expressing it so well that even I can understand it (and you make it fun too! that's always a plus). Many Thanks!!

Can you make videos about wheel theory or quarternions?

An explanation using effective mathematics. I'm impressed.

So how would you have that distinction in a language with no articles, like Russian?

Ooh, are we heading towards sheaves?