Learn Topology with Our Ongoing Video Series - Perfect for All Levels!

[QuanPenguin]

Hello, I am currently creating an ongoing series of Topology video lectures and looking for students of an appropriate level for some accurate feedback. They are much in the style of the popular Khan Academy.

Link

Thank you for your time.

 

[micromass]

Ow, nice. I'll check them out later today...

 

[micromass]

Wow man, this is great stuff! I actually teach topology at university and I'm surely going to recommend this to all my students. I'm looking forward to your new videos.

Anyway, here are some things I'd like to mention
- I think you have a nice balance between theory and examples. I also like that there are many counterexamples. It's really nice... Some more examples of topologies would be appreciated tho, since the concept really many good examples to be understood.

- There are some very small mistakes somewhere. For example, in video 2, at 6:13 you forgot the square (but it isn't really that important). And in video 7, something weird happens at 6:49: the video just jumps back in time and repeats itself. But its all minor.

- I think you should also have defined closed sets, interiors or closures for metric spaces. It might be to abstract if you suddenly introduce it for topological spaces. On the other hand, the metric piece is already very long, so I don't really know...

- I think the prood that open balls are open would have been quite instructive. But that's just me.


Overall, you did a great job. Is there something I can help you with in making these videos? I'd really like to contribute somehow (:

 

[Ulagatin]

Hi Quan,

Take my note with a grain of salt: I'm just a Year 12 student in Australia with a love of pure mathematics. I've seen a little bit of topology before, but only enough to have a few general ideas about it.

I watched all 10 of your videos in the series today and they are excellent! Very well done, very clear. Starts off simple and very intuituve (basic distance metrics) and works through to notions of continuity - really mind-bending stuff! Woah!

Congratulations on a job well done, definitely work on more! :)

Davin

 

[QuanPenguin]

Thank you so much for the encouragement, I will certainly be dedicating more time to this.

I'm really not sure what to ask for in the way of contributions, obviously I've only been doing this a very short time so right now I'm just trying to get some exposure and some constructive feedback. I'm very grateful for the error corrections, youtube actually has a very good caption system for leaving overlays over the video to point out small corrections without having to reprocess the whole video, so anything you spot I can remedy (when I figure out how :D). I'm also currently reuploading #7 to fix that. As for help, if you like the videos there are a couple of things you can do to help me move a little higher up the search list. Leaving comments on the videos themselves, clicking a few "like" buttons, and subscribing to my channel all help in some small way to do this, so if you'd like to that would be very much appreciated. I hope one day to be able to do this kind of thing full time.

I did think about introducing closure/interior ect within the scope of metric spaces, but I assumed that most people watching would have completed some kind of real analysis course, but I'll make some effort to analogise it with metric diagrams. It's really great that I have come across as understandable to a Year 12 student, you must have some amazing passion for maths, so I'd love further feedback as I progress through my course. I have about 100 videos worth of content planned in Topology, and what I really like about this video format is I can diverge into things not normally covered on a standardised course, like for example separation axioms beyond just Hausdorffness, and perhaps different strengths of compactness and connectedness. Is there anything you've wanted to put in a topology course but couldn't for lack of time? Suggestions like that would be fantastic!

I suppose my long term goal is to do a series on measure theory and linear analysis after finishing the topology series, and hopefully progress into topological groups and harmonic analysis.

 

[micromass]

Well, the thing I mostly miss about topology courses, is that doesn't always lay a connection with other fields.

So I'll love seeing videos about
- the Zariski topology
- matrix groups (like O(n) is compact, SO(n) is connected, GL(n) is open,...)
- Stone representation
- the Baire theorem and some basic applications

- Also I would like videos explaining all separation axioms (of course T0 - T6), but also soberness, etc.
- The relation between Alexandroff spaces and pre-orders
- some cool compactifications: Alexandroff, projective, Cech-Stone,...
- Forms of compactness: supercompact, Lindelof, Bolzano-Weierstrass property,...
- Quasicomponents, extremal disconnectedness, when are all open sets closed,...
- the relation between filters and ideals of a ring (i.e. the filters on P(X) are ideals in a special ring)

there's much more I could mention

 

[Theaumasch]

Hi! So far I think your video's are very nice, yet I have a question about the 4th video. When discussing balls in a discrete metric space, you say that only {c} is contained in the ball if r is less than or equal to 1.
Now, there are two possibilities in my head. The first is that this is a small error, and it should have been just less than one. The other is that B_r(c) is defined with d(c,x)<r. Which one is correct?

Thanks in advance.

 

[micromass]

B_r(c) is defined as d(x,c)<r

 

[QuanPenguin]

Amazing suggestions, I guess 200 videos would be appropriate :D

Just wondering if you managed to introduce the videos to your students yet, I imagine given the point in the semester you're probably far far ahead of what I've covered so far. I guess I need to speed up to get enough videos out in time for exam crunch!

 

[micromass]

Yeah, the topology class is already to far ahead to have a use of the videos, I'll still suggest them tho. But I've also introduced them to people who first use about metric spaces.

But if there are a lot of videos next year, then I'll certainly suggest them to the new topology classes.

I think there are so much possible topological subjects that you could possible make over 1000 videos

 

[micromass]

Alright, I've watched upto bases. It's still really great stuff. I really liked the analogy with the bases of linear algebra. Such analogies with other math fields makes videos great, so I'm really happy you included it.

Some things I would like to mention:
- in video 15, at 8:04, u wrote X\V, but that should be Y\V. That's the only mistake I found
- To bad you didnt include the following equivalence. B is a basis of (X,T) iff
\forall U\in \mathcal{T}:~\forall x\in U:~\exists B\in \mathcal{B}:~x\in B\subseteq U.
I know it's trivial to show, but you actually use this a lot.

I also had an idea. What if you post every now and then some videos containing exercises. So you present some fun exercises which the viewer should solve. I can provide you with some cool exercises if you want (e.g. prove that there are infinity many prime numbers using topology, with some hints of course).

Anyways, great videos!

 

[Buri]

They seem really good! I actually just ordered a copy of Munkres Topology book (arriving this Monday) because I wanted to read it on my own and now I have your videos to watch as well! :) Thanks a lot for these! I'll be subscribing to your account (I'm Buri1750 on YT) :)

 

[QuanPenguin]

Thanks for the corrections, I'll get on that.

I was having a similar idea actually, I was thinking that around exam time I would ask for sample questions, mostly things that people are struggling on or have seen in a practise paper and try and whizz through a bunch of them on video. Now that you mention it though, it might be a way cooler to take a load of sample questions, (some from students, and some just plain evil ones) and just write them out on video, let the comments frenzy over them for a week, and then write out the solutions on video a week later. That could be a lot of fun!

Your idea touches on another distant idea I had. I recently picked up the book Topological Groups and Related Structures by Alexander Arhangel'skii, which is a quite recent expository on basically everything we know about Topological Algebra (it's huge), and in there are hundreds and hundreds of bitesized open problems, small enough to explain in 5-10 minutes and many are very interesting. Now obviously I haven't covered even remotely enough material for the expected audience to understand them, but I figured that once I had I could just throw out one a week, talk a little about it, and see if the combined viewer base could have a genuine crack at it!

Also, I'd like your opinion on the best order to go through limit points and closure. My next three videos are going to cover neighbourhood basis and then countability axioms but not separability, and then I would cover separability at the same time as density. First I figured since I already defined closed sets abstractly I could just go right ahead with the intersection of all closed sets approach, but then how do you link it back to limit points without it seeming forced? Should I instead start at sequences in metric spaces? Might take forever, and I don't want to revisit metric spaces too heavily. Maybe I could just talk about nets first. It just seems any way I try to order it some rigor is sacrificed. Where do you think is the best place to start?

 

[micromass]

Hmm, tough call.

I always thought that topology should be seen as a generalization of metric spaces. So imo, it's best to discuss sequences in metric spaces. Dont take to much time for it tho, you would just have to discuss adherence, convergence and some easy notions. But I understand that you wouldn't like to do much metric spaces...

If you don't like doing metric spaces again, then maybe you should start with doing nets immediately. Most people will already know sequences in metric spaces, so nets shouldn't be to difficult to understand.

Doing limit points without sequences or nets is also possible, but I wouldn't do it. People won't be able to link back to what they know in the metric context and it will probably sound a bit forced.

What I suggest is that you spend a little time doing sequences (probably half a video or something) and then immediately generalize to nets. Imho, that would be best.

On a related note, are you also going to do filters and ultrafilters? Nets are easier to grasp, but I've always loved the filter approach. It doesn't have to be right away, but I would be sad if you won't mention filters at all

Anyway, I'm signing up to youtube just to become fan of your channel (:

 

[micromass]

I've watched until neighbourhood bases now. Really good stuff! I liked the allegory you did in video 19, with the little man moving around in an open set.
I also liked the equivalence between bases and neighbourhood bases.

Carry on, youre doing great!

 

[QuanPenguin]

Thanks! was a fun week, exhausting though. I did retake after retake of the metric equivalences video because I kept convincing myself it was wrong, by the time I'd redone it about four times I sounded so bored I had to retake it again :D

Anyway, I sketched up a plan for next week, wondering if you can spot anything I'm missing. I think I will swap #25 with the example half of #24, but I could probably use another few minutes of something else interesting. I think I managed to get away with enough interesting sequence stuff without hitting metric spaces too hard, anyway tell me what you think.

#21 Countability Axioms plus example Cofinite topology, also some example of 1c but not 2c, example of 2c.
#22 Sequences in metric space -> sequence in topological space how d(x_n,x) -> 0 generalises to balls for n > N and then to neighbourhoods with diagrams ect
#23 Sequence terminology "eventually, frequently, (more?)" , examples of sequences with multiple limits, example only eventually constant sequences converge in discrete ect
#24 Sequential Continuity, cont => seq cont, counterexample for converse id : cocountable topology -> usual topology on R.
#25 Proof that first countable spaces seq cont => cont [this is too short]
#26 Directed Sets /Examples
#27 Nets, subnets, convergence, (might be short)
#28 Continuity in terms of nets
#29 Limit Points/dervied set in terms of nets and neighbourhoods
#30 Closure (finally)!

If anything is too long I can split it up into two videos and just bump closure to next week, funny how I was originally planning to do it at #15!

 

[micromass]

Yeah, that sounds cool.

For 21, I would also show that all metric spaces are 1c (but I think you were already going to do that )

If 25 is to short, maybe you can add this two theorems:
1) A set G in a first countable space is open iff every x in G and for every x_n --> x, x_n is eventually in G.
2) A set F in a first countable space is closed iff, whenever a sequence in F is converging to x, then x is in F.

And if the video is still to short, you can maybe add that such a space is called sequential.


For 27, maybe you can say a bit about ultranets and there existence.


I'm really looking forward to next week, I've never decently learned about nets so I really want to know what you have to say about it.

 

[QuanPenguin]

So much to add, I'm going to draw up a long term plan, it'll probably be around 100 videos before I even touch compactness. Amazing

 

[micromass]

Alright, I've watched everything upto directed sets.
Some remarks:
- in video 21 at 51seconds, you say that the cardinality of N is at least the cardinality of B(x). Is this correct? I'm not a native english speaker, but I found that weird...

Also, I kind of missed a counterexample of a directed set. But then again, it's not really vital...

It would be nice if you would talk about the specialization pre-order sometimes. This is a really handy concept in algebraic geometry, so it would be nice to see a video about it, or just to see it mentioned...

Did you make a long-term planning for the video series? I'm quite interested in that

 

[QuanPenguin]

Hello there. Sorry about the lack of updates but I've had the combination of not having a job, being very ill and moving home in the last week. I should be able to pump out videos starting early next week. I also want to make sure I introduce nets perfectly, and make sure that all my definitions of limit/adherence in terms of neighbourhoods are perfectly sound and understandable.

Also yeah, that countability thing does sound a little strange, oh well, I hope most people watching will have some set theory understanding anyway.

 

[CoolPro]

These are awesome! Just so you know I'm a first year undergraduate student and I can understand you PERFECTLY! Except for the bits about countability which I've never seen before but I think I get the idea. A countable set is just one that you can write as a sequence right?

 

[quasar987]

Very well done. I'm sure many an undergrad is grateful to you for these clear, concise and colorful explanations.

(CoolPro: You're right that a set is called countable if it can be written as a sequence. But when talking about topological spaces, the word countability usually refers to a property of the topology. There various type of countability properties a topological space can have, and these are generally referred to as "the countability axioms". A given topological space may satisfy some of these axioms or none. You can surely find out what these are on wikipedia.)

 

[ThoughtSpace]

Hello all, after a few months of personal problems I can finally revive and hopefully complete this series. I have two new videos out with more on the way. Thanks to everybody for their support in the past, I have a new determination to see this through.

I'm still open to suggestions on how best to introduce closure, which will likely be in a couple of videos time.

http://www.youtube.com/user/ThoughtSpaceZero

 

[micromass]

Ah nice, I quite missed your videos I really like the videos about nets by the way!

How do you best introduce closure? Hmm, that's not easy. There are three ways to do so:
- the set of all points of closure
- the smallest closed sets containing the original set
- the set of all convergence points of nets

I think the easiest way is the second. But the first way is more often used...

 

[dendub]

What happened to these?