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love this video! I do hope the video is a bit louder though

Pretty dope theorem

Nicely explained

At the end, wouldn't 0 itself be in the intersection though? would love to be corrected

my goat

Thanks a lot. Great video.

Amazing visualization. When I first studied topology, I thought of R^w (or R^N) as the set of all real-valued sequences and R^w1 (or R^R) as the set of all real-valued functions. Later I learned about how this concept is called an Exponential in a Cartesian Category. Studying category theory really helped me go back and re-learn abstract algebra, topology, etc. through a new lens.

Wonderful. Thanks

Great video. I spent so much time trying to understand this. Your video really helps. Will you make more videos?

It's really good! Thanks for sharing 😊

Ok I just have to say this before I watch the whole video the bagel and the Coco milk looks delicious 😋

This is the box topology not the product topology according to my modules material..

Such a cool proof, and very helpful!

On 5:30 you have a mistake when you say x* => x_n for all n=>N. Consider a monotone decreasing sequence x_n=1+1/n. Then x*=limsup(x_n)=liminf(x_n)=lim x_n = 1. Howevere x_n > 1 = x* for all n. The conclusion you derive later is correct but for the wrong reason. You need to conser the sequence X_n=sup{ x_m | m=>n}. Then x*=inf(X_n). Then you can say that by definition of inf there exists N such that x*+eps> X_N and since for every n=>N we have that X_N is the upper bound of {x_n | n=>N } we conclude that x*+eps > x_n for all n>N.

Using the concept of open balls without using the numerical meaning to drive the definition of continuity in topological spaces: got it!

Pretty good video. i never thought of it like this before

I like the idea that if a set is connected the only disjoint sets from the topology that form X are the entire space and the empty set. Or in other words, the only two clopen sets that can form the topology is the space itself and the empty set. Otherwise its not connected. thanks viro. I used this graph idea recently on this set of { (x, y in R^2 | x > 0 and y = sin(1/x)}. its graph is {x, f(x)} which is homeomprhic to the real line, and is thus path connected. i love this stuff. you an also use this connected property to show if something isnt a homeomorphism. for eample R -> S1 isnt a homeomorphism since if we take a point away from the line R we have created two connected componets. However if we remove a point from S1, its still connected! BLEW MY MIND, way easier than proving a homeomorphism typically.

I wonder how do you animate these things?

I just finished learning about co-vectors which are the set of linear functions from R^n and has size R^n. I wonder what the size of other restricted sets of functions like polynomial functions or continuous functions ect.

I should have seen this video when I started studying topology on infinite cartesian products, it would have made much more sense to me. Great video, as always :)

Hello sir, I just want to know that the prove you have shown is Constructive or Non Constructive.

Thank you so much for taking the time to draw out the visuals as well. It's really difficult to visualize without a representation sometimes so all the graphs are a godsend.

Why are 1's in these formulas? i.e. I'm not clear why the formulas all involve either +1 or -1 in some fashion. Why 1 and not some other number? I'm picturing light shining through a transparent sphere, and I wouldn't have thought something as clearcut as a simple factor of 1 would be at play, I would have thought it'd be some complicated mess involving pi or something. Thanks I appreciate it

Thank you so much! This helped me understand directional derivatives a lot more with the visualizing.

You have no idea how helpful these videos on complex analysis are to me. Thank you for your efforts

Wonderful explanation

An amazin explanation. Respect

This video makes the world a better place

Incredible video

This really helped me understand, thankyou very much keep it up

Very helpful!! Thank you

You cannot find the first N just as you cannot find the last real number closes to the origin.

This is so cool!!!! I love math

I'm watching all the videos in this playlist without headphones and my mum went: "He has such a beautiful voice" lolol and I agree tbh. Jokes aside thank you for sharing these, life-saving. ❤

Great vid bro

Thanks! I just encountered the phrase "commuting diagram" for the first time and this video was a great sanity check!

Thank you! It was very helpful :D

Great video, thanks

Thanks

You saved my life😢😢😢!Very clear and helpful! Thank you!

thank you

This is great

😀

I think this deserves a sequel that explains intuitionistic logic.

Really amazing ,can you teach complex number, like this

Your lectures are helping me tremendously in my Analysis course!

Georg Cantor was German. Also if you use German names, could you pronounce them German?

The Banach-Tarski construction produces sets without measure, so it is not a problem at all.

Ever since I learned out it when I was 12, I was in love with it. It makes perfect sense, and no-one could adequately explain the problem.

So it sounds like the Axiom of Choice isn't that you can pick an element from a set, it's that you can pick an element from a set and _know what it is._ For example, using the well-ordered principal, you can always find the smallest value in a set, so you can always choose the smallest element. Without the well-ordered principal, it's like trying to choose a sock when you can't tell which one is the left or right. That doesn't work with a bag of red marbles, because there's no order to them, so you can't pick the 'smallest'. If you could distinguish them by the number of atoms in each one, you could order them. If you had a bag of marbles of all different colors, you might order them by the wavelength of light that they reflect, and thus choose the one reflecting the shortest or longest wavelengths. But without some means of ordering (which requires additional information about the marbles other than "red"), whatever gets picked is arbitrary, not defined by a rule, and thus seems to fail the assertion because there is no function to get you the result. Or more accurately, a specific, repeatable result, because a function with a given input must always produce the same output. (This assumes that the different red marbles are in fact different entities, which is required by another comment describing how probability works.)