The precise claim is that every *_closed_* surface (compact connected no-boundary 2-manifold) is determined by its Euler characteristic *_and_* whether it's orientable or not.
Sir: on your opinion, is 'rubber sheet geometry' a good description of general topology, whose spaces sometimes are not even T1? I am suspicious of the beauty of general topology being shortchanged.
Topology does not apply only to manifolds in R^n. Do these 'stretching' analogies apply to non T1 spaces? I ask because I am suspicious of 'rubber sheet geometry' being used as a description of topology per se.
Group theory is important only to algebraic topology, not general topology. And the latter does not even require real analysis. (Though an understanding of metric spaces can certainly motivate.)
Hey small issue with the video... I think. Continuous deformation l is not a homeomorphism.. It's called a homotopy. A homeomorphism is just a bicontinuous bijection and in general is a much much less demanding map. For example, a trefoil knot and a circle are homeomorphic, but there is no continuous deformation possible between the two. Cheers!
'every surface is homeomorphic to either a sphere, torus, double torus etc..' What about an annulus, double annulus, etc? Toruses contain a 2D hole (the space in the middle) but annuli do not (sorry if incorrect terminology). Surely they are not topologically equivalent? (I'm pretty new to topology but if anyone could explain I'd be really grateful)
This is late, but I'm confused about the earth "obviously" being simply connected. If we were coming from the perspective of having never seen space images of the earth, how would we figure out that all loops can be adjusted to a point? Sorry if this is stupid, but I'm stuck on that. I've been listening to math lectures while on a road trip today, so my brain is a bit tired. I came to this video to get an explanation of how a coffee cup is a torus (I kind of get that).
Get a really long string with its ends joined, then move it until it is not taut, and continue in that direction /s In seriousness, you could also use triangulation to find the Euler characteristic of the Earth, which itself has practical applications in cartography, so there’s an additional incentive to do it
@@videostar75 Yes but that uses 'external information from space', and he said we can demonstrate Earth is simply connected without any external information. Also is it obvious that any loop on earth can be shrunken to a point? Have we looked at every possible loop on the surface of earth? maybe there is some loop we have not yet come across that can't be shrunken to a point (which would given evidence of a toroidal surface).
Two surfaces having the same euler characteristic does not garantee that they are homeomorphic. It is a required condition but it is not sufficient. In general the video is only about orientable surfaces, for which this is true, but there are also non orientable surfaces.
This is just wrong. You can deform a ring in a circle without cutting nor terring yet they are not homeomorphic. The ring is arch-connected when removing two points and the circle is not.
Also euler's charactistic is a topological invariant, this means that if two spaces are homeomorphic to eachother they will have the same euler's characteristic but the reciprocal statement doesn't hold. It is not an if and only if
That is a bad definition of homeomorphism. What matters is that there is a one to one function that assigns one set to another set. Euler characteristic is only one of an infinite number of choices of such a function.
Two topological spaces X and Y are homeomorphic if there exists a bijection f : X -> Y such that both f and f^-1 are continuous, I think is the best way of defining it
you made me understand topology in 22 seconds. I think I heard the actual click in my mind
27 dislikes are from flat earthers, because you casually proved the shape of the globe just using topology 😂
That the earth is a globe us an unproved lemma. Work harder.
@@zapazap what??? The earth being a globe is proven
@@TD-iy8us The commenter presented the claim without proof.
@@zapazapGreenland has a special property, (how?)
go to space oh wait you cant
@@guidinglight1lul If you know that can't go to space, then why did you advise me to go there?
Are you engaging in good faith sir?
This is the first video of topology I ever watched. Thank you for sparking my interest.
The precise claim is that every *_closed_* surface (compact connected no-boundary 2-manifold) is determined by its Euler characteristic *_and_* whether it's orientable or not.
Finally! A video with simple explanation on the concept of genus!
they... didn't even say the word genus
@@martyguild LMAOO
Awesome video!!! Even I can't tell in words how helpful it is for me.Please make videos about topology of glueing,cutting etc.
This is interesting, it makes me wanna learn topology
Wonderfully explained. Thanks a lot!
Mind-blowing! Quality over quantity (5:00 min)!
And logical-mathematical psychoanalysis. started from lacan analytic discourse. Thanks for the video!!!!!!
My college needs you as a teacher!
Excellent visual demonstration of useful applications!
Make more, more, more !! =)
This is a fabulous video. Incredibly clear and helpful. Bravo!
This explanation provides very good insight. A very good video.
I am PHD in Topology, and this is the simplest explanation for laymen
Sir: on your opinion, is 'rubber sheet geometry' a good description of general topology, whose spaces sometimes are not even T1?
I am suspicious of the beauty of general topology being shortchanged.
clear and crisp intro to the concept.....
easiest explanation found till now great
thanks for your simple explaination
This video has saved my masters
Very insteresting subject. Excellent explanation. Thank you so much!
Hey nice video! I really enjoyed your intuitive explanation. You made it real interesting and good luck bro!!
super .....create more videos like this....with a picturized explanation .....one can easily understand .....next part pls😊 😊
Topology does not apply only to manifolds in R^n.
Do these 'stretching' analogies apply to non T1 spaces? I ask because I am suspicious of 'rubber sheet geometry' being used as a description of topology per se.
Study Group theory and real/complex analysis before touching topology. The concepts in algebra and analysis naturally lead to topology
Group theory is important only to algebraic topology, not general topology. And the latter does not even require real analysis. (Though an understanding of metric spaces can certainly motivate.)
I like the concept of deformation in telling a lot about what the future is preparing for us to discover
Hi Jack great video. Any chance of and introduction to manifolds? Peter
Awsm..am speechless ..cant use wordz for prase on your presentation on topology
Wah! I think I got the idea, thanks a lot, much better than reading a book!
This is interesting, it makes me wanna learn clarinet
Jack Li's videos: Music, music, music, music.. TOPOLOGY!!
Thanks for the awesome video! Now when my friends talk intuitively about topology, I know what to say.
Hey small issue with the video... I think. Continuous deformation l is not a homeomorphism.. It's called a homotopy. A homeomorphism is just a bicontinuous bijection and in general is a much much less demanding map. For example, a trefoil knot and a circle are homeomorphic, but there is no continuous deformation possible between the two. Cheers!
Is 'rubber sheet geometry' a good description of general topology, whose spaces sometimes are not even T1?
'every surface is homeomorphic to either a sphere, torus, double torus etc..' What about an annulus, double annulus, etc? Toruses contain a 2D hole (the space in the middle) but annuli do not (sorry if incorrect terminology). Surely they are not topologically equivalent?
(I'm pretty new to topology but if anyone could explain I'd be really grateful)
Thank you very much for sharing your knowledge freely.
In my religion this has a big reward for you from Allah.
Thank you again.
Outstanding Dear!!!!!!!!!!!!!!!!1 waow!!!!
Great video!
This is late, but I'm confused about the earth "obviously" being simply connected. If we were coming from the perspective of having never seen space images of the earth, how would we figure out that all loops can be adjusted to a point? Sorry if this is stupid, but I'm stuck on that. I've been listening to math lectures while on a road trip today, so my brain is a bit tired. I came to this video to get an explanation of how a coffee cup is a torus (I kind of get that).
Get a really long string with its ends joined, then move it until it is not taut, and continue in that direction
/s
In seriousness, you could also use triangulation to find the Euler characteristic of the Earth, which itself has practical applications in cartography, so there’s an additional incentive to do it
Hey, can you turn a torus into a kline bottle? Both euler characteristics are 0, but I believe you can’t.
Aye yo my g big ups man
Excellent !
I will now use this knowledge to debate flat-earthers. Earth must be a sphere!
very good!
@ Jack Li ...what program did you use to create your presentation? Thanks
Very cool video, thanks
very interesting and straight forward, however I guess the word "verticies" is a wrong spelling
Great video - I understand it
each bridge, window, dor tunnel, .... i am not quite sure what the euler caracteristic of earth is....
Orcodrilo Orquial p
"Next time you're out with your topologists friends..." 😂
great video!! thanks a lot
Every surface is homeomorphic to a ball, a donut, an eight or a fidget spinner. Got it
The human body is homeomorphic to a 7-holed donut unless you decide to pierce it
This will not get you to the surfaces surrounding knots.
How do you know that Earth is simply connected?
He explains why at 4:30. You can shrink any loop to a point without cutting or glueing
@@videostar75 Yes but that uses 'external information from space', and he said we can demonstrate Earth is simply connected without any external information. Also is it obvious that any loop on earth can be shrunken to a point? Have we looked at every possible loop on the surface of earth? maybe there is some loop we have not yet come across that can't be shrunken to a point (which would given evidence of a toroidal surface).
@@videostar75 That holds for a sphere. To say it holds for the Earth requires more work.
There's a lot of homeomorphism in San Francisco.
Are you thinking of homorphisms?
Its not important to worry about why maths is important. One can assume it isn't and prove it is always.
Splendid
Very nice video. Thanks a lot :-)
Why this video is recommended while I'm trying to studying the topic of math?
Amazing👌🏻
Two surfaces having the same euler characteristic does not garantee that they are homeomorphic. It is a required condition but it is not sufficient.
In general the video is only about orientable surfaces, for which this is true, but there are also non orientable surfaces.
that was a homotopy and the iff statement with euler characteristic doesn’t hold
You have a typo at 1:19. It should be vertices. Other than that, thank you for the introduction.
"Separable" was spelled wrong too. ;)
Great tutorial... there’s “a rat” in separate -> separable. )
PLEASE PART TWO
Still donut understand
I see some crossovers to Calc 3.
Awesome
No words!!!
bro thats so facts
This is content.
This is just wrong. You can deform a ring in a circle without cutting nor terring yet they are not homeomorphic. The ring is arch-connected when removing two points and the circle is not.
Also euler's charactistic is a topological invariant, this means that if two spaces are homeomorphic to eachother they will have the same euler's characteristic but the reciprocal statement doesn't hold. It is not an if and only if
And you missed the projective planes when talking about clasification, this just holds for orientable ones
Affine planes aee also simply connected, you missed the compact part
That is a bad definition of homeomorphism. What matters is that there is a one to one function that assigns one set to another set. Euler characteristic is only one of an infinite number of choices of such a function.
Two topological spaces X and Y are homeomorphic if there exists a bijection f : X -> Y such that both f and f^-1 are continuous, I think is the best way of defining it
@@asparkdeity8717 All knots are homomorphic. Are they all homeorphic?
top
so
Laughs in blender
It's pronounced, You-ler.
No
dude. this is money. have a donut.
Does this guy have a cold or allergies or something??
( mathematical term for a donut 😂)
cant stop pretending😭