Introducing Topology - Topology #1
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- čas přidán 5. 11. 2019
- This video is an informal introduction to some topological concepts. I discuss the notion of a topological space and motivate this with a few examples, before discussing the notion of topological invariants. I emphasise how these invariants can be used to conclude whether two spaces are NOT homeomorphic (meaning they cannot be continuously deformed into eachother).
I then explore how we realise topological spaces as sets by giving a simple one dimensional example, the circle S^1. In the next video I will extend this further by constructing more complicated spaces using the cartesian product (uploading soon!).
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Excuse me(very videos and i have some questions)
1.do circles with different radius have different topological invariant
and they cannot described by only angle (because you need two variable to uniquely specify it
thank you
i
1) Circles with different radii would all be considered topologically equivalent since they can be simply deformed into eachother (all circles are isomorphic) - since this is the case the analysis of any circle can be reduced to analysis of the unit circle and this is often left implied. Furthermore since when working with topology and topological spaces we do not have a well defined notion of distance (until it is chosen), so at this level of pure topological spaces (before they become metric spaces) there would be no notion of distance to assign a radius - all circles in the space are isomorphic and there is no notion of circles of different sizes and shapes.
It is important to remember that what we are dealing with here and calling a circle is simply an interval with identified endpoints - the pictures we draw representing this as a round connected line segment are all being created inside additional structure that we are not considering (namely the fact the circles exist inside a higher dimensional space and have a notion of distance implied from their physical size). So really there is only one (topological) circle - S^1, which is implicitly the 'unit' circle when expressed in any distance system. Then depending on the geometric structure we add to this topological circle we produce one of the many possible geometric representations of S^1!
2) It might feel like a sphere could be deformed into a circle but unfortunately this just isn't possible because of the dimension, because a sphere is inherently 2D there's no possible way to condense that information into a one dimensional space (without cutting and gluing) - you could squash a sphere until it becomes completely flat forming a disk, but there's no way this disk can be contracted to form a circle - since the circle is the boundary of the disk, removing the disk would count as cutting the space and is hence not allowed. Hope that helps!
@@WHYBmaths i see. All circles are isomorphic because you can map each elements on the circle into other circle and there exists a inverse function between them(which implied one to one and onto).
Nice explanation
thank you so much !
i would recommand this lecture/videos to other people
what can i do for supporting your videos
hope other people can watch this and there are not much people find these videos.
@@garytzehaylau9432 Thankyou! Any recommendations of my videos you can make are really helpful thankyou! And just keep watching and asking questions to support me I appreciate it!
@@WHYBmaths today I've spent almost all day trying to find something which will help me to understand why will cartesian product of two circles ever be a torus, but you did it, so I really appreciate it, thank you so much! :) all explanations on math stackexchange (for example) are too complicated and kinda implicit, but your one is good, thank you) you actually may try answering some questions there (i mean on math stackexhange, but youtube is fine too, so thanks again), because that would be pretty helpful for others i guess!
The dog is a good listener 👀
Excellent lesson. You have a natural talent for teaching.
Love your dog chilling with the torus and spectating.
Great video, and great dog, too!
These videos are so much better than any other lectures I've seen
Whoopee! I saw you on Reddit. Can't wait to dive in 🥳🥳🥳🥳
These videos are great you're an amazing teacher.
Thanks a lot @WHYBMATHS. Today I can say I have really learned something.
Essentially nice video
These videos are great! Love the dog, lol.
Fantastic
Cheers!
amazing video man. Thanks
how is the dog doing?
Thanks a lot
I didn't realize Morrissey knew so much about mathematics
I think he looks a bit like a young Pete Nice from 3rd Bass. "Brooklyn Queens!"
thank youuu
I am a subscriber
I like how the dog is looking quietly...
What if you were to reshape the circle in such a way that it has two point on the same angle. (e.g. by stretching a part of the unit circle and forming some bends with the stretched loop, similar to a toadpole or sperm cell bending its tail fully).
I would image describing the circle by only angles causes equivalence of elements that should not be the same.
Could you perhaps shed some light on how such a shape would be described?
Kind regards,
Jarek
Wish they gave my topology lectures like you. It was horrible.
Yes part of my motivation for making these videos is how awful most lectures (particularly physics based) are at explaining this stuff! Glad they are helpful:)
@@WHYBmaths Your videos helped me so much with physics at university!!! And inspired me to want to learn more about pure maths!!, thank you:))
Are topological sets always open sets??
In terms of topological spaces being open sets I can see why you are not allowed to cut and re-join topological spaces!! Because in the process of cutting you create boundaries, if you cut an open set you may retain its being an open set but you could not re-join what you cut?? A circle is an open set, if you cut a circle you create endpoints, but even if you somehow kept it an open set you could not then re-join them in the absence of endpoints!!
What is doggo's name?
Lula! instagram.com/luladogislush
Can it be said that Man is invariant to monkey?
Shouldn't the Theta belong to [0, 2pi) rather than [0, 2pi]
Yes in most cases, but since I wanted to emphasise that the zero and 2pi points are the same I included them in the interval initially, but yes for the circle we usually just take S^1 theta ~ [0, 2pi) since the endpoint 0 and 2pi are equivalent!