Topology (What is a Topology?)
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- Äas pĆidĂĄn 20. 08. 2024
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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information. Viewers should always verify the information provided in this video by consulting other reliable sources.
Integral Tricks Teachers Wont' Tell You!
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Which videos would you recommend for me to watch before this one so i can understand the terminology a bit better?
Hi Prof, can you tell me what application/ device u used for the maths writing ( I mean writing on the screen and how do you video it) ? Actually I want to teach maths online (tuition) but I dont know what apps to be used, and i dont think writing on a white board is interesting.
So.... what does this have to do with doughnuts morphing into coffee cups exactly?
You're really good at writing backwards, with your left hand. 10/10
I'm afraid I have to give the credit to video editing!
The video is mirrored.
@@BriTheMathGuy đ afraid of what?đ
Leonardo Da Vinci would be proud!
@@yizhang7027 you're out of the magic circle.
Bro, thanks so much man. You made a series of math definitions for this newbie very digestible and easy to understand :)
You're very welcome!
Thank you for this video, as someone self studying topology and running into various abstract examples of topologies, it is nice to see a simple example of a topology.
Thanks so much, this is much clearer than a whole session at college
Glad you thought so and thanks for watching!
BriTheMathGuy youâre welcome! Do you have more videos about topology?
@@ratanasorn8080 Not right now unfortunately
First motivate me to study topology; why topology; for what topology. This is the actual thing that most of the teacher forget to explain.
You should not study pure mathematics then
@@ujjalmajumdar618 đ€Ł
@@ujjalmajumdar618 was going to say the same thing
@@ujjalmajumdar618
You are definitely a crammer!
@@atiurrahman7907 itâs because they donât want to understand it they just want to regurgitate what they learned from someone else. itâs really like learning poetry in a foreign language without understanding the meaning to try to impress other people.
Simple and to the point. Thanks. What about making more videos on what is everything we do not understand. Like for example, what is Abstract Algebra, and do it in the same fashion? Would we all become mathematicians of some sort? Great way to go!
Next series is topology?
I donât know that Iâll make it a series, but I may make more videos on the subject
I appreciate that you didn't just go "Uh.. *points* ... torus?"
Providing the mathematical definition makes something much easier to understand. Thanks.
Thanks very much for watching! Have a great day.
What a great explanation! Thanks for the content!
Youâre welcome, have a great day!
@@BriTheMathGuy Would really appreciate if you can make more content on topology. Looking forward for this :)
Nice and clear explanation, Thanks!
You're welcome!
Soon I'm going to do an exam about topology.
You have really helped me to understand topology.
Thank you
I'm a year 7 and my maths teacher told me if I watched this video and understood the mathematical components to a topology we would talk about it 8nstad of doing work this was very helpful thank you. However I have one question how do you find the set?
the 'set' is given.
Why is a cute guy explaining advanced mathematics so hot. I am in awe. Thank you for helping me pick Topology as a course I want to take
More video please on Topology
I really like this. Does this open to something else or it just expands from this?
Thanks from India . Nice explanation
Thank you for explanation!
You are welcome!
very accessible introduction :)
looking forward to other videos , on real or complex analysis , or whatever you're passionate about :P
crazy (or then again not so much) writing a paper on cognitive linguistics brought me here. Thanks for such a crystal clear explanation
Did you write backwards on glass in front of you ?
The Topology rules remind me of the rules for Rings and Fields when I studied Modern Algebra (also called abstract algebra). Is a Topology an example of a Field?
A topology is not a field. In fact, a topology is not any kind algebraic object. A topology is just a set, and for it to form an algebraic object (by that I mean a group, ring, field, vector space, module, etc.), it would need an additional component called a binary operation. This additional structure is what makes an object âalgebraicâ, and a topology lacks this additional structure
Although the opening sound of your video was very loud thank god my ears survived I still feel it, but the concept was so good, I will forget itâŠ.man thank you for this great videoâŠ
Whoever made this video used a good technique of mirroring this
and here I thought he was just writing everything backwards :P
Very nice video. Please make a series on topology and also the applications of topology
Maybe one day I will :)
Thank you so much for the explanation đ
Glad it was helpful!
I thought Topology was to do with shapes?
It does, but so much more :)
Excellent video!
Thank you very much!
how do you write backwards so fluently
Courage, a steady hand , and the power of video editing.
@@BriTheMathGuy how tho, looks so... natural? don't look like simple mirror-ing footage to me.
marco pivetta he writes normally then reflects the screen
Where are the other topology videos?
Brian, Thanks for explaining beautifully by taking such a simple example. Can you correlate it by taking some surfaces and curves?
thanks for clear beginning
Glad it was helpful!
I did this in my maths degree. Never really understood it. Failed it.
I'm sorry to hear that. Topology is certainly an abstract/different subject. Hope things worked out in end!
Bro it's so trippy when you write backwards on the whiteboard
Video editing is powerful :)
Just awesome đđ
Thank you! Cheers!
There more vıdeo or not about measurable the f inverse the sigma or borel
this guy is a genius
Something seems to be missing in the definition. T should be a collection of subsets of X.
Man, is that a field cricket in the background?
Thanks Bri!
You bet!
I kind of understand now what it is but still don't understand why it is or what it does.
I came here after seeing a video about cords tangled around handles being untangled
And this guys writing back wordsâŠ.Iâm so behind đ€Ł
Somebody show me where the L in âdrawâ is
What is the purpose of a Topology?
Thank youuu so much! đđ§Ąđ
thank you
You're welcome! Have a nice day.
What can I use topology for in the field of computer science.
I canât speak for computer science. A google search would tell you more than I could. Best of luck and thanks for watching!
@@BriTheMathGuy will you make some topology course for us? I really love the way you explain abstract mathematics concept.
David Olaboye thank you very much! I may make more topology videos in the future, though Iâm not sure if I will make a series at this point.
3D modeling?
How do you determine and prove if script B is a basis for a topology?
Nice!
Thanks! Have a great day.
please more topology
the study of tops?
Best explanation.
Glad you think so!
it's that easy??!!! why couldn't my teacher explain it?
thanks for the video. i never would've understood it otherwise
Very nice.
Thank you! Cheers!
Quick question like what age group do you think topology is taught in
Also how hard is it
end of undergrad/ graduate level
so 22+
difficulty is rather subjective
I didn't got that why second condition required. .bcz intersection of two sets is always contained by both set..and if both sets r in T then obviously there intersection is in T.
No, for eg consider {1,2,3} as X. If we take a family's of subsets of X ,say empty set,X,{1,2} and {2,3}. Clearly it's not a topology as intersection of last two sets gives {2} which is not in the collection.
@@sreelakshmivb2580 yeah got it.i went on wrong way . thanks
please change your background colour like black or dark blue. explanation is really good
I'll do my best in the future!
How {a} and X = {a}.......,..,.wouldnt it be a, not {a} ?
Topology; studying surfaces in reference to holes
Bottomology; studying holes in reference to surfaces
like top and bottom are making their runways..oopss
Thx ı understand always see your lesson
Glad you enjoyed it!
Cool
Thanks! Have a great day!
will you expand this series one day? i am interested in learning about manifolds/homology theory but there isn't much content i can study with in video format beyond general topology...
Excellent
Thank you! Cheers!
Good explanation â€ïžâ€ïž
Glad you think so!
intro sound to too large
So, what does it has to do with coffee mug and a donut being equivalent?
Equivalence between those things is understood as a continous invertible mapping between them. Topology gives us the notion of continuity alternative to metric definition. Intuitively we say that map is continuous when it maps points that are close to each other to points that are close to each other. If we have a metric we usually define this by Δ,Ύ bounds on distances between points and in topology we define beign close to each other by belonging to the same open set (element of topology). More strictly a map between topological spaces is continuous if inverse image of open set is open.
At first glance this definition should be put in different direction, but take for example a function from real numbers to real numbers that is equal to 0 for x less than 0 and 1 for x bigger or equal to 0. In most common topology on R, open intervals are open sets, so interval (1/2, 3/2) is open. The inverse image of this interval through our function is set of x bigger or equal to 0. It is not an open set and that says that this function is not continuous.
My internet connection is ok but I can't watch this video... Why It's happening with me!!!
I hope you got it working!
It's so confusing, why can't he just write on paper
X must be {{a},{b},{c}} , because a != {a}
đ€Ż
Definition without motivation or implications is near worthless. Those examples are worthless too. Answer the âso whatâ.
*i was supposed to see secure contain protect - scp foundation video. But why math :'(*
Mathematics is a discipline that is very harmful to health. It can cause nervous and mental disorders and great discomfort. Therefore, after the ninth Ńlass (Ńn a Soviet school with 11 classes) mathematics is required to be a sport for prodigies, or to study it in some laboratories in the course of work, starting with a laboratory assistant... I want a healthy young generation to grow up, not tortured by mathematics. And let the future Lobachevskys, PoincarĂ© be trained by mathematics clubs, as the future Alyokhins, JosĂ© RaĂșl Capablanca - chess clubs. Why play stupid shows when some pretend to teach mathematics, , strength of materials, theoretical electrical engineering.............. and others pretend to study these subjects. It even looks indecent. Strength of material, theoretical electrical engineering are needed by a very limited circle of engineers.
ndiwe chikali
my brain is a set of my body, my heart is also a set of my body, so I guess they are the union of my body, oh, very interesting mathematical concepts.
laugh now and be happy...
What is a non-algebraic topologist?
A person who can't tell his a$$ from two holes in the ground.
Referencing the www.storyofmathematics.com: "Topology: the field of mathematics concerned with spatial properties that are preserved under continuous deformations of objects (such as stretching, bending and morphing, but not tearing or gluing)."
I don't see how your math and set theory examples explains topology. You're expecting your audience to be familiar with set theory as a start. If not, then they are lost. Can you provide a much simpler explanation of a topology please.
If you can't understand very basic set theory then Topology will be way out of your reach
This video was not about motivation, but more about definition of topology. Often intuitive notions like limit of a function get very complicated when put in abstract language. Imagine solid ball in three dimensional space. You can think of so called open ball, which are those points of the ball that are not on its surface. If the centerpoint of this ball is called O and it has a radius of 1, then open ball would be defined as those points X in three dimensional space that are at distance less than 1 from point O (||X-O||
So you can probably think of open set as a set of points that are close to each other in some sense without using notion of distance. If we have a topology on set X we can show that a set U is open if for every point x inside U there is an open set V such that x belongs to V and V is contained in U. This says that points that are close to x (points from V) are also close to points that are close to x (points in U).
If we get used to this meaning of open set we can proceed with the notion of continuous mappings. The most popular example is a torus and a surface of a mug. You imagine them as made of strechy rubber. You can change the shape of this rubber making torus into the surface of a mug or vice versa. We intuitively can think that such mapping is continuous if it does not tear the surface during transformation. So it must send points that are close to each other to points that are close to each other. When we have a metric we define being close as some bound on distance between points (usually we assign letters Δ,Ύ for those bounds). In topology we use open sets. Strict definition is a bit misleading at first glance, so you need to work it out with examples. We say that a mapping between topological spaces is continuous when inverse image of open set is an open set. It looks like it should go in opposite direction, but it doesn't. If the inverse image of open set (so of points that ore close to each other) is not open, then there can be a point that is close to points in inverse image, but its image would be outside the initial open set (so it would be far apart). Therefor we would have some points close to each other sent to points that are far from each other making the mapping discontinuous.
I once asked my friend, who has a PhD in Math, "Which branch of Math is completely useless?"
Him: "Every Math class has cry-babies who whine 'what applications does this have?' When will I use this? Wah-wah-wah. F***ing babies."
So, I said: "I want to take the most useless Math class there is & ace that B**-yach!"
Him: "I think you'll dig Topology then."
Me : "Cool."
Him: "Ok, but people are still going to ask you about it's practical uses; any thoughts on how you'll address them?"
Me: "Oh, I'm just gonna smack them in the mouth. .....like, hard af!!!"
Him: "what if a girl asks you?"
Me: "I'll be smooth; I'll use my Topology book to roll a blunt or cut some rails for her and I to snort Molly off of it and then I'll say 'how's that for practical use?'"
Him: "That's actually pretty cool. I bet no one's ever used that particular textbook b4 for that."
Me: "Well there was that time when I sniffed 'H' off of Smith, Eggen, St. Andre."
Him: "Refresh my memory?"
Me: "The book you gave me, "A Transition to Advanced Mathematics," which I read ALL the time... it's by Smith, Eggen, St.Andre!!!!"
Him: "Oh, that's right! Wow... You're so rare."
Me: "Yes indeed. An Eagle's got nothing on me. In fact, if I ever see an Eagle I will use it's eggs to make breakfast. I'll call it 'The Eagle Omelette.' cuz I don't **** around."
Him: "I could go for some Bob Evans all of a sudden."
Btw, to whom it may,
I'm in Calculus III now at Wayne, after that is "Linear Algebra" & "ODE/PDE," and there's no more Math classes higher than that; I'll have to tranfer to UofM or some other University, (even though I learned Gaussian, Row Reduction, back subbing, Cramer's Rule, Transpose, Cofactor, Adjoint, Inverse, Determinants of 2x2 & 3x3, Eigenvalues, Eigenvectors, LU Decomposition, and basic operations like addition, subtraction, multiplication, powers, on my own. I'll have to repeat it.... even though Calculus III so far has been ALL vectors and dot product, cross product, which I know!!!! Grrrr.... I just hate this notation: âš a, b, c â© = ai + bj +ck notation!! WTF??? Just stack them vertically, and draw a square or rectangle around them!! Geez, f***ing ***holes do it all the hard way!