Topology (What is a Topology?)

[@BriTheMathGuy]

🎓Become a Math Master With My Intro To Proofs Course! (FREE ON KZbin) kzbin.info/www/bejne/aZTdmJl-irGNedU

[@phyein4815]

So.... what does this have to do with doughnuts morphing into coffee cups exactly?

[@hellstormangel]

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.

[@sonovadob]

You're really good at writing backwards, with your left hand. 10/10

[@BriTheMathGuy]

I'm afraid I have to give the credit to video editing!

[@yizhang7027]

The video is mirrored.

[@graceope-oluwa7816]

@@BriTheMathGuy 😅 afraid of what?😁

[@joefuentes2977]

Leonardo Da Vinci would be proud!

[@sonovadob]

@@yizhang7027 you're out of the magic circle.

[@ratanasorn8080]

Thanks so much, this is much clearer than a whole session at college

[@BriTheMathGuy]

Glad you thought so and thanks for watching!

[@ratanasorn8080]

BriTheMathGuy you’re welcome! Do you have more videos about topology?

[@BriTheMathGuy]

@@ratanasorn8080 Not right now unfortunately

[@atiurrahman7907]

First motivate me to study topology; why topology; for what topology. This is the actual thing that most of the teacher forget to explain.

[@ujjalmajumdar618]

You should not study pure mathematics then

[@Anteater23]

@@ujjalmajumdar618 🤣

[@dilshadimon4402]

@@ujjalmajumdar618 was going to say the same thing

[@atiurrahman7907]

@@ujjalmajumdar618 You are definitely a crammer!

[@christopherdan611]

@@atiurrahman7907 it’s because they don’t want to understand it they just want to regurgitate what they learned from someone else. it’s really like learning poetry in a foreign language without understanding the meaning to try to impress other people.

[@karungijane7490]

Soon I'm going to do an exam about topology. You have really helped me to understand topology. Thank you

[@AIphilosophy101]

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!

[@asifmahmud5646]

What a great explanation! Thanks for the content!

[@BriTheMathGuy]

You’re welcome, have a great day!

[@asifmahmud5646]

@@BriTheMathGuy Would really appreciate if you can make more content on topology. Looking forward for this :)

[@maxpercer7119]

very accessible introduction :) looking forward to other videos , on real or complex analysis , or whatever you're passionate about :P

[@poojitha]

Nice and clear explanation, Thanks!

[@BriTheMathGuy]

You're welcome!

[@fishersofmen5124]

More video please on Topology

[@Gurujanana]

Thanks from India . Nice explanation

[@georgepp98]

Thank you for explanation!

[@BriTheMathGuy]

You are welcome!

[@th1441Atgirl]

crazy (or then again not so much) writing a paper on cognitive linguistics brought me here. Thanks for such a crystal clear explanation

[@timothystudies2753]

Next series is topology?

[@BriTheMathGuy]

I don’t know that I’ll make it a series, but I may make more videos on the subject

[@solidstatejake]

I appreciate that you didn't just go "Uh.. *points* ... torus?" Providing the mathematical definition makes something much easier to understand. Thanks.

[@BriTheMathGuy]

Thanks very much for watching! Have a great day.

[@yakam7918]

thanks for clear beginning

[@BriTheMathGuy]

Glad it was helpful!

[@BriTheMathGuy]

Integral Tricks Teachers Wont' Tell You! kzbin.info/www/bejne/eYXZaWptjNRqnJY

[@siyabongaskhosana7782]

Which videos would you recommend for me to watch before this one so i can understand the terminology a bit better?

[@anasmikael5753]

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.

[@Victual88]

Thanks Bri!

[@BriTheMathGuy]

You bet!

[@donnaguerrero5588]

Thank you so much for the explanation 💖

[@BriTheMathGuy]

Glad it was helpful!

[@ashishlulla974]

I really like this. Does this open to something else or it just expands from this?

[@jennabrown2318]

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

[@akbarahmed3078]

Excellent video!

[@BriTheMathGuy]

Thank you very much!

[@nagendras9696]

Very nice video. Please make a series on topology and also the applications of topology

[@BriTheMathGuy]

Maybe one day I will :)

[@Cst-bv7fx]

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?

[@maxpercer7119]

the 'set' is given.

[@yohaan.kukreja1191]

Did you write backwards on glass in front of you ?

[@---00764]

Where are the other topology videos?

[@commirevo89]

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?

[@mattbrannock127]

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 “algebraic”, and a topology lacks this additional structure

[@michaelaswartz9844]

Whoever made this video used a good technique of mirroring this

[@carterswafford2222]

and here I thought he was just writing everything backwards :P

[@nosheenarif1920]

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 it….man thank you for this great video…

[@haojiangliu685]

please more topology

[@gehadsalah6575]

thank you

[@BriTheMathGuy]

You're welcome! Have a nice day.

[@kevinagee5085]

this guy is a genius

[@mathstoinfinityclassinTamizhla]

Just awesome 👍😊

[@BriTheMathGuy]

Thank you! Cheers!

[@centerpoint2844]

Bro it's so trippy when you write backwards on the whiteboard

[@BriTheMathGuy]

Video editing is powerful :)

[@xbenzemin1911]

Best explanation.

[@BriTheMathGuy]

Glad you think so!

[@lenethsajulga5524]

Thank youuu so much! 💖🧡💛

[@ashutoshkumarjha41]

Brian, Thanks for explaining beautifully by taking such a simple example. Can you correlate it by taking some surfaces and curves?

[@BrienDunn]

Nice!

[@BriTheMathGuy]

Thanks! Have a great day.

[@achw5721]

Something seems to be missing in the definition. T should be a collection of subsets of X.

[@blugreenrt3]

How do you determine and prove if script B is a basis for a topology?

[@munaredha7317]

There more vıdeo or not about measurable the f inverse the sigma or borel

[@YFanoush]

Man, is that a field cricket in the background?

[@ImaginaryAsh]

I did this in my maths degree. Never really understood it. Failed it.

[@BriTheMathGuy]

I'm sorry to hear that. Topology is certainly an abstract/different subject. Hope things worked out in end!

[@papytheo-maths9234]

Very nice.

[@BriTheMathGuy]

Thank you! Cheers!

[@stevenzheng5459]

Topology; studying surfaces in reference to holes Bottomology; studying holes in reference to surfaces

[@anasmikael5753]

like top and bottom are making their runways..oopss

[@syamalchattopadhyay2893]

Excellent

[@BriTheMathGuy]

Thank you! Cheers!

[@lakshdeep3]

Quick question like what age group do you think topology is taught in Also how hard is it

[@fnsdjkovnsdkvn]

end of undergrad/ graduate level so 22+ difficulty is rather subjective

[@viandcupcait]

I came here after seeing a video about cords tangled around handles being untangled

[@121sayer]

I kind of understand now what it is but still don't understand why it is or what it does.

[@munaredha7317]

Thx ı understand always see your lesson

[@BriTheMathGuy]

Glad you enjoyed it!

[@marcopivetta7796]

how do you write backwards so fluently

[@BriTheMathGuy]

Courage, a steady hand , and the power of video editing.

[@marcopivetta7796]

@@BriTheMathGuy how tho, looks so... natural? don't look like simple mirror-ing footage to me.

[@marquez2390]

marco pivetta he writes normally then reflects the screen

[@lucyoliva7818]

it's that easy??!!! why couldn't my teacher explain it? thanks for the video. i never would've understood it otherwise

[@jaydee9953]

And this guys writing back words….I’m so behind 🤣

[@chasereiter4760]

Somebody show me where the L in “draw” is

[@davidolaboye5549]

What can I use topology for in the field of computer science.

[@BriTheMathGuy]

I can’t speak for computer science. A google search would tell you more than I could. Best of luck and thanks for watching!

[@davidolaboye5549]

@@BriTheMathGuy will you make some topology course for us? I really love the way you explain abstract mathematics concept.

[@BriTheMathGuy]

David Olaboye thank you very much! I may make more topology videos in the future, though I’m not sure if I will make a series at this point.

[@vegalight196]

3D modeling?

[@Anteater23]

I thought Topology was to do with shapes?

[@BriTheMathGuy]

It does, but so much more :)

[@YourHostNathanBurke]

What is the purpose of a Topology?

[@Diaming787]

So, what does it has to do with coffee mug and a donut being equivalent?

[@andrzejkarolak3766]

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 ε,δ 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.

[@andrzejkarolak3766]

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.

[@axog9776]

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...

[@tonypang1577]

Cool

[@BriTheMathGuy]

Thanks! Have a great day!

[@TheVoiceOfQurann]

Good explanation ❤️❤️

[@BriTheMathGuy]

Glad you think so!

[@MeetPatel-pg5hj]

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.

[@sreelakshmivb2580]

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.

[@MeetPatel-pg5hj]

@@sreelakshmivb2580 yeah got it.i went on wrong way . thanks

[@KingGisInDaHouse]

the study of tops?

[@keishamselex1689]

please change your background colour like black or dark blue. explanation is really good

[@BriTheMathGuy]

I'll do my best in the future!

[@gwapdamathtutor2108]

How {a} and X = {a}.......,..,.wouldnt it be a, not {a} ?

[@matveyshishov]

I love topology! But unfortunately for some reason there isn't a good explanation for what it is. Why sets? What else can there be as "not belonging to the same set"? Why do we care? If topology is so fundamental, how come we also have geometric algebra? And wait, are you saying there's algebraic geometry? What's that, Lie algebra, abstract algebra, category theory, OMG, I thought topology was fundamental?! Haha, so yeah, I mean, I love math, but I got involved when I had my own questions and problems and tried to solve them and found out about the subfields where I discovered a treasure chest of ideas of those who came before me. Not a single time I have encountered an explanation, however, which would actually be approachable for me, had I not known about what I wanted in the first place and why I was reading that textbook. Hopefully one day somebody comes and fixes this, and more people are welcome in math!

[@adrian_zombturtle148]

It's so confusing, why can't he just write on paper

[@naimurpranto3403]

My internet connection is ok but I can't watch this video... Why It's happening with me!!!

[@BriTheMathGuy]

I hope you got it working!

[@STUWIETN]

*i was supposed to see secure contain protect - scp foundation video. But why math :'(*

[@dwayniebacon2960]

🤯

[@gregaizi]

X must be {{a},{b},{c}} , because a != {a}

[@shkim295]

intro sound to too large

[@weinerblut6869]

Definition without motivation or implications is near worthless. Those examples are worthless too. Answer the “so what”.

[@nm1978]

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...

[@edwinhapunda5321]

ndiwe chikali

[@andrijgonchar5010]

Mathematics is a discipline that is very harmful to health. It can cause nervous and mental disorders and great discomfort. Therefore, after the ninth сlass (іn 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, Poincaré be trained by mathematics clubs, as the future Alyokhins, José Raúl 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.

[@xyzct]

What is a non-algebraic topologist? A person who can't tell his a$$ from two holes in the ground.

[@Pete-Prolly]

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: ⟨ a, b, c ⟩ = 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!

[@ninemilliondollars]

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.

[@SamyTessier]

If you can't understand very basic set theory then Topology will be way out of your reach

[@andrzejkarolak3766]

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||

[@andrzejkarolak3766]

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).

[@andrzejkarolak3766]

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 ε,δ 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.