Intro to Topology 

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@@eliaswenner7847 me 😭

Same here

This may be the best simple description.

Except it's a play doh statue and the police are watching to make sure you don't break or glue it.

That statement is stupid... but whatever.

play doh but you can’t make or remove holes

you summed up topology in 6 words

This deserves an award!!

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If you make them, I will buy them, lol.

@@wynstansmom829 if buy then i will it them

Ok buddy topologist

Yep! A donut with a few fluid-filled internal compartments, but still. :)

nolifeonearth a three-holed donut, I think, since the nasal sinus connects to the mouth.

no there is more than one hole

nolifeonearth فوزي موزي

@@malcomthonger more than one hole? that's the ULTRA DELICIOUS donut!

i dont think this channel is reviving back any time soon :cc

Oh, when I say “triangle” I mean the boundary of a 2-simplex, and those are homeomorphic to circles. It’s true that my some of my language here was hand-wavy - this is an introductory video, and when I made it I was laying the groundwork for motivating the rigorous definition of homeomorphisms in a later video. I never got to it, but maybe one day.

I meant to make a video explaining this eventually, and maybe I will one day, but in the meantime: The technical terms for the squishiness are “homotopy” and “homeomorphism”. When we say “You can squish one shape to turn it into another”, the technical way of saying that is either “those two shapes are homeomorphic” or “those two shapes are homotopy equivalent.” (The second statement allows for more intense squishing, where you can squish a ball into a single point, for example.) I don’t know what your background is or whether you’ve taken a topology class, and I’m afraid that may be an unhelpful answer if not, but those are the terms where that intuition connects to technical definitions. Thanks for your question, regardless.

@@AlternatingSum holy shit I didn't think you'd reply lol. I am a math major and I've studied topology before (introductory) i do understand the homomorphism analogy. What I meant in my question is how does the definition of topology makes us see shapes from the perspective of "how many holes there are"

@@user-cd8yi4dk6p Ha sure, I wanted to reply because I initially thought to make a topology series to address exactly the disconnect you’re talking about, between very intuitive introductory videos and rigorous but arcane-seeming topology lectures. Since I only made one video I didn’t get very far towards that goal, alas. Re: number of holes - if two spaces are homeomorphic then they have the same number of holes, the same kinds of holes, and those holes interact with each other in the same way. Really what I mean is: They have the same fundamental group and the same homology groups. Introductory topology classes usually don’t get to homology, and often don’t get to the fundamental group either, which I think is a little unfortunate - they’re kind of laying the groundwork for concepts but never quite getting there. Algebraic topology classes are where you really get into that stuff. I do like actually like a lot of point-set-topology on its own, the weird pathological examples like the topologists’s sine curve are fun. It can be a nice exercise in stretching your intuition about how space can behave, but it’s not everyone’s cup of tea (or donut of tea, I suppose).

Gitonga Mwaniki in topology you cant make/mold together holes. in the 1 hole and 2 hole donut example, streching the donut and pulling the middle together to connect counts as making a hole. as such you can conclude that they are not topologically equal. if it was allowed everything would be the same and there would be no point in topology

( ͡° ͜ʖ ͡°)

That's all true except that you can close holes with continuous transformations. You just can't "open" a hole through tearing because those points at the seam/cut will move a different distance than the neighboring points.

You're still right though, these would not be topologically equivalent because while you can close holes technically, the tearing is what eliminates the inverse. Without an inverse function, we can't preserve continuity.

@@benjaminhanson6137 actually I don't think you can close a hole because then you would be making the surface difference, if you just shrunk the hole until it "disappeared" then the hole might not look like it's there, but its inner lining would have to still be there unless I'm completely misunderstanding topology, because there would be no way to manipulate the shape to have the inner lining disappear.

I love dmt

Various concepts have several different applications. Here is the one application of topology. So in topology you have topological spaces, some topological spaces are very useful and they're known as manifolds(basically surfaces in higher dimensions). In general theory of relativity, spacetime is a 4 dimensional manifold that allows one to understand gravity and how a lot of things in the universe work. This is just one application. It even has application in quantum computing.

сферическая геометрия

hmm

@@darwinvironomy3538 Poke-ology, Advanced Concepts in Void

@@ryangunnison38 you made it up or that's real?

@@ryangunnison38 it's a cool subject but i don't get the application and if you define something consistent

@@darwinvironomy3538 I was thinking about it for a scifi concept but I dont think there are real world applications, and the names are just me having fun

pstuddy Might have something to do with meshes.

Topology is useful in animation

Topological data analysis is a hot topic in modern science. Basically, one imagines that all of the data points had to come off of some topological shape (called a manifold) and then there is some math to try to figure out what type of manifold the data points came from. One of my coworkers did a few years at Sandia Labs and found out that topological data processing methods were much better at figuring out where to put the drill to get oil and natural gas than previous partial differential equations methods were. One reason why this is the case is that the PDE method had to try to figure out the approximate shape of the caves that the oil was in, but the topological method only cared about the topological types of caves that the oil was in.

Hole* There’s definite answer yet, mostly the scientific consensus is it’s just a really dense ‘object’ not is wormhole

Eventually! I decided to make the "Distance, Dimension, and Space" playlist first, since that material is a prerequisite for the next topology video. I'll release a video on 4-dimensional space soonish, then one on distances in high dimensional spaces. Then I'll be ready to resume the topology series. :)

Great, thanks for replying, subbed!

shavan merja

undergraduate math programs usually have topology as a 3rd or 4th year subject, mandatory or elective depending on the university

@@lordspongebobofhousesquare1616 thank u spongebob

simp

Clarify?