This is Why Topology is Hard for People #shorts

This is Why Topology is Hard for People #shorts

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[valoraz]

The difficulty of mentally dissociating it from Euclidean geometry

[grmpf]

Topology would probably be hard for me because I do not study mathematics, and yet somehow, this video was recommended to me.

[joaorodriguezmarcondes]

I think the level of abstraction also makes it a very hard subject to learn for the first time

[Entropize]

I think an underrated source of difficulty for the course is that the definition of a topology is left completely unmotivated by most professors. The class is often taught as "here's this definition, now let's do some problems and prove some things with it" rather than as a rubber sheet geometry course backed by intuition.

[MatematicasNuevoLeon]

Topology is usually one of the first really abstract courses in the curriculum. I mean. It's very hard to grasp the notion of a non-metric space (but with a topology). There is simply no good intuition, compared with all you have seen before. I remember completing proofs of theorems in my first topology course and thinking at the end... what the hell is this really means? It's like... I know this is "true", but don't know what it is.

[Brien]

In my Analysis 2 class we covered a lot of topology specifically differentiable manifolds. I found the concepts rather easy yet the sheer amount of theorems building upon the previous one’s made it easy to get lost. Especially proofs concerning the classifications of 1 manifolds or borders made my head explode as a vast amount of preexisting knowledge is needed to understand them.

[surrealistidealist]

For me, what makes Topology hard is not knowing anything at all about what it is, since I'm still just brushing up on my Algebra! 🤣

[pgray]

Bizarre counterexamples that I could never come up with. When I was a student, a book called Counterexamples in Point Set Topology helped.

[paulrogowski]

I think the first great thing that topology clarifies is the idea that given a point in a continuum, there is no "next point", but there are only adjacent sets of points. This makes precise the idea of "gradual change" of a function which is fundamental in models of physics.

[mansurdaschaew]

I totally agree with you. Due to its very elementar structure, topology covers so many objects so it can be very unintuitive. There are even books like "Counterexamples

[sanjursan]

Like abstract algebra, topology demands one must be willing to memorize definitions with perfect clarity.

[jaimemenapadilla]

I love topology and found it quite easy, what gets me every time is any form of analysis be it real, complex, multivariate, or functional. It gets me every time.

[SmileyHuN]

It's so intuitive yet so technical

[sydneyteacherjobrightmind]

Hi, Maths Socerer, it's nice of you to introduce new subjects based on a lot of your personal experiences to maths learners. By encouraging discussion, I can obtain more information from your followers.

[gregwong-dgjq]

I think the hardest bit about learning topology is developing a feel for how to find a perfect balance of geometric, analytic and algebraic intuition to solve each problem or understand each concept.

At the beginning (usually the first half of your first topology class is on point set), you are usually working with very many analysis-esque ideas (for instance, stuff on connectedness/ compactness/ metrisability where a lot of intuition from real analysis can be ported over very well to this more general setting) so for most people this is their initial feel for what kind of mathematical skill topology needs. 

When the focus starts shifting onto an intro to alg top (usually the second half of your first topology class) through stuff on fundamental groups and covering spaces though, because suddenly all the notation and terms start to get so dense (which is natural since you are beginning to ask questions about much "nicer" geometric structures such as the sphere, torus, klein bottle etc so you necessarily work with a stronger set of prior assumptions), there needs to be a significant mindset shift within the student to start borrowing a lot more intuition from their knowledge of geometry and to rigorise this intuition via their knowledge of category theory and algebra or else they are at risk of getting way too bogged down in tiny details to understand the big picture or to have any clue at all on how to solve problems. A lot of students who weren't really prepared for this change of gear start really finding it difficult to have a deep conceptual understanding of what was going on (which was the case for me until I relearnt the stuff in my own time). 

This ports over to one's second course in topology (which is usually an introduction to homology/ cohomology and higher homotopy), except here some of the concepts naturally involve a lot of work in what I like to call "exact sequence acrobatics" which demands a higher level of fluency in algebraic reasoning so here one again has to rebalance one's intuition towards that a little more. Moreover, because of the cumulative nature of maths classes, you are often required to go back and forth between these different modes of inspiring intuition and arguing your reasoning from various mathematical fields in order to do the problem sets in a typical second topology class. It's kind of like expecting a jazz musician in a band to be able to play three/four different types of instruments and to improvise something that sounds good on the spot by playing any combination of these different instruments wherever necessary. Nevertheless, I feel as though I am someone who is very easily bored, which is why doing this (at least for me) is really fun and interesting, which combined with the fact that topology is just such a beautiful field of maths with such wacky results makes it in my opinion the coolest part of the subject.

[moli]

I totally agree. Enroll in a topology class when you are fully prepared. I learned that from the few weeks when I was in one. It definitely feel desperate

[dhickey]

Nice! Just finished studying proofs with limits. Now I know why we do this!

[readjordan]

When i took the course, i was surprised and it was difficult because it wasnt like the YouTube videos at all. The professor was the dean of the math department, and there was not one mention of coffee cups and donuts. Not one mention of mobius strips. Not one mention of anything i seen online really. After class i would mention these concepts and the professor smiled saying exactly, these are applications of the class theorems and good intuitions behind what were doing in class and why. But really it was all just rigorous logic and proving things are continuous, exist at all, what happens when spaces change (but purely from the analytics, no visual ever.) it was just raw analytics, calculation, and proofs. No calculator, no graphic depiction, no description of what were doing, just that were doing it. Not to mention the professors writing was the most unique writing ive ever seen. It was harshly sharp, and triangular, all letters were rotated towards the bottom right, like -π/6 and sometimes -π/2 and all spoken instruction (aside from proper nouns) were in Mandarin. I passed the class and all, and the professor is one of the friendliest people youll meet. But i definitely took it raw like WWE.

[jamesbra]

The topology of it is just so hard to navigate for me.

[michaelwalker]

You seem like a pretty nice guy (professor). I've taken up to Calculus 3. Took differential equations, linear algebra and discrete mathematics. According to your advice i should work on Set Theory before studying Topology. Oh, and I'm in my 40's and well out of school. My interest in this field lies in creating geometrical shapes with woodworking. Strange, i know. But i always had a fascination with the complex shapes I'd see in advanced mathematics text books. Do you know anyone who does this kind of thing? Or do you have any thoughts on this?