Topology Illustrated (2015) | Hacker News Reader

Topology Illustrated (2015)

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122 pointsKoshkin21 comments

What are the prerequisites for learning the basics, or developing a rudimentary understanding of this subject? I find the idea of it (as far as I can understand what that even might be) fascinating, but also exotic and forbidding. I had a couple of friends who reconsidered their major after taking topology :(

Just some basic algebra and calculus is the main prerequisite. You can probably get away with just linear algebra if you don't plan on going too deep into the subject.

There aren't really any mathematical prerequisites, except for a basic understanding of writing proofs. Most of topology is about defining a set of axioms (specifically, a topology is a set X together with a collection T of subsets such that...) and studying the consequences of those axioms. Studying topology is a great way to build up mathematical maturity.

However, I'd say it's very difficult to see the "point" of topology without taking a proof-based real analysis class (where you use epsilons and deltas to prove things about sequences, limits, derivatives, integrals, etc.). The concepts in topology generalize everything that is typically covered in a real analysis class, and many of the standard examples rely on some familiarity.

(in a similar way, it's very hard to understand the "point" of category theory until you've noticed that many proofs across group theory, linear algebra, real analysis, etc. are all suspiciously similar)

As for how to study: Topology is great because you really can build it all from the ground up. Find a book with good exercises, and do all the exercises! That way, you'll be forced to learn for yourself how all the pieces fit together. Unfortunately I don't have a book to recommend since I learned from a set of unpublished lecture notes.

EDIT: I dug up the old lecture notes, here [1] you go! Authored by Harrison Bray (no relation to me afaik) for University of Michigan, MATH 490, Fall 2016.

[1] https://github.com/benrbray/benrbray.github.io-source/blob/m... [2] http://www-personal.umich.edu/~hbray/

> in a similar way, it's very hard to understand the "point" of category theory until you've noticed that many proofs across group theory, linear algebra, real analysis, etc. are all suspiciously similar)

Would you be able to give an example of such a proof cluster?

One example I'm aware of is Lawvere's fixed point theorem, of which Gödel's (first) incompleteness theorem, the undecidability of the Halting problem, and Cantor's theorem are all special cases.

Not really related to "group theory, linear algebra, real analysis, etc.", but interesting nevertheless.

It's quite a wide generalisation which really just captures the nature of diagonalisaion arguments, but it does formally tie together various proofs/theorems which "smell the same".

Not the previous poster, and not a maths expert by any means, but for example there are parallels between the rank-nullity theorem in linear algebra, the orbit stabiliser theorem in group theory, the splitting lemma in homological algebra, and the Fredholm alternative in functional analysis. They can all be seen as manifestations in some form or another of the first isomorphism theorem in abstract algebra.

TY for the notes!!

Since the subject is pretty abstract, you need a certain level of mathematical maturity to grok the concepts. A previous course in abstract algebra or geometry (involving proofs) would be useful.

Introductory texts in topology tend to divided into two main subareas - point set topology (the low level axioms of how you define a space and its connectivity properties), and algebraic topology, which focuses on the global mathematical structures of these topological spaces. I found point set topology extremely dry when I first learned it, but it's important to understand those low level basics before moving on to the "cool stuff" in algebraic topology. I recommend "Topology" by Munkres to learn the fundamentals.

I took an experimental class in high school which taught topology and knot theory to people with a pre-calculus background. We used an early draft of a textbook (not sure of the title or if it’s even been released - our teacher got access because she knew the author) and a lovely knots textbook called “The Knot Book.”

I think we gained a really good intuitive understanding of homeomorphism, neighborhoods, and homology groups. Knowing about these topics has been super useful in my studies and career.

On the other hand, some of the algebraic components of topology went totally over my head. So I do think there’s a limit to how much you can understand without diving into other fields.

Also, we had the benefit of a trusty excellent mathematician as our teacher, which helped quite a bit. Good luck diving into this field, I hope you find it rewarding!

Topology is a canonical example of "mathematics without numbers" (and indeed of "qualitative" rather than "quantitative" mathematics), so for anyone approaching it with an undergrad or lesser background in math, learning it feels a bit like starting with a clean slate. As benrbray mentions in a sibling comment, there aren't any prerequisites in the usual sense, as you won't need to draw on any reserve of mathematical facts built up in learning previous topics while studying it. But learning it does require the ability to assimilate axioms (and to intuit why they make sense as a starting point), and then the ability to reason about those axioms rigorously to draw out the interesting consequences.

That said, a background in some mathematical topics is useful for learning topology. The subject is rooted in a particularly abstract notion of distance, so it's useful to have some experience reasoning with somewhat less abstract notations of distance like metrics, the most familiar of which is the quite numerical Euclidean metric involving the familiar square root of the sum of squares. Familiarity with metric spaces in general (that is, spaces equipped with any axiomatically acceptable notion of a metric distance) is even more useful because it requires a similar sort of axiomatic reasoning as does topology.

Kolmogorov & Fomin's classic text on analysis, the Dover edition of which is under $12 on Amazon [0], has a good (albeit austere) introduction to topology, leading up to it via axiomatic set theory and metric spaces in the book's first chapters. On the one hand, I hesitate to recommend this book, because it was the text for a course that mercilessly exposed the shortcomings in my mathematical education to that time; on the other hand, I recommend it for precisely that reason. Having later assimilated the material, I do consider the book a very good introduction for anyone who either already has the mathematical maturity to do the book properly or for anyone who wants to gain that ability in the way that everybody who has done does: by staring at the same page for hours on end while working everything out on paper, down to the axioms if necessary, until you stop misunderstanding, and then start understanding.

That said, there surely are gentler introductions to the subject if you just want to get a rough idea of it.

[0] https://www.amazon.com/Introductory-Analysis-Dover-Books-Mat...

> so for anyone approaching it with an undergrad or lesser background in math, learning it feels a bit like starting with a clean slate

Did you mean without? Algebraic topology would be a pretty standard 4th year undergrad course in the UK.

Contrary to what others are saying here, the author says the following:

> A rigorous course in linear algebra is an absolute necessity. In the early chapters, one may be able to avoid the need for a modern algebra course but not the maturity it requires.

This is a requirement that is rather unusual for a first book in topology, many of which have not a drop of linear algebra. The reason for this is presumably that the book dives immediately into homology which is often left for an algebraic topology course. It's an interesting choice.

I've been looking for a good calculus book. I was curious if this worked well, and I found:

1) It costs $300 for a paper copy 2) There are no previous. I have no idea if it's actually illustrated

I'm looking for something which is intuitive, and illustrated. A lot of pictures, good example, and perhaps less math and text (in this case, assuming the educator will be talking through most of this, and the learner doing things together).

I don't know if they deliver outside the UK but it's listed here for £57.44:

https://blackwells.co.uk/bookshop/product/Topology-Illustrat...

Don't let the domain name fool you, this is not a calculus textbook. It's a topology textbook. It looks like an awesome topology textbook and I think I'll buy it, but I don't think it's what you're looking for.

I don't really have a suggestion for a good calculus textbook. My school used Stewart and it's okay. I'm also, apparently, a huge math nerd and sailed through those classes; thus I don't think my opinions are terribly general.

But in my experience, learning calculus must come hand in hand with tons of exercise. Worked solutions are really crucial for a good book for self-directed study.

It's less directed self-study as tutoring and mentorship.

My experience is a bit different; the trick is to:

1) first expose kids to the general concepts

2) build them out in as broad a range of contexts as possible, slowly and gradually. Write some code. Do some engineering problems. Etc.

3) Go back and formalize it once the intuition is there

The earlier you start, the better. But you want a different calculus book for a college student, as for a high school student, as for a middle school student, as for an elementary school student. The earlier you go, the more you want pictures and intuition (and the more time you then have to build out the formalism later).

i learned from this back in college:

https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/091409...

one of my favorite text books ever. (tho it was paired w/ one of my favorite teachers ever, which i'm sure helped a lot.)

I think this would be a good book in a year or two. For now, if I can introduce basic concepts like slopes, derivatives, limits, and similar, simply and intuitively, that would be ideal.

Thus far, my favorite is Cartoon Guide to Calculus.

If it's helpful at all to you, I've had success with the list of books from this website: https://mathblog.com/mathematics-books/

Most of them likely have significant previews on amazon. The ones I was interested in which didn't have previews ended up being available one way or another online, which I used to assess whether they'd be worth it for me or not (instead of wasting everyone's time and energy on ordering a book and returning it within five minutes).

Calculus Made Easy is exactly what you are looking for. http://calculusmadeeasy.org/

Thank you!

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