5 ms·

Topology seems like a really cool field. Can I jump into this book without any topology knowledge? I have not taken differential equations yet and wonder is th

Topology seems like a really cool field. Can I jump into this book without any topology knowledge? I have not taken differential equations yet and wonder is that is a prerequisite.

myth_drannon 8y ago
From Goodreads review: " This is not so much a math textbook as an extended infomercial for the modern field of topology, with applications and intuition stressed over formal results.As the author's intention appears to be to give a broad overview of the field and its potential uses which will encourage readers to pursue deeper study, ..." So looks like the answer is yes
nikofeyn 8y ago
you will understand very little of the math in this book without a substantial background in mathematics. however, and i think the author would agree, you should jump in anyway. you'll pick up pictures, applications, vocabulary, and some ideas, all of which could help pave the way as you learn the material more traditionally. the book sets goalposts so to speak, especially since it is written in a more expository manner with lots of example applications. i don't understand anywhere near everything in the book (much less than that probably), and i still get something from it. in terms of learning some topology and geometry, i recommend topology by james munkres and an introduction to manifolds by loring tu. both are accessible to junior and senior undergraduates and beginning graduate students.
- throwawaymath 8y ago
  I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra. If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).
  - bigmit37 8y ago
    Thank you. I am actually a self- learner and have only take calculus, linear algebra, probability, statistics. Currently learning bayesian statistics and reading “first course in abstract algebra” by John B Fraleigh. I am guessing I should do Real Analysis and then the Munkres book to get an idea of topology ?
    - magoghm 8y ago
      Right now, I'm self-studying analysis with Stephen Abbott's "Understanding Analysis". I strongly recommend it. It has clear explanations that help to develop the intuition about each concept and what I think are well chosen exercises.
      - nikofeyn 8y ago
        there is a book by jay cummings called real analysis: a long-form mathematics textbook that i highly recommend checking out. used with a more "advanced" book, it could be a nice complement. Real Analysis: A Long-Form Mathematics Textbook https://www.amazon.com/dp/1724510126/ https://www.amazon.com/dp/1724510126/
  - drilldrive 8y ago
    I personally really enjoy Gamelin and Green's Topology, and (if you need it) the first chapter goes through topology of metric spaces. The rest of the book is well written; I couldn't quite find myself to like Munkres.
  - nikofeyn 8y ago
    agreed. i had it in mind the person was a math major. although i think topology is approachable without analysis with some dedication, it certainly helps provide context for the abstraction. i listed another nice book in another comment, but metric spaces: iteration and application by victor bryant is a pretty fun introduction to metric spaces and analysis at a very approachable level.
- ajudson 8y ago
  +1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.
- adenadel 8y ago
  I know Munkres is the go-to standard for topology, but I think that Essential Topology by Martin D. Crossley is a better book. It's only 200 pages, packed full of great examples, and covers all the (as the title says) essential topics in undergrad topology (topological spaces, continuity, connectivity, compactness, Hausdorff property, homeomorphisms, homotopy, Euler number, homotopy groups, simplicial homology, etc.)
earthicus 8y ago
The book has an appendix, labeled 'background' which lays out more precisely the formal prerequisites from point-set topology and algebra which you should review. The algebra is surely going to be a bigger obstacle than differential equations.
- dajohnson89 8y ago
  for an expository book like this, i suggest having the pdf on one screen, and google/wikipedia on another.
ambicapter 8y ago
Here is the first two sentences of the first chapter, first section. > A topological n-manifold is a space M locally homeomorphic to R^n. That is, there is a cover U = {U_{alpha}} of M by open sets along with maps \phi_{\alpha} : U_{\alpha} -> R^n that are continuous bijections onto their images with continuous inverses.

Topology seems like a really cool field. Can I jump into this book without any topology knowledge? I have not taken differential equations yet and wonder is that is a prerequisite.

myth_drannon 8y ago

From Goodreads review: " This is not so much a math textbook as an extended infomercial for the modern field of topology, with applications and intuition stressed over formal results.As the author's intention appears to be to give a broad overview of the field and its potential uses which will encourage readers to pursue deeper study, ..." So looks like the answer is yes

nikofeyn 8y ago

you will understand very little of the math in this book without a substantial background in mathematics. however, and i think the author would agree, you should jump in anyway. you'll pick up pictures, applications, vocabulary, and some ideas, all of which could help pave the way as you learn the material more traditionally. the book sets goalposts so to speak, especially since it is written in a more expository manner with lots of example applications. i don't understand anywhere near everything in the book (much less than that probably), and i still get something from it. in terms of learning some topology and geometry, i recommend topology by james munkres and an introduction to manifolds by loring tu. both are accessible to junior and senior undergraduates and beginning graduate students.

throwawaymath 8y ago
I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra. If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).
- bigmit37 8y ago
  Thank you. I am actually a self- learner and have only take calculus, linear algebra, probability, statistics. Currently learning bayesian statistics and reading “first course in abstract algebra” by John B Fraleigh. I am guessing I should do Real Analysis and then the Munkres book to get an idea of topology ?
  - magoghm 8y ago
    Right now, I'm self-studying analysis with Stephen Abbott's "Understanding Analysis". I strongly recommend it. It has clear explanations that help to develop the intuition about each concept and what I think are well chosen exercises.
    - nikofeyn 8y ago
      there is a book by jay cummings called real analysis: a long-form mathematics textbook that i highly recommend checking out. used with a more "advanced" book, it could be a nice complement. Real Analysis: A Long-Form Mathematics Textbook https://www.amazon.com/dp/1724510126/ https://www.amazon.com/dp/1724510126/
- drilldrive 8y ago
  I personally really enjoy Gamelin and Green's Topology, and (if you need it) the first chapter goes through topology of metric spaces. The rest of the book is well written; I couldn't quite find myself to like Munkres.
- nikofeyn 8y ago
  agreed. i had it in mind the person was a math major. although i think topology is approachable without analysis with some dedication, it certainly helps provide context for the abstraction. i listed another nice book in another comment, but metric spaces: iteration and application by victor bryant is a pretty fun introduction to metric spaces and analysis at a very approachable level.
ajudson 8y ago
+1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.
adenadel 8y ago
I know Munkres is the go-to standard for topology, but I think that Essential Topology by Martin D. Crossley is a better book. It's only 200 pages, packed full of great examples, and covers all the (as the title says) essential topics in undergrad topology (topological spaces, continuity, connectivity, compactness, Hausdorff property, homeomorphisms, homotopy, Euler number, homotopy groups, simplicial homology, etc.)

earthicus 8y ago

The book has an appendix, labeled 'background' which lays out more precisely the formal prerequisites from point-set topology and algebra which you should review. The algebra is surely going to be a bigger obstacle than differential equations.

dajohnson89 8y ago
for an expository book like this, i suggest having the pdf on one screen, and google/wikipedia on another.

ambicapter 8y ago

Here is the first two sentences of the first chapter, first section. > A topological n-manifold is a space M locally homeomorphic to R^n. That is, there is a cover U = {U_{alpha}} of M by open sets along with maps \phi_{\alpha} : U_{\alpha} -> R^n that are continuous bijections onto their images with continuous inverses.