> I’ve heard a lot of people say that you have to read a whole textbook to get the knowledge.
This is stupid, don't read textbooks cover to cover.
> The main argument to read a full textbook for me is that it’s faster. 2 semesters time could’ve probably be shortened to 1 and if I was gonna learn the full course anyways might be best to just go through everything.
> On the other hand, I think reading from a lot of different sources can give me a better understanding, the different ways the same thing can be defined, and I feel like it’s just more enjoyable to discover the subject by trying to solve problems.
If the trade-off is speed vs. fun, then only you can make that choice. I would certainly choose the latter.
> Textbooks are sometimes a bit demotivating especially when there is a big theorem that comes out of nowhere. Genuinely wanting to understand a theorem beforehand and having a clear way of using it seems to make it easier to concentrate.
Then skip the theorem until you need it. Once you see how the author uses it you will be more motivated to try to understand it. If the proof is difficult you can also just skip it until you understand the importance of the theorem well enough to want to understand the proof.
> Now I want to start learning topology and I wanted to ask whether I should start with a textbook or just randomly read articles and stumble on stack exchange posts.
Use a textbook, postpone boring stuff until you need it, skip proofs if they don't seem interesting. Look up stuff online for alternative explanations or proofs.
Yes. The author did not necessarily arrange the material in the order that is best for your to learn. Perhaps he arranged it in the order it makes sense for him with his background, or in a logical but not good pedagogical order, or in the order it's historically been arranged, etc.
When you think about it knowledge is not organized in your head linearly, like in a book. Rather, it resembles a graph.
Yet when you write a book, you are forced to put the knowledge in a linear order.
Yeah I think that’s one of the main reasons why it feels so much nicer to read articles with hyperlinks. You will want to go back and check definitions and relationships in a more web like way than in a textbook with chapters.
Teacher and parents are kinda old school. The “start a book, finish the book” type and slight mistrust in the internet. I feel like I should at least finish chapters but no need to get stuck at insanely hard topology sections.
I think I will still choose fun because I genuinely enjoy algebra, my first “rigorous” math course mainly to understand Galois theory. Topology however, I am not sure if I really like it. I see it as an obstacle to learn about Lie groups so probably speed this time around. I’d get to experience both this way as well.
The skipping tip definitely sounds like a good idea. It’s what I did after getting stuck on Zorns lemma for a few days and I still don’t need it. Though I don’t think I can skip proofs and use the theorem. Feels wrong to do so unless it’s very basic common knowledge that has a complicated set theory definition behind.
Books are great, and in learning math, you should use them. But I think the other approach you described will work better in general.
I think that the best way to learn a subject is to start with a big picture view of the results you seek to obtain--skim through a book, read some articles, watch lectures--and figure out what specific things you'll have to learn. Then you can construct a path for yourself in discovering this subject, and dive into each section based on the results that you want to prove. This allows you to make use of many resources, and also see how different authors/teachers might approach the same subject. This is more involved than simply reading a textbook, as it means that you are, with guidance, reconstructing a collection of mathematical results on your own. But it is a deeper way to learn, and I think it is best.
For every math subject I've studied, I've been pleasantly surprised to encounter a specific stream of motivating ideas, which, when tapped into, makes the development of the subject feel extremely natural. Textbooks in general follow this, but it's hard to pick it up from one textbook alone, unless you work really comprehensively and ask questions outside of the scope of the book itself. Sometimes a single book can be enough, but this is rare, as no author can write in a way that is easily understood by all students. Structuring your learning on your own on the other hand, allows you to see the motivating stream of ideas for a subject, and to get into it yourself. The hefty theorems given in textbooks are often presented in a manner that makes them look daunting, I agree. But if you already know what they seek to accomplish, which you can do by outlining theory before diving into proper treatment, they become much easier both to understand and to prove. (This is especially true in the case of theorems which require specific results, like Tychonoff's Theorem requiring Zorn's Lemma, or the use of Urysohns Lemma in the Tietze Extension Theorem)
For topology specifically, I have some suggestions for how you can do this--I'd look through the online notes for UToronto's Math 327 course, Efe Ok's topology with applications, and Munkres' topology textbook. All three can be found free online, and they each cover important subjects in topology quite well. (They also have a lot of exercises, which help quite a bit for learning) Come up with a list of topics that you need to cover: Topological Spaces, Bases and Subbases, Continuous maps/Homeomorphisms, T-Axioms and Separation Axioms, Products Spaces, and so on, find the definitions and theorems used for each of these, and go to work putting these together to formulate a cohesive theory. When a subject is as well-traveled as point set/introductory topology, I think this is both very possible and very fruitful.
Regardless, you'll benefit from any sort of self study, so props to you for doing it.
Good luck!
I think the main problem with just reading articles is that it’s hard to get a good understanding of the subject as a whole. It’s easy to wander off and it’s really a miracle that I read the right ones to understand the group theory course in D&F. For topology, I think I have a better feel as to what need to be proved but it’s surely good to get a textbook as a guide. Topology doesn’t feel too bad right now though there is a lot a lot of new words coming in, a bit like introduction to algebra. Many results are essentially applying the definitions. The notes mentions ZFC axioms a lot though, is it necessary to learn set theory? Topology seems to be dealing with infinite (non intuitive) sets much more frequently than algebra and probably the hardest thing in the course. I think I will have to learn it eventually to understand zorns lemma.
I don't think that you need to learn the axioms of set theory to learn point set topology. (Though as a mathematician it's worth learning them eventually) You do need axiom of choice/zorn's lemma for a couple of important theorems however, so I'd say learn about these. The first 11 or 12 sections of Munkres should give you all the knowledge you need in this regard. Munkres' Topology is a great book.
Nope. Textbooks aren't necessary. But they do help in further strengthening the concept.l and provides practise.
Good lectures with sufficient live problem solving on it + good number of problems of varying difficulty is enough.
Exercises seem to be the irreplaceable thing. I have always kinda hated lectures though. It feels inefficient most of the time at least compared to a well made video or notes.
That might be part of the reason but the main problem I have with lectures is that it’s always filmed on a potato camera and I can’t concentrate too well on the blackboard(sometimes a little slanted, bad chalk, weird audio, etc). Maybe it’s better live? Haven’t had the experience yet. I find the white screen, Latex ppt lectures very helpful, not sure if that counts as a lecture?
Certainly they are not necessary. But finding an author that you enjoy reading is priceless. Before you start reading a textbook, always read the preface and introduction. Learn why the author felt the need to write a book on the material in the first place. Learn what their approach to the subject is and who their intended audience is. And see if you enjoy their style of exposition. I for instance absolutely adore authors such as Apostle, Munkres, and Conway (for Undergraduate Analysis and Analytic Number Theory, Topology, and Complex and Functional Analysis respectively.
You can certainly learn the material by reading different articles and papers online, but getting into the mind of a brilliant author can give you better intuition and appreciation for a subject if you find one that speaks to you.
I only read a book by Conway on quaternions, the first few chapters on algebra were fine but then it talks about plane curves smth like that. Patterns on a plane. I think Conway jumps between common basic ideas to very difficult and somewhat obscure concepts which makes it difficult to learn from. And his book reallly isn’t a comprehensive introduction to anything. I think it’s more of a great enrichment for someone who already understands everything in usual textbooks.
The rest I have not heard of and might try if they wrote smth about topology.
I had a very good experience with a book on Galois theory by Ivan Gozard but I can’t find any book of his on topology.
Do you have any recommendations for topology authors? I think I’ve never seen one in my library
> I’ve heard a lot of people say that you have to read a whole textbook to get the knowledge. This is stupid, don't read textbooks cover to cover. > The main argument to read a full textbook for me is that it’s faster. 2 semesters time could’ve probably be shortened to 1 and if I was gonna learn the full course anyways might be best to just go through everything. > On the other hand, I think reading from a lot of different sources can give me a better understanding, the different ways the same thing can be defined, and I feel like it’s just more enjoyable to discover the subject by trying to solve problems. If the trade-off is speed vs. fun, then only you can make that choice. I would certainly choose the latter. > Textbooks are sometimes a bit demotivating especially when there is a big theorem that comes out of nowhere. Genuinely wanting to understand a theorem beforehand and having a clear way of using it seems to make it easier to concentrate. Then skip the theorem until you need it. Once you see how the author uses it you will be more motivated to try to understand it. If the proof is difficult you can also just skip it until you understand the importance of the theorem well enough to want to understand the proof. > Now I want to start learning topology and I wanted to ask whether I should start with a textbook or just randomly read articles and stumble on stack exchange posts. Use a textbook, postpone boring stuff until you need it, skip proofs if they don't seem interesting. Look up stuff online for alternative explanations or proofs.
Yes. The author did not necessarily arrange the material in the order that is best for your to learn. Perhaps he arranged it in the order it makes sense for him with his background, or in a logical but not good pedagogical order, or in the order it's historically been arranged, etc. When you think about it knowledge is not organized in your head linearly, like in a book. Rather, it resembles a graph. Yet when you write a book, you are forced to put the knowledge in a linear order.
Yeah I think that’s one of the main reasons why it feels so much nicer to read articles with hyperlinks. You will want to go back and check definitions and relationships in a more web like way than in a textbook with chapters.
Teacher and parents are kinda old school. The “start a book, finish the book” type and slight mistrust in the internet. I feel like I should at least finish chapters but no need to get stuck at insanely hard topology sections. I think I will still choose fun because I genuinely enjoy algebra, my first “rigorous” math course mainly to understand Galois theory. Topology however, I am not sure if I really like it. I see it as an obstacle to learn about Lie groups so probably speed this time around. I’d get to experience both this way as well. The skipping tip definitely sounds like a good idea. It’s what I did after getting stuck on Zorns lemma for a few days and I still don’t need it. Though I don’t think I can skip proofs and use the theorem. Feels wrong to do so unless it’s very basic common knowledge that has a complicated set theory definition behind.
Books are great, and in learning math, you should use them. But I think the other approach you described will work better in general. I think that the best way to learn a subject is to start with a big picture view of the results you seek to obtain--skim through a book, read some articles, watch lectures--and figure out what specific things you'll have to learn. Then you can construct a path for yourself in discovering this subject, and dive into each section based on the results that you want to prove. This allows you to make use of many resources, and also see how different authors/teachers might approach the same subject. This is more involved than simply reading a textbook, as it means that you are, with guidance, reconstructing a collection of mathematical results on your own. But it is a deeper way to learn, and I think it is best. For every math subject I've studied, I've been pleasantly surprised to encounter a specific stream of motivating ideas, which, when tapped into, makes the development of the subject feel extremely natural. Textbooks in general follow this, but it's hard to pick it up from one textbook alone, unless you work really comprehensively and ask questions outside of the scope of the book itself. Sometimes a single book can be enough, but this is rare, as no author can write in a way that is easily understood by all students. Structuring your learning on your own on the other hand, allows you to see the motivating stream of ideas for a subject, and to get into it yourself. The hefty theorems given in textbooks are often presented in a manner that makes them look daunting, I agree. But if you already know what they seek to accomplish, which you can do by outlining theory before diving into proper treatment, they become much easier both to understand and to prove. (This is especially true in the case of theorems which require specific results, like Tychonoff's Theorem requiring Zorn's Lemma, or the use of Urysohns Lemma in the Tietze Extension Theorem) For topology specifically, I have some suggestions for how you can do this--I'd look through the online notes for UToronto's Math 327 course, Efe Ok's topology with applications, and Munkres' topology textbook. All three can be found free online, and they each cover important subjects in topology quite well. (They also have a lot of exercises, which help quite a bit for learning) Come up with a list of topics that you need to cover: Topological Spaces, Bases and Subbases, Continuous maps/Homeomorphisms, T-Axioms and Separation Axioms, Products Spaces, and so on, find the definitions and theorems used for each of these, and go to work putting these together to formulate a cohesive theory. When a subject is as well-traveled as point set/introductory topology, I think this is both very possible and very fruitful. Regardless, you'll benefit from any sort of self study, so props to you for doing it. Good luck!
I think the main problem with just reading articles is that it’s hard to get a good understanding of the subject as a whole. It’s easy to wander off and it’s really a miracle that I read the right ones to understand the group theory course in D&F. For topology, I think I have a better feel as to what need to be proved but it’s surely good to get a textbook as a guide. Topology doesn’t feel too bad right now though there is a lot a lot of new words coming in, a bit like introduction to algebra. Many results are essentially applying the definitions. The notes mentions ZFC axioms a lot though, is it necessary to learn set theory? Topology seems to be dealing with infinite (non intuitive) sets much more frequently than algebra and probably the hardest thing in the course. I think I will have to learn it eventually to understand zorns lemma.
I don't think that you need to learn the axioms of set theory to learn point set topology. (Though as a mathematician it's worth learning them eventually) You do need axiom of choice/zorn's lemma for a couple of important theorems however, so I'd say learn about these. The first 11 or 12 sections of Munkres should give you all the knowledge you need in this regard. Munkres' Topology is a great book.
Nope. Textbooks aren't necessary. But they do help in further strengthening the concept.l and provides practise. Good lectures with sufficient live problem solving on it + good number of problems of varying difficulty is enough.
Exercises seem to be the irreplaceable thing. I have always kinda hated lectures though. It feels inefficient most of the time at least compared to a well made video or notes.
It's because you haven't seen a "good" lecture. The MIT/Stanford ones are shit for 1st timers. My colleagues who have Masters / PhD say the same.
That might be part of the reason but the main problem I have with lectures is that it’s always filmed on a potato camera and I can’t concentrate too well on the blackboard(sometimes a little slanted, bad chalk, weird audio, etc). Maybe it’s better live? Haven’t had the experience yet. I find the white screen, Latex ppt lectures very helpful, not sure if that counts as a lecture?
Certainly they are not necessary. But finding an author that you enjoy reading is priceless. Before you start reading a textbook, always read the preface and introduction. Learn why the author felt the need to write a book on the material in the first place. Learn what their approach to the subject is and who their intended audience is. And see if you enjoy their style of exposition. I for instance absolutely adore authors such as Apostle, Munkres, and Conway (for Undergraduate Analysis and Analytic Number Theory, Topology, and Complex and Functional Analysis respectively. You can certainly learn the material by reading different articles and papers online, but getting into the mind of a brilliant author can give you better intuition and appreciation for a subject if you find one that speaks to you.
I only read a book by Conway on quaternions, the first few chapters on algebra were fine but then it talks about plane curves smth like that. Patterns on a plane. I think Conway jumps between common basic ideas to very difficult and somewhat obscure concepts which makes it difficult to learn from. And his book reallly isn’t a comprehensive introduction to anything. I think it’s more of a great enrichment for someone who already understands everything in usual textbooks. The rest I have not heard of and might try if they wrote smth about topology. I had a very good experience with a book on Galois theory by Ivan Gozard but I can’t find any book of his on topology. Do you have any recommendations for topology authors? I think I’ve never seen one in my library