Topology Without Tears(topologywithouttears.net)
topologywithouttears.net
Topology Without Tears
https://www.topologywithouttears.net/
6 comments
This is my complaint with a lot of math circles, such as the Math Stackexchange, where the amount of policing the motives of askers makes the site a very hostile place to newer users. Treat users and students in general like adults. Assuming bad faith at the outset makes math communities feel elitist and stifling rather than welcoming and playful.
I’ve observed the phenomenon and I think it boils down to being poorly communicated on both sides.
An essential part of reasoning in mathematics comes from the experience of being repeatedly stuck.
Walking through an exciting “maze” for 20 minutes or a few hours most sufficiently motivated people can manage but it get’s harder and harder to endure when days or even weeks pass by and you feel like you have done an exhaustive inventory of every single item of your reasoning, uncountable times, feeling utterly lost. For my part I cannot even imagine being stuck on a single problem for years on end.
So, when there seems to be a reluctance to give away the answer, it is because a big part of mathematics is building up an arsenal of strategies to tackle different problems, and this is best taught through a variety of of build-up problems themselves which really challenge you at your current skill set. This of course widely differs.
It’s easier to get a good estimate if you know something about the person’s background but I can see this inquisitiveness coming off as judgemental and elitist. I’ve mostly found when you clearly can state ‘where’ exactly you feel stuck and which approaches you have tried you are heartwarmingly helped. To try to exactly pinpoint “where” your difficulty lies and patiently hitting the wall (building up a tolerance against immediate satisfaction/frustration) is how you will unlock the problems rather sooner than later.
That being said if someone asks me specifically for a solution which I happen to know, I provide it. Mostly because I don’t think you can force the aforementioned insights and I don’t want to put people into the general atmosphere - which I myself despise - of “explaining oneself”.
An essential part of reasoning in mathematics comes from the experience of being repeatedly stuck.
Walking through an exciting “maze” for 20 minutes or a few hours most sufficiently motivated people can manage but it get’s harder and harder to endure when days or even weeks pass by and you feel like you have done an exhaustive inventory of every single item of your reasoning, uncountable times, feeling utterly lost. For my part I cannot even imagine being stuck on a single problem for years on end.
So, when there seems to be a reluctance to give away the answer, it is because a big part of mathematics is building up an arsenal of strategies to tackle different problems, and this is best taught through a variety of of build-up problems themselves which really challenge you at your current skill set. This of course widely differs.
It’s easier to get a good estimate if you know something about the person’s background but I can see this inquisitiveness coming off as judgemental and elitist. I’ve mostly found when you clearly can state ‘where’ exactly you feel stuck and which approaches you have tried you are heartwarmingly helped. To try to exactly pinpoint “where” your difficulty lies and patiently hitting the wall (building up a tolerance against immediate satisfaction/frustration) is how you will unlock the problems rather sooner than later.
That being said if someone asks me specifically for a solution which I happen to know, I provide it. Mostly because I don’t think you can force the aforementioned insights and I don’t want to put people into the general atmosphere - which I myself despise - of “explaining oneself”.
You’re assuming these people care about getting good at solving math problems. Maybe they just want to understand what other people have discovered.
Treating people like adults (should) imply being kind rather than nice. While it could certainly be seen as "nice" to just give people solutions to problems, it's not very kind, as it deprives them of working through to the solution. IMO, hints should be encouraged, but only after the hint seeker describes what they've done and how they've approached the problem so far.
Why do I have to care about learning to solve math problems? Why can’t I just be interested in learning about what mathematicians have discovered?
Why are you attempting to solve math problems if you just want to learn about what's been discovered?
As this is primarily a text on point-set topology, I recommend reading this article on the Texas Topology approach, which was taught to me when I was in undergrad
https://www.sciencedirect.com/science/article/pii/S031508600...
Basically the idea is for the students to discover and prove the key theorems themselves -- with the instructor giving them hints and some feedback. No cheating and no textbooks -- the founder, R.L. Moore, would screen students to his classes to weed out those who already knew too much.
This approach was quite tough, and lasted from the about 1970 to the 1980s, at which point there was a bit of a revolution -- according to the story told me, the younger professors moved all the furniture of the older professors onto the lawn, and declared they would go back to a more standard approach to teaching topology.
At the same time, it's incredibly rewarding to discover and prove something like the Baire category theorem all by yourself. No matter how clumsy and inefficient your approach ends up being. It's a bit of a shame that no university does this anymore, but I think an enterprising student can still persuade a professor to work with them like this.
https://www.sciencedirect.com/science/article/pii/S031508600...
Basically the idea is for the students to discover and prove the key theorems themselves -- with the instructor giving them hints and some feedback. No cheating and no textbooks -- the founder, R.L. Moore, would screen students to his classes to weed out those who already knew too much.
This approach was quite tough, and lasted from the about 1970 to the 1980s, at which point there was a bit of a revolution -- according to the story told me, the younger professors moved all the furniture of the older professors onto the lawn, and declared they would go back to a more standard approach to teaching topology.
At the same time, it's incredibly rewarding to discover and prove something like the Baire category theorem all by yourself. No matter how clumsy and inefficient your approach ends up being. It's a bit of a shame that no university does this anymore, but I think an enterprising student can still persuade a professor to work with them like this.
While I admire the intention of this book's author to make topology accessible earlier in someone's mathematical education, I believe that if you are at the intended level of this book, then perhaps topology is not something you should study right now. Studying abstract topological spaces without studying metric spaces or even analysis over Euclidean spaces first is doing injustice to your intuition and provides very less motivation to appreciate the need for topological abstraction. The book "Introduction to topology and modern analysis" by Simmons is perhaps a better introduction with this viewpoint in mind. Usually subjects like combinatorics and algebra also provide the necessary mathematical maturity to handle more abstract subjects like topology. And once someone is mature enough Topology by Munkres is the best introduction.
The content of a book is always more important than its form, but we cannot get to its content if not through its form. What I don't like here is the general Microsoft Word-like feeling of the book. What I like is the use of colors and layout that makes it easier to find definitions and other concepts.
An expert shared, for free, a great and well-respected book in the mathematics community, a book that often makes top-n lists and is often the recommendation for starting off in topology. Yet.. the main complaint on HN is about the fonts, colors, and layout?
How anti-intellectual, sad and absurd.
This is probably how book reviews read in Idiocracy.
How anti-intellectual, sad and absurd.
This is probably how book reviews read in Idiocracy.
I see two sides of a same coin. Consumer of information, or maybe consumer in general, should be conscious about the quality being consumed is roughly determined by the expertise of producer and effort being put in. Hence the quality ladder is roughly book > blog > youtube video > social media posts > tiktok videos. OTOH, a producer should know their audience, and lower the entry barrier for their content to be consumed sometimes. That's also why, in my opinion, mostly used software are often not those with best qualities or features, but those with nice UIs, and good marketing money.
This is a book on topology. This is not a "consumer" product for people who are just browsing. Most readers will have a good chunk of a math or similar undergrad or are grad students. And you will likely spend at lest a few months mastering its material.
The writer knows their audience very well. People who want to learn topology. The kinds of people who have learned that you don't make a decision on a book based on colorful it is. Otherwise you will read a lot of trash.
In general. You can't be an expert in a field by choosing what you read based on how pretty it is. "Oh sorry, we didn't read the reference handbook for the plane we're flying because the fonts they used in the 70s were too ugly", "Yeah, I'm a heart surgeon but I only read papers in Times New Roman with at a minimum 4 colors".
This is pretty much the dumbest shallowest comment on a technical book I can possibly imagine. "The words are hard to evaluate. The fonts aren't great and but it's just about colorful enough. 5/10"
The writer knows their audience very well. People who want to learn topology. The kinds of people who have learned that you don't make a decision on a book based on colorful it is. Otherwise you will read a lot of trash.
In general. You can't be an expert in a field by choosing what you read based on how pretty it is. "Oh sorry, we didn't read the reference handbook for the plane we're flying because the fonts they used in the 70s were too ugly", "Yeah, I'm a heart surgeon but I only read papers in Times New Roman with at a minimum 4 colors".
This is pretty much the dumbest shallowest comment on a technical book I can possibly imagine. "The words are hard to evaluate. The fonts aren't great and but it's just about colorful enough. 5/10"
3B1B YouTube and Mathologer's PowerPoint -based YouTube are better than most books.
The important thing is that colors in a book should have more to do with content than markup.
OTOH it's still relatively common for interesting mathematical work to be put online as scans of handwritten slides...
it looks as though it's been done with latex, but a real nasty font and other formatting.
I chose Metric Spaces for my optional lesson as if i didn't already failed Introduction to Topology last year. So there will be some tears when i couldn't graduate. But i really like the book. I think it will be really helpfull. Thanks!
My complaint with this is the audience reading this book aren’t students in classes and likely a lot of them don’t have access to mathematicians or professors. Working with others is a great way to learn how to properly do proofs when first starting out. I think the FB groups / discord groups or whatever communities of learners should discuss their proofs and suss out difficulties or logical errors with each other.
This approach instead seems to assume the learner will know whenever they develop a correct proof but often one can just fool themselves into thinking their proofs are correct or even get utterly stuck. Or it assumes seeing solutions will “rob” the readers of learning.
Also, who really cares? Most of the people working through this book aren’t getting a grade from it. If they want to rob themselves of learning something just by seeing solutions without thinking first, then that’s on them.
Otherwise, beyond that complaint this seems like a good resource and it’s impressive it’s all free. I’d also recommend another topology book that’s free https://topology.mitpress.mit.edu/, albeit it’s more advanced.