from Hacker News

Topology Without Tears

by arkj on 11/16/22, 4:01 AM with 20 comments

• by 363849473754 on 11/16/22, 11:19 AM

  >“ There is a facebook group called "Topology Without Tears Readers" where readers of the book can communicate with each other. But this is definitely not a place to ask others to solve your homework problems. If you ask questions like "How do you solve the problem ...." your post will be removed and you will probably be blocked from this group.”

  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.

• by rsj_hn on 11/16/22, 8:41 PM

  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.

• by euler1729 on 11/17/22, 7:11 AM

  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.

• by rob74 on 11/16/22, 9:17 AM

  Tears as in tearing up a book in frustration, or tears as in crying? Possibly both...

• by squaredot on 11/16/22, 9:27 AM

  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.

• by neslisah on 11/16/22, 7:50 PM

  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!