This is maybe my favorite math book altogether.
https://www.amazon.com/Introduction-Topology-Applied-Colin-Adams/dp/0131848690
For that problem, you probably want to use the definition of limit point directly. What definition does your book use?
Munkres' "Topology" is a decent book that is considered to be a standard in many math curricula. The book has many details, and it covers quite a lot of point-set topology and a little bit of algebraic topology in the end. It is highly recommended for those planning to go to a grad school in math. My only concern is that it might contain too much material for a beginner who is self-studying the subject, and it is also quite expensive ($95 on Amazon, although you can probably find a way to get a cheaper international edition somewhere).
Thereis a book published by Dover that looks promising: It is Mendelson's "Introduction to Topology." It covers all of the essential topics on point-set topology in about 200 pages, and it's very cheap ($9.59 on Amazon as of now). It also discusses metric spaces before general topological spaces (metric space is a nice example of topological space, and this is why you read comments saying you should study real analysis first). I have personally not read this book, but the reviews of this book looks promising.
If you are interested in studying algebraic topology later on, I can also recommend another book that I have read(aside from Munkres), which is Kosniowski's "A First Course in Algebraic Topology." The book covers just enough point-set topology needed at the beginning to study algebraic topology.
Introduction to Topology: Pure and Applied is a really neat book. The author explains concepts clearly and includes easy to follow proofs and theorems. Also, as the title suggests, there are some sections on the applications of Topology, including some cool stuff like Cosmology, Knots, Dynamical Systems and Chaos. You normally don't see that in the standard Topology textbook.
More elegant but not necessarily better
Mendelson's "Introduction to Topology" is a good one. I picked up a copy a decade ago and it really helped with my understanding of the subject.
https://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523/
Okay thats fair.; Exact category is often debated.
The less debated fields don't have a replication crisis per se: They have a falsifiability crisis.
https://blogs.scientificamerican.com/cross-check/how-physics-lost-its-fizz/
Just want to comment, that physics dont have to be beautiful minimalistic which is also discussed in the science community and also found its way into popular books like Lost in Math: How Beauty Leads Physics Astray.
Imo it is quite unscientific to rule out solution which are not "mathematically beautiful" or "simple and elegant". Or not searching for solutions which do not meet this criterias.
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Thanks a lot! I'm currently trying to get through Bert Mendelson's Intro to Topology book. Would you say this is a good enough start in your opinion?
The fundamental thing about quaternion, octonions etc. is, they involve extradimensions of sort. The quaternion-based Maxwell theory anticipated quantum and scalar wave effects, which the current Maxwell theory cannot. So there are good reasons for utilizing the octonion math within contemporary physics. The opposite problem is, such a vector math is too dependent on right-angled Cartesian system. And once the number of dimensions increases, the things stop to become right angled anymore - the extradimensions violate Euclidean geometry too. The octonion-based theory will be still usable, but overly complicated, over-parametrized and as such suboptimal.
This is btw also the fact, which ruined string theory too and a general problem of every hyperdimensional theory: they apply only in flat 3D space-time following the Cartesian system and their validity scope is thus constrained to very subtle phenomena, which don't violate the dimensionality of space-time in which they reside too much (i.e. dark matter fluctuations). Even the best brains on the planet don't know what to do with it (a renormalization problem of switching extrinsic and intrinsic perspectives). As famous blogger L. Motl noted the octonion based math is the same case of fancy but void formal approach, like this one criticized recently by Hossenefelder - who indeed had the string theory of L. Motl on mind instead... ;-)
The fundamental thing about quaternion, octonions etc. is, they involve extradimensions of sort. The quaternion-based Maxwell theory anticipated quantum and scalar wave effects, which the current Maxwell theory cannot. So there are good reasons for utilizing the octonion math within contemporary physics. The opposite problem is, such a vector math is too dependent on right-angled Cartesian system. And once the number of dimensions increases, the things stop to become right angled anymore - the extradimensions violate Euclidean geometry too. The octonion-based theory will be still usable, but overly complicated, over-parametrized and as such suboptimal.
This is btw also the fact, which ruined string theory too and a general problem of every hyperdimensional theory: they apply only in flat 3D space-time following the Cartesian system and their validity scope is thus constrained to very subtle phenomena, which don't violate the dimensionality of space-time in which they reside too much (i.e. dark matter fluctuations). Even the best brains on the planet don't know what to do with it (a renormalization problem of switching extrinsic and intrinsic perspectives). As famous blogger L. Motl noted the octonion based math is the same case of fancy but void formal approach, like this one criticized recently by Hossenefelder - who indeed had the string theory of L. Motl on mind instead... ;-)
The topos of music is a famous example of this. Topos theory is as abstract as you can get in mathematics and there seems to be a lot to be said using this language.
I can't really comment on this since I haven't read it, but maybe someone else can chime in.
Any graduate student should know basic topology already, just to be clear. Also this could be another very cheap and accessible textbook.
Ghrist's book makes a great overview of not only a bunch of topics in algebraic and differential topology, but also has a bunch of applications. I don't think it would be very good as a first introduction to topology, but it's certainly good for browsing and getting a general idea of things.
For a textbook, you might be best getting Munkres and working through that. Another book I really like that is shorter than Munkres is Armstrong's topology book.
I like Munkres but maybe a better option for you is Adams and Franzosa's Introduction to Topology: Pure and Applied. It's a little bit slower and less dense than Munkres, but most importantly it uses a lot of applied examples to illustrate their points, (though a few of them are a bit contrived). I learned topology from that book and I've taught from it too and I think its better for less experienced students.
Beyond Boas and Arfken I would also suggest looking at The Geometry of Physics by Frankel. It covers Geometric and Topological methods across a wide range of areas and is worth checking out.
I'm currently working through "Introduction to Topology" by Mendelson. I really like it. It a dover book, and you can get it pretty damn cheap.
I wish I was only taking those two. I've also got Abstract Algebra II (Ring Theory), and teaching the one class on top of that. This is my "tough" semester. The next two I'll probably only be taking 2 classes each semester, plus teaching.
What book are you using for Topo? We're using Munkres.
And what are you using for Real Analysis? I know Baby Rudin is sort of the standard, but we're using Ross.
pfff first semester theory student? come back when you've passed your quals. btw this is the kind of thing mathematical physics means now; no one has called numerical pdes mathematical physics for at least 50 years - it's called computational physics now.
If you're not familiar with basic proof strategies and set theory, pick up a book like How To Prove It (or if you like free stuff, this looks pretty good) and work through it until you can prove things about sets and functions without too much difficulty. I'd say that's the only prerequisite for topology, though, other than basic knowledge of how real numbers work.
As for a textbook, this book is incredibly cheap and covers just about exactly what you'd expect to find in a one-semester undergraduate topology course.
Falconer's book is very good. You should be able to find a copy in a library. It has exercises and is mathematically oriented. It's also decently self-contained.