I think people generally agree that analysis, topology, and abstract algebra together provide a pretty solid foundation for graduate study. (Lots of interesting stuff that’s accessible to undergraduates doesn’t easily fall under any of these headings, e.g. combinatorics, but having a foundation in these headings will equip you to learn those things quickly.)
For analysis the standard recommendation is baby Rudin, which I find dry, but it has good exercises and it’s a good filter: it’ll be hard to do well in, say, math grad school if you can’t get through Rudin.
For point-set topology the standard recommendation is Munkres, which I generally like. The problem I have with Munkres is that it doesn’t really explain why the axioms of a topological space are what they are and not something else; if you want to know the answer to this question you should read Vickers. Go through Munkres after going through Rudin.
I don’t have a ready recommendation for abstract algebra because I mostly didn’t learn it from textbooks. I’m not all that satisfied with any particular abstract algebra textbooks I’ve found. An option which might be a little too hard but which is at least fairly comprehensive is Ash, which is also freely legally available online.
For the sake of exposure to a wide variety of topics and culture I also strongly, strongly recommend that you read the Princeton Companion. This is an amazing book; the only bad thing I have to say about it is that it didn’t exist when I was a high school senior. I have other reading recommendations along these lines (less for being hardcore, more for pleasure and being exposed to interesting things) at my blog.
For analysis the standard recommendation is baby Rudin, which I find dry, but it has good exercises and it’s a good filter: it’ll be hard to do well in, say, math grad school if you can’t get through Rudin.
I feel that it’s only good as a test or for review, and otherwise a bad recommendation, made worse by its popularity (which makes its flaws harder to take seriously), and the widespread “I’m smart enough to understand it, so it works for me” satisficing attitude. Pugh’s “Real Mathematical Analysis” is a better alternative for actually learning the material.
For point-set topology the standard recommendation is Munkres, which I generally like.
I would preface any textbook on topology with the first chapter of Ishan’s “Differential geometry”. It builds the reason for studying topology and why the axioms have the shape they have in a wonderful crescendo, and at the end even dabs a bit into nets (non point-set topology). It’s very clear and builds a lot of intuition.
Also, as a side dish in a topology lunch, the peculiar “Counterexamples in topology”.
I think people generally agree that analysis, topology, and abstract algebra together provide a pretty solid foundation for graduate study. (Lots of interesting stuff that’s accessible to undergraduates doesn’t easily fall under any of these headings, e.g. combinatorics, but having a foundation in these headings will equip you to learn those things quickly.)
For analysis the standard recommendation is baby Rudin, which I find dry, but it has good exercises and it’s a good filter: it’ll be hard to do well in, say, math grad school if you can’t get through Rudin.
For point-set topology the standard recommendation is Munkres, which I generally like. The problem I have with Munkres is that it doesn’t really explain why the axioms of a topological space are what they are and not something else; if you want to know the answer to this question you should read Vickers. Go through Munkres after going through Rudin.
I don’t have a ready recommendation for abstract algebra because I mostly didn’t learn it from textbooks. I’m not all that satisfied with any particular abstract algebra textbooks I’ve found. An option which might be a little too hard but which is at least fairly comprehensive is Ash, which is also freely legally available online.
For the sake of exposure to a wide variety of topics and culture I also strongly, strongly recommend that you read the Princeton Companion. This is an amazing book; the only bad thing I have to say about it is that it didn’t exist when I was a high school senior. I have other reading recommendations along these lines (less for being hardcore, more for pleasure and being exposed to interesting things) at my blog.
I feel that it’s only good as a test or for review, and otherwise a bad recommendation, made worse by its popularity (which makes its flaws harder to take seriously), and the widespread “I’m smart enough to understand it, so it works for me” satisficing attitude. Pugh’s “Real Mathematical Analysis” is a better alternative for actually learning the material.
I would preface any textbook on topology with the first chapter of Ishan’s “Differential geometry”. It builds the reason for studying topology and why the axioms have the shape they have in a wonderful crescendo, and at the end even dabs a bit into nets (non point-set topology). It’s very clear and builds a lot of intuition.
Also, as a side dish in a topology lunch, the peculiar “Counterexamples in topology”.