What follows are the topics in pure math that I have self-studied so far, in the order that I studied them. This could serve as a study guide for anyone interested in self-studying math. Each topic has an associated text book that I used to self-study, which I researched ahead of time to find a well regarded book that ideally also has solutions available somewhere online. I also include links to solutions manuals, which are very useful when self-studying to check your solutions to exercises.
Speaking of exercises, one thing I want to emphasize is that, to get a solid understanding of concepts in higher math, it is not sufficient to simply read about them. It requires working through many exercises. This is explained very nicely in this quote:
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
-James Munkres in his text “Topology”
To elaborate a little more, working exercises enables the student to become more comfortable with the concepts and to “see them from different angles” to develop a more complete mental model of them. Additionally, in many texts the exercises guide the reader in further developing often crucial parts of the theory, as implied by Munkres.
So without further ado, here are the topics that I have studied so far, in order: