What are some good introductory books that leave many proofs as exercises? I have been self studying analysis by reading Tao's two fantastic books which eventually leaves most of the (easier) proofs as exercises, and I am wondering if there are similar books for other topics (as of right now I am thinking about introductions to groups and/or topology). Thanks!
I personally quite like "Fundamentals of General Topology" by Arhangelski'j and Ponomarev. All theorems are exercises; answers are included in the back. Do then in order: there is a build up. Notation is quirky (Russian style, but it's good to get used to different styles of notation anyway).
Robin Hartshorne's Algebraic Geometry textbook leaves many proofs and parts of proofs as exercises. To cite one example off the top of my head, the fundamental result in chapter III that every flasque sheaf of abelian groups is acyclic is given a proof citing several exercises from chapter II (where those parts are given as self-contained exercises on the general notion of sheaf).
A Hilbert Space Problem Book. by P.Halmos. An introduction to the theory of Hilbert spaces, in which most of the proofs are left to the reader. The author has included proofs in the back of the book.
General Topology. by R. Engelking. Proofs in the main body are detailed and thorough and $not$ left to the reader. But an enormous amount of extra material is in the Exercises and Problems, with complete references to the original publications. And there is an excellent index of topics (including index-referrals to Problems). E.g. the topic of ordered spaces is covered only in Exercises & Problems in several chapters. Combined (& with proofs) they could make a sizeable chapter.
I'm gonna say Munkres, and Kelley, for Topology.
Many theorems are proved, but, as is typical, alot is expected of the reader. There's plenty left to work out on your own. Even though perhaps not most theorems are left as exercises, I think these are good books for self study. Especially if you have a lot of time on your hands.
I remember studying them both, years ago, when starting out. And I think they helped.
Kelley is rather fancy (even the typeset). While you may find it difficult, the exposure should be good.
Both are good books.