When you first start out in math you won't even know the basics like e.g. "proof by contrapositive" but a basic "proof course" perhaps accompanied by "Linear Al

When you first start out in math you won't even know the basics like e.g. "proof by contrapositive" but a basic "proof course" perhaps accompanied by "Linear Algebra by Hoffman and Kunze" will get you pretty far. You can never ever ever ever learn too much linear algebra. It is pivotal in the development of e.g. "field extensions" and "differential geometry" to name two random items. That said, partway through Hoffman you should start an algebra book like Dummit and Foote so you can see groups and rings. *After you know how to prove things you could read an analysis text instead of Linear Algebra Lastly, any serious math requires some topology, but it's less about what a topology is and more about knowing how to quickly apply the basics so certain statements are easy to state/prove/think about