TITLE: Concepts in mathematics which are referred to as 'generalizations'
QUESTION [20 upvotes]: I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).
For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.
I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by  mathematicians.

REPLY [2 votes]: Lawvere's Fixed Point Theorem generalizes Cantor's Theorem, Turing's Halting Problem, Gödel's First Incompleteness Theorem, Tarski's Undefinability of Truth Theorem, the Recursion Theorem and more.
References:
Diagonal arguments and cartesian closed categories.
A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.
Fixed points in cartesian closed categories.