Conley's fundamental theorem of dynamical systems
In 1978, Charles Conley classified all continuous dynamical systems. His theorem, dubbed the "fundamental theorem of dynamical systems" states that the orbits of any continuous map on a compact metric space fall into two classes: gradient-like and recurrent. When the recurrent part is factored out, the dynamics appear to be gradient-like. While one might wonder how a theorem that applies to every continuous map could be of any use, it plays a foundational role in many deep results.