The Gelfand–Naimark Theorem

Given a compact topological space X, the space C(X) of continuous functions on X can be viewed as abstract algebra satisfying certain axioms, even if we forget about the topology on X. Namely, it satisfies the axioms of a C*-algebra. It turns out that these axioms capture much of the structure of C(X). The commutative Gelfand-Naimark theorem states that all commutative C*-algebras arise as algebras of continuous functions on topological spaces.