Stone–Weierstrass Theorem

A characterization asserting that a subalgebra A of C(X), the real-valued continuous functions on a compact Hausdorff space X, is dense in the uniform topology if and only if A contains the constants and separates the points of X (with an added *-closure condition in the complex-valued case).

A theorem giving conditions under which a subalgebra A of C(X) (continuous real- or complex-valued functions on a compact Hausdorff space X) is uniformly dense in C(X): typically A must contain the constants and separate points, and in the complex case be closed under complex conjugation (a *‑subalgebra).