Let $X$ be a compact space and let $C^0(X,\mathbb{R})$ be the algebra of continuous real functions defined over $X$ . Let $\mathcal{A}$ be a subalgebra of $C^0(X,\mathbb{R})$ for which the following conditions hold:

1. $\forall x, y \in X, x \ne y, \exists f \in \mathcal{A} : f(x) \neq f(y)$
2. $1 \in \mathcal{A}$

This theorem is a generalization of the classical Weierstrass approximation theorem to general spaces.