Let F be a topological field,
which means in particular Hausdorff and commutative. The “Stone-Weierstrass”
theorem for F would state that if X is a compact space and 𝒜 an algebra of
continuous functions from X to F which contains the constant functions and
separates points, then 𝒜 is uniformly dense in the algebra of all continuous functions
from X to F. This is true for the reals but false for the complexes, and is
commonly regarded as a special property of the reals. In fact, however, it is the
complex field which is exceptional: the Stone-Weierstrass theorem and its
characteristic corollaries hold for all valuable fields other than the complex
numbers.
