Approximation from the topological viewpoint

Abstract

Let X be a metric space. A family H of continuous functions of several variables of X with values in X is said to be generating if, whenever A ⊂ C( K, X) separates points and H operates on A, then A is dense in C( K, X). (For example, the family H = { x + y, xy, constants} in C( R 2, R) is generating (for R) by the Stone-Weierstrass theorem.) We identify metric spaces which admit generating families (not all do), and among those, we search for spaces X that admit generating families in C( X 2, X)—such as R. (This may be considered a topological version of Hilbert's 13th problem.) Once we know this, we try to identify some (small) generating families in C( X 2, X). (This is done in particular when X = R.) As a fringe benefit we obtain a “topological” proof of the Stone-Weierstrass theorem.