Hiblert's 13th Problem

The 13th Problem from Hilbert's famous list [16] asks whether every continuous function of three variables can be written as a superposition (in other words, composition) of continuous functions of two variables. Let Χ be a space. A family Φ ⊆ C(Χ) is said to be basic for Χ if each f in C(Χ) can be written as linear superposition for some functions from in Φ and some one-variable real functions. A family Ψ is elementary in dimension m if the family of maps generated by Ψ by addition is basic for Χ*…*Χ . Kolmogorov and Arnold [18, 4] showed that the closed unit interval has a finite elementary family in every dimension, thereby solving Hilbert's 13th Problem.Define a new cardinal invariant basic(Χ ) = min {|Φ|: Φ is a basic family for Χ}. It is established that a space has a finite basic family if and only if it is finite dimensional, locally compact and separable metrizable (or equivalently, homeomorphic to a closed subspace of Euclidean space).Such a space has dim(Χ) ≤ n if and only if basic(Χ) ≤ 2n+1. Separable metrizable spaces either have finite basic(Χ) or basic(Χ) equal to the continuum. The value of basic(K) for a compact space K is closely connected with the cofinality of the countable subsets of a basis B for K of minimal size ordered by set inclusion.It is proved that a space has a finite elementary family in every dimension m if and only if it is homeomorphic to a closed subspace of Euclidean space. It is further shown that there is a finite elementary family for the reals in each dimension m consisting of effectively computable functions, and effective procedures for representing any continuous function of m real variables as a superposition of these elementary functions and other univariate maps