Approximation from the topological viewpoint (Q2639282)

Let X be a topological Hausdorff space, K a compact Hausdorff space, T a set and \(A\subset C(K,X)\). A function \(h\in C(X^ T,X)\) is said to operate on A if \(\{f_ t:\) \(t\in T\}\subset A\) implies \(h((f_ t))\in A\) where \(h((f_ t))(k)=h((f_ t(k))),\) \(k\in K\). One says that a subfamily H of \(\cup_{n\geq 1}C(X^ n,X)\) operates on A if every \(h\in H\) operates on A. The family H is called generating if for every compact K and every \(A\subset C(K,X)\) separating the points of K, A is dense in C(K,X) whenever H operates on A. Taking \(X=R\) and \(H=\{x+y\), x.y, constants\(\}\) it follows that A is a subalgebra of C(K) separating the points of K and containing the constants and applying the Stone- Weierstraß theorem one concludes that H is generating. The family \(\cup_{n\geq 1}C(X^ n,X)\) is generating for absolute retracts and for zero dimensional spaces X. The family \(C(X^ 2,X)\) is generating if X is a Banach space. If p(x) is a nonlinear polynomial on \({\mathbb{R}}^ 2\) then the family \(\{\) p(x)-y, constants\(\}\) is generating in \(C({\mathbb{R}}^ 2,{\mathbb{R}})\) but \(H'=\{p(x)+y\), constants\(\}\) is not generating provided the leading coefficient of p is an integer. The problem is also related to the Hilbert's 13th problem on representation of continuous functions of several variables by continuous functions of fewer variables.