Interpolation at a few points (Q1279886)

Let \(X\) be a Hausdorff topological space and \(C(X)\) the space of real-valued continuous functions defined on \(X\). A subspace \(F\) of \(C(X)\) is called \(k\)-interpolating \((k\geq 2)\) if for every system of \(k\) distinct points \(\{x_1, \dots,x_k\}\) from \(X\) and every system of \(k\) real numbers \(\{a_1, \dots, a_k\}\) there exists \(f\in F\) such that \(f(x_i)=a_i\), \(i=1,2,\dots,k\). Using the principle of the invariance of the domain the author obtains characterizations of \(k\)-interpolating finite dimensional subspaces of \(C(X)\). Also he obtains shorter proofs of some results of B. Shekhtman.