The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions

The closed subspaces of finite codimension of the space $C(X)$ of all continuous real-valued functions on a compact Hausdorff space $X$, for which the set of elements of best approximations of every function $f\in C(X)$ is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space $X$ is infinite, then $C(X)$ has no subspace of a finite Codimension $n>1$ which has a nonempty set of elements of the best approximation for an arbitrary function $f\in C(X)$ and which has an upper-semicontinuous metric projection.