On the existence of compact metric subspaces with applications to the complementation of $c_{0}$

Abstract:

A topological space X has property $\sigma - {\text {CM}}$ if for every countable family F of continuous scalar valued functions on X there is a compact metrizable subspace M of X such that $f(X) = f(M)$ for every f in F. Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property $\sigma - {\text {CM}}$. Every continuous image of an arbitrary product of spaces having property $\sigma - {\text {CM}}$ also has property $\sigma - {\text {CM}}$. If X has property $\sigma - {\text {CM}}$, then every copy of ${c_0}$ in $C(X)$ is complemented in $C(X)$. If a locally convex space E belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of ${c_0}$ in E is complemented in E.