If X is σ-compact Polish, then has a σ-closure-preserving base

We prove that if X is a σ-compact Polish space, then the space C k ( X ) of all continuous real-valued functions on X with the compact-open topology is a μ-space, and hence is M 1 , i.e., it has a σ-closure-preserving base. We also construct an explicit σ-closure-preserving base for C k ( X ) .