A characterization of the existence of a fundamental bounded resolution for the space \(C_c(X)\) in terms of \(X\) (Q1624146)

The main object of the paper is the space \(C_c(X)\) of real-valued continuous functions on a non-empty, completely regular Hausdorff space \(X\). The space \(C_c(X)\) comes equipped with the compact-open topology. The author proves that for metrizable \(X\), the space \(C_c(X)\) has a fundamental bounded resolution if and only if \(X\) is \(\sigma\)-compact. If \(X\) is completely regular and \(C_c^*(X)\) denotes the subspace of \(C_c(X)\), consisting of bounded functions, then \(C_c^*(X)\) has a fundamental bounded resolution if and only if \(X\) is a \(cn\)-space.