Characterization of barrelled, d-barrelled, and m\(\sigma\)-barrelled spaces of continuous functions (Q1071215)

Let T be a completely regular Hausdorff space and C(T) the space of all continuous real-valued functions on T. Let \({\mathcal P}\) be a family of subsets of T on each of which every \(f\in C(T)\) is bounded. Let \(C_{{\mathcal P}}(T)\) be C(T) provided with the topology of uniform convergence on members of \({\mathcal P}\). The barrelled, d-barrelled and \(\sigma\)-barrelled spaces of type \(C_{{\mathcal P}}(T)\) were characterized in the book.'' Espaces de Fonctions continues'' of \textit{J. Schmets} (1976; Zbl 0334.46022). The present author makes a similar study of \(C_{{\mathcal P}}(T)\) viewed as a topological vector lattice in the sense of \textit{J. Jameson} [Math. Z. 103, 139-150 (1968; Zbl 0173.153)] where this kind of structure is called an M-space. He defines the m-barrelled, md-barrelled and \(m\sigma\)-barrelled M-spaces and characterizes the spaces \(C_{{\mathcal P}}(T)\) of this type. It turns out that \(C_{{\mathcal P}}(T)\) is m-barrelled (resp. md-barrelled) if and only if it is barrelled (resp. d-barrelled). The author notes that it is unknown whether the analogous result holds for the \(\sigma\)- barrelledness property.