Buchwalter-Schmets theorems and linear topologies (Q1283395)

A subset \(S\subseteq X\) of a Hausdorff completely regular topological space \(X\) is called bounding if \(f(B)\) is bounded for each \(f\in C(X)\). The authors define the space \(C_\sigma(X)\) as the space \(C(X)\) topologized by taking as a base of neighborhoods of the origin the family \(\{f\in C(X):K\cap \text{supp }f=\emptyset\}\), for each finite subset \(K\) of \(X\). They prove that \(C_\sigma(X)\) is linearly barrelled if and only if each bounding subset of \(X\) is finite. They also show that the following statements are equivalent: (i) \(X\) is replete. (ii) \(C_\sigma(X)\) is a linearly bornological space. (iii) \(C_\sigma(X)\) is the linear inductive limit of the family formed by its countable-dimensional subspaces.