Barrelled and Bornological Function Spaces

For a completely regular space X and a normed space E letC X, E.be the set l of all E-valued continuous maps on X endowed with the topology of uniform convergence on the members of l, where l is a family of closed and t-bounded subsets of X such that l is a cover of X and it is invariant with respect to finite unions. Necessary and sufficient conditions in the terms of the topology of X are given in order C X, E.to be barrelled, quasi-barrelled, bornological, or ultra- l bornological space