On the structure of the vector lattice of all real uniformly continuous functions (Q1902760)

Let \(L\) be a set of real functions defined on a set \(X\) and let \(\mathcal H\) be a directed family of subsets of \(L\). Let \(u_{\mathcal H}\) be the weakest uniformity on \(X\) for which all sets \(H \in \mathcal H\) become uniformly equicontinuous with respect to the uniformity \(u_{\mathcal H}\). Certain conditions on \(L\) and \(\mathcal H\) are given that guarantee that \(L\) is the set of all uniformly continuous functions on \(X\).