Generalized Dini theorems for nets of functions on arbitrary sets (Q259807)

The authors prove the following characterization of the uniform convergence of pointwise monotonic nets \(F_\Delta=(f_\delta)_{\delta\in\Delta}\) of bounded real-valued functions \(f_\delta:S\to\mathbb{R}\) defined on an arbitrary set \(S\) (without a topological structure): A net \(F_\Delta\) converges uniformly to \(0\) iff the closure (in the product space \(\mathbb{R}^\Delta\)) of the set \(F_\Delta(S)=\{ (f_\delta(s))_{\delta\in\Delta}: s\in S\}\) is contained in the class \(c_0(\Delta)\) of all sequences \((r_\delta)_{\delta\in\Delta}\in \mathbb{R}^\Delta\) with \(\lim_{\delta\in\Delta}r_\delta=0\). A similar result is obtained for nets of functions with relatively compact range in a Hausdorff topological ordered vector space \(X\).