Generation of the uniformly continuous functions. (Q1426514)

Let \(X\) be a set and \(\mathcal F\) be a family of real-valued functions defined on \(X\). A function \(f\colon X\to {\mathbb R}\) is uniformly continuous with respect to \(\mathcal F\) if for each \(\varepsilon>0\) there exists \(\delta>0\) and a finite subfamily \(\{ g_1,\ldots, g_n\}\subset {\mathcal F}\) such that the condition \(| g_i(x)-g_i(y)| <\delta\), \(i=1,\ldots, n\), implies \(| f(x)-f(y)| <\varepsilon\). Let \(U(\mu_{\mathcal F}X)\) be the collection of all real valued \(\mathcal F\)-uniformly continuous functions over \(X\). Two problems are considered in the paper: (1) when \(\mathcal F\) is uniformly dense in \(U(\mu_{\mathcal F}X)\); (2) how to generate \(U(\mu_{\mathcal F}X)\) from \(\mathcal F\). In particular, the authors find an internal condition on \(\mathcal F\) in order to be uniformly dense in \(U(\mu_{\mathcal F}X)\).