On the lattice of prerealcomplete uniformities (Q1314905)

If \((S, {\mathcal U})\) is a uniform space, \((\mathbb{R}, {\mathcal R})\) is the real line with its standard uniformity, let \(C({\mathcal U})\) denote the set of the uniformly continuous functions \(f : (S,{\mathcal U}) \to (\mathbb{R},{\mathcal R})\). A subset \(A\) of \(\mathbb{R}^ S\) is said to be uniform if there is a uniformity \(\mathcal U\) on \(S\) such that \(A = C({\mathcal U})\). A uniformity \(\mathcal U\) is called pre-realcomplete if for each \(U \in {\mathcal U}\) there exists a finite number of uniformly continuous functions \(f_ 1, f_ 2, f_ 3, \dots, f_ n : S \to\mathbb{R}\) and an \(\varepsilon > 0\) such that \(| f_ i(x) - f_ i(y) | < \varepsilon\) \((i = 1,2,\dots,n)\) implies \((x,y) \in U\). For example, the \(n\)-dimensional Euclidean space with the usual uniformity \((\mathbb{R}^ n, {\mathcal R}^ n)\) is prerealcomplete. The following results are proved in this paper: 1. A subset \(A\) of \(\mathbb{R}^ S\) is uniform if and only if (1) \(A \neq \emptyset\), (2) if \(g\) is a function from \(S\) into \(\mathbb{R}\) and for each \(\varepsilon^* > 0\) there are functions \(f_ 1,f_ 2,\dots,f_ n\) from \(A\) and \(\varepsilon > 0\) such that \[ | f_ i(x) - f_ i(y)| < \varepsilon\quad (i = 1,2,\dots,n) \Rightarrow | g(x) - g(y)| < \varepsilon^* \text{ for each }x, y \in S, \] then \(g \in A\). 2. For each uniform subset \(A\), there is a largest uniformity \(\mathcal V\) such that \(C({\mathcal V}) = A\), and these uniformities are exactly the prerealcomplete ones on \(S\). 3. For a given set \(S\), the set \(S^*\) of the prerealcomplete uniformities on \(S\) is a complete lattice with respect to the fineness among filters. The intersection in \(S^*\) coincides with the intersection (\(\wedge\)) among filters, while the union \(\vee\) in \(S^*\), in general, is different from the union among uniformities. Several examples are given at the end of the paper displaying different situations, for instance, an example shows that the union of prerealcomplete uniformities may not be prerealcomplete.