Cauchy nets and open colorings (Q2774639)

If \(g:\mathbb N\to\mathbb N\) is a function such that \(\lim g(n)=\infty\) (\(n\to\infty\)), then \(U_g\) denotes the set of all pairs \(\langle x,y\rangle\) of functions in \(\mathbb R^\mathbb N\) such that \(|x(n)-y(n)|< g(n)+M\) for all \(n\in\mathbb N\) and some fixed \(M\in\mathbb N\). The collection of all sets \(U_g\), \(g\) as above, is a base of a pseudo-uniformity \(\mathcal U\) on \(\mathbb R^\mathbb N\). The main result of the paper (Theorem 2.1) says that assuming the Open Coloring Axiom OCA, introduced by \textit{S. Todorčević} [Partition problems in topology, Contemp. Math. 84 (1989; Zbl 0659.54001)], the space \((\mathbb R^{\mathbb N},\mathcal U)\) is complete. This result strengthens a recent result of J. Steprans that the Proper Forcing Axiom implies completeness of \((\mathbb R^\mathbb N,\mathcal U)\). Notice also that S. Watson proved that under CH this space is not complete.