When lattices of uniformly continuous functions on \(X\) determine \(X\) (Q500935)

For a uniform space \(X\) the lattice of all uniformly continuous real-valued functions on \(X\) is denoted by \(U(X)\). The authors investigate the following question: under which conditions does a lattice isomorphism of \(U(X)\) and \(U(Y)\) imply that \(X\) and \(Y\) are uniformly homeomorphic? They prove that this is true in the following cases: (1) \(X\) and \(Y\) are products of complete metrizable spaces, (2) \(X\) and \(Y\) are sums of complete uniformly zero-dimensional spaces having monotone uniform bases, (3) \(X\) and \(Y\) are locally fine, proximally fine and uniformly realcomplete, (4) \(X\) and \(Y\) are locally fine, proximally fine and complete having no uniformly discrete subspace of Ulam measurable cardinality. The paper contains also three open problems.