Lattices of uniformly continuous functions determine their sublattices of bounded functions (Q2254720)

For a topological space \(X\), \(C(X)\) means the lattice of all continuous real-valued functions on \(X\) and \(C^{\ast} (X)\) the sublattice of bounded functions. Similarly, \(U(X)\) is the lattice of all uniformly continuous real-valued functions on a uniform space \(X\) and \(U^{\ast} (X)\) the sublattice of bounded functions. The author proves the following interesting theorems: { Theorem A.} If \(U(X)\) and \(U(Y)\) are lattice isomorphic, then \(U^{\ast}(X)\) and \(U^{\ast}(Y)\) are lattice isomorphic, too. { Theorem B.} Two complete metric spaces \(X\), \(Y\) are uniformly homeomorphic iff their lattices \(U(X)\), \(U(Y)\) are isomorphic.