When lattices of uniformly continuous functions on X determine X

Abstract

Several Banach-Stone-like generalizations of Shirota's result for metrizable uniform spaces are proved. Namely, if complete uniform spaces X, Y have isomorphic lattices U(X), U(Y) of their real-valued uniformly continuous functions, and both X, Y are either some products of spaces having monotone bases (metrizable or uniformly zero-dimensional), or are locally fine and of non-measurable cardinality, then X and Y are uniformly homeomorphic. .

Keywords

• Lattice of functions
• Uniform continuity
• Banach-Stone theorem