We define lattice-valued Cauchy spaces. The category of these spaces is topological over SET and cartesian closed. Special examples are lattice-valued uniform convergence spaces and probabilistic Cauchy spaces. We further define completeness and give a completion for these spaces which has the property that Cauchy continuous mappings between spaces can be extended to Cauchy-continuous mappings between their completions.

Keywords: L-fuzzy convergence; L-topology; L-filter; L-limit space; L-Cauchy space; L-Cauchy filter; L-uniform convergence space; probabilistic Cauchy space; completeness; completion.

Quaestiones Mathematicae 33(2010), 53–74.