Abstract The concept of a ⊤-Convergence space has recently been introduced and studied. These spaces are related to the top level space of a lattice-valued convergence space. In order to consider completions, ⊤-Cauchy spaces are defined and investigated in the present work. Each ⊤-Cauchy space possesses a completion and, moreover, each topological ⊤-Cauchy space has a topological completion. Completions of ⊤-uniform limit spaces are also introduced.