In this paper, we develop a theory of ⊤-nets and study their relation to ⊤-filters. We show that convergence in strong L-topological spaces can be described by both ⊤-nets and ⊤-filters and both concepts are equivalent in the sense that definitions and proofs that are given using ⊤-filters can also be given using ⊤-nets and vice versa. Manuscript profile

We define and study two completeness notions for saturated L-quasi-uniform limit spaces. The one, that we term Lawvere completeness, is defined using the concept of promodule and lends a lax algebraic interpretation of completeness also for saturated L-quasi-uniform limit spaces. The other, termed Cauchy completeness, is defined using saturated Cauchy pair prefilters. We show that both concepts coincide with related notions in the case of saturated L-quasi-uniform spaces and that also for saturated L-quasi-uniform limit spaces, both completeness notions are equivalent. Manuscript profile