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We define level structures for lattice-valued uniform spaces and latticevalued uniform convergence spaces. We show that these structures characterize the spaces from which they are obtained and use them to show that the categories of probabilistic uniform convergence spaces, of probabilistic uniform spaces and of approach uniform spaces are isomorphic to subcategories of the category of latticevalued uniform convergence spaces.

Keywords: L-fuzzy convergence; L-topology; L-filter; L-uniformity; L-uniform convergence space; probabilistic uniform limit spaces; probabilistic uniform space; approach uniform space

Quaestiones Mathematicae 31(2008), 255–277