Using the notion of remote neighborhood, we dene the separation axioms T0 and T1 in L-fuzzy topological spaces (L-fts). The relations between our denitions, Hutton and Reilly's, and Wang's are discussed, and the separations of Hutton's fuzzy unit interval and Gantner's fuzzy real line are examined. Characterizations of these L-fts are proved and a series of properties of them are investigated. Moreover, some results on minimal T0 L-fts and T1 L-fts are established, which will be used immediately in our subsequent discussion.

This paper is devoted to the study of the inverse limits in LTop, the category of L-fuzzy topological spaces and continuous L-valued Zadeh-functions between them, where L is a completely distributive complete lattice. Structures of such limits are described and a series of their properties are investigated. Furthermore, the mappings between two arbitrary inverse systems is de ned and some of their properties are discussed for the case of category LTop.

This paper is devoted to the study of H1 topological molecular lattices, a generalization of the Hausdor fuzzy topological spaces. Some primary properties of such topological molecular lattices are discussed. For any complete and completely distributive lattice L, the existence and uniqueness of minimal H1 co-topologies on L are investigated, and the characteristic theorem of minimal H1 co-topologies on L is established.

In order to provide a categorical framework for fuzzy topology, this paper proposes the concept of weak topological category, a weaker form of topological category. Properties of weak topological categories are investigated, and some examples of such category are provided. Moreover, the relationship between a useful weak topological category LTop, the category of L-fuzzy topological spaces and continuous L-valued Zadeh functions, and some of its subcategories is studied.