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.

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.

The aim of this paper is to find further connections between two different current researches on fuzzy topology: the one based on the concepts of neighborhood, filter and Zadeh function, and the other based on the concepts of remote neighborhood, ideal and order-homomorphism on fuzzy lattices. To approach this, the paper studies two weaker forms of continuous order-homomorphisms, the semi-continuous order-homomorphism and the weakly semi-continuous order-homomorphism between L-fuzzy topological spaces. These order-homomorphisms can be characterized in each of the notions: remote-neighborhood, neighborhood, ideal, filter and molecular net. In addition, some relations connecting weakly semi-continuous order-homomorphisms, semi-Hausdorff L-fts and Urysohn L-fts have also been established.

This paper studies local strong F1 connectedness and local Q-connectedness of L-topological spaces. It is shown that every L-interval J (L) and its weak induci.cation J(L) are both local strong F1 connected andlocal Q-connected, and that the local Q-connectedness of an L-topological space is equivalent to that of its induced I (L)-topological space.

This paper is a sequel to Li (1999). For some classes F of L-valued Zadeh functions, we will discuss two problems connected with inverse limits in LTop, K1HLTop and K2HLTop: (1) Will the condition that all L-valued Zadeh functions between factor L-fts of two given inverse systems are in F imply that the induced limit LTop-morphism is in F? (2) Will the condition that all bonding L-valued Zadeh functions are in F imply that all projective L-valued Zadeh functions are in F?