For L a closed sets lattice, we prove the existence and a characterization theorem of the minimal elements of T 3LX, where T 3

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.

Based on the consideration to the layer structures of fuzzy lattices and the level topologies of L-fuzzy topological spaces, a connectedness is dened for an arbitrary L-fuzzy set in this paper. This denition reects the degree of connectivity of an L-fuzzy set. It is shown that the connectedness of L-fuzzy topological spaces is an L-good extension, multiplicative and preserved under continuous L-valued Zadeh functions, and that the inverse limit of continuums is a continuum. General L-fuzzy intervals and H-intervals are dened and connectedness of them are proved. It is also shown that the connectedness of an L-fuzzy topological space is equivalent to the connectedness of its induced I (L)-fuzzy topological space.

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 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?

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.

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.

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.