Liu Ying-Ming

Fuzzy topology. II. Product and quotient spaces

In [6], after introducing the fundamental concepts of fuzzy points which take crisp singletons or ordinary points as special cases, q-relations and Q-neighborhoods of fuzzy points, we built up a satisfactory theory of neighborhood structures and generalized many fundamental notions and theorems, especially those theorems in the theory of Moore-Smith’s convergence such as those contained in [5, Chaps. I, II]. The purpose of the present paper is to generalize all the ehtorems concerning product spaces and quotient spaces contained in [5, Chap. III]..Our treatment will sharpen the related results contained in the literature [l-4] so that these problems in fuzzy topology will be solved to the same degree as the corresponding ones in general topology. The present work is the continuation of [6] and hence all the conventions in [6] still hold good in the present paper, especially that (X, r) and (Y, %) d enote fuzzy topological spaces.

Structures of fuzzy order homomorphisms

Abstract First the concept of fuzzy order homomorphism is introduced which is a variety of the generalized order homomorphism given in [7]. Then we show that the fuzzy order homomorphism possesses a nicely local structure. Furthermore, we establish a necessary and sufficient condition under which the Fuzz function is a Zadeh type function. The present condition is simpler than that given in [6]. Finally. for each fuzzy order homomorphism f, we yield some characterizations about ( f −1 ) −1 = f which sharpen the corresponding theorem in [6].

L -fuzzy version of Stone's representation theorem for distributive lattices

Abstract In this paper the category of L-fuzzy locales is introduced which is proved to play the same role with respect to stratified L-fuzzy topological spaces as the locales play for topological spaces. Secondly, Stones representation theorem for distributive lattices is generalized to the L-fuzzy case, and the notion of L-fuzzy spectrum of distributive lattices is introduced and its properties are systematically studied; moreover, the harmony between the localic version of Stones representation theorem and spectral spaces of distributive lattices is also valid in the L-fuzzy case.

Lattice-valued Hahn-Dieudonné-Tong insertion theorem and stratification structure

Abstract In this paper, we shall generalize the Hahn-Dieudonne-Tong Insertion Theorem, a classical result on semicontinuous functions, to the case that the ranges are a certain kind of lattices L. We consider the family of all lattice-valued (lower) semicontinuous mappings as a topology, and then solve this problem via reducing it to a problem on a certain kind of separation property. Therefore, as a by-product, the following theorem is obtained for a suitable range L: An induced space is normal if and only if the underlying space is normal. The relative counterexamples show that the limitation for L is necessary. These results and counterexamples point out that normality of induced spaces and the existence of latticed-valued inserting mappings close relates to the internal characterization of the range L. The previous proofs of the classical insertion theorem are analytic and skillful; compared with them, the present method determining mappings via the stratifications is very natural and conceptual. The success of the method is based on the study in depth on the topological relations among the stratifications. We may hope that the method, which determines the mapping via inductively getting its stratifications, will be further applied in other aspects.

On fuzzy convergence classes

Abstract The concept of fuzzy point and its quasi-coincidence neighborhood have been introduced in [1]. Using these notions, we give the theory of fuzzy convergence classes. Especially a characterization of fuzzy topology via the fuzzy convergence classes and relative analysis are given.

Lattice-valued mappings, completely distributive law and induced spaces

The lattice-valued mappings whose value domain is a completely distributive lattice are investigated. We describe the analytic condition - the semicontinuity of lattice-valued mappings on a topological space X via the algebraic property - the completely distributive law and conversely, we use the semicontinuity to depict the completely distributive law. The family of all the upper semicontinuous lattice-valued mappings on a topological space X (or a part of them) constitutes a L-fuzzy topological space on X; a series of global properties of these lattice-valued mappings is presented. In particular, we investigate the impact of N-compactness on the semicontinuity. Using these results, we solve some problems on fuzzy Stone-Cech compactification.

STEENROD'S THEOREM FOR LOCALES

We give an explicit description of the construction of inverse limits in the category of locales and show a localic version of Steenrods theorem without the Axiom of Choice; as an application we generalize the classic result by Steenrod.

A localic L-fuzzy modification of topological spaces

Abstract An L-fuzzy modification of the category of the topological spaces is obtained in this paper by establishing a theory of L-fuzzy locales, which is meant to bridge the study of L-fuzzy topologies and that of locales.

T D property and spatial sublocales

As one of main backgrounds of locale theory, topologies have close connections with locales. But locales have other backgrounds such as algebra, mathematical logic, etc. So there are many differences between locales and topologies. Spatiality is an important localic property to investigate the connections between locales and topologies. TheT D property is a special separation property which plays an important role in this kind of investigations. Just as it will be proved in this paper, theT D property often appears as the lowest requirement for many topological spaces such that they can be described with localic properties and vice versa. In this paper, we show these special properties of theT D axiom and investigate some other interesting and important problems ofT D -spatiality of locales.

An Analysis on Fuzzy Membership Relation in Fuzzy Set Theory

Abstract In the previous paper (Liu, 1982 b) we gave some mutually equivalent systems of axioms, which seem intuitively to be evident, for determining the neighborhood structure in fuzzy topological spaces from the standpoint of to oology and fuzzy set theory. We also proved the theorem that the unique neighborhood structure satisfying one of these systems of axioms is exactly the Q-neighborhood structure (Lu and Liu, 1980a; Liu, 1933) which has already been playing and will continually play a significant role in the research of fuzzy topology, but the neighborhood structure is determined by a fuzzy membership relation betweeη the fuzzy points and the fuzzy sets. Therefore we presently pay attention to this relation are analysie it from the view point of fuzzy set theory. The four principles for determining the fuzzy membership relation are established. We also show taat the unique fuzzy membership relation is exactly the Q-relation which determines the Q-neighborhood structure mentioned above.