L-fuzzy Topology Research Articles

Urysohn separation property in topological molecular lattices

The theory of topological molecular lattices, which is a generalization of the theories of point set topology, fuzzy topology and L-fuzzy topology, is established by Wang. In this paper, the concepts of some kinds of Urysohn separation axioms in topological molecular lattices are introduced and systematically discussed. Several important results, such as the U i (i =−1,0,1,2,3,4) , Urysohn regular and Urysohn normal separation properties are both hereditary and topological invariant, etc., are given.

Induced I( L)-fuzzy topological vector spaces

In this paper we introduce and study the induced I( L)-fuzzy topological vector spaces. The following main result is proved: If ( L X , δ) is an L-fuzzy topological vector space, then ( I( L) X , ω I( L) ( δ)) is an I( L)-fuzzy topological vector space, where I( L) is the fuzzy unit interval and ω I( L) ( δ) is the induced I( L)-fuzzy topology of the L-fuzzy topology δ. Furthermore, we also prove that the induced I( L)-fuzzy topological vector spaces ( I( L) X , ω I( L) ( δ)) preserve the product and quotient space as well.

On stratiform [formula omitted]-fuzzy topologies and their application

For a given L-fuzzy topological space and a fixed prime element r of fuzzy lattice L, the concept of L-fuzzy r-open sets is introduced, and it is proved that all r-open sets for L-fuzzy topological space form a new L-fuzzy topology, which is called stratiform L-fuzzy topology. Making use of the stratiform L-fuzzy topology, a series of new characterizations of L-fuzzy continuous mapping and L-fuzzy open mapping and three kinds of separateness (Hausdorff, regular and normal) are presented.

The Group of L-Fuzzy Homeomorphisms and the Group of Permutations of the Ground Set

In this paper we study some relations between the group of L-fuzzy homeomorphisms of an L-fuzzy topological space X and the group of permutations of the ground set X. In contrast with the results in general topology, it is proved that the subgroups generated by cycles and proper normal subgroups can be represented as groups of L-fuzzy homeomorphisms for some L-fuzzy topology on X. The relation between the group of L-fuzzy homeomorphisms and the group of homeomorphisms of the associated topology is also discussed.

On Convergence Classes inL-fuzzy Topological Spaces

Convergence theory is a primary topic in topology. In fact, topology and so-called convergence class are characterized by each other. In fuzzy topology (L-fuzzy topology), more than 40 papers published in the last ten years were concerned with convergence theory. Among these papers, the problem of convergence class was solved for the case ofL=[0,1] [7]. Since the neighbor structure, so called “quasi-coincident neighborhood system,” of anL-fuzzy point in anL-fuzzy topological space is in general not directed under the inclusion order, the conditions of convergence class in [0,1]-fuzzy topology will not be valid any longer in the case of lattice. Moreover, quite different from the cases of {0,1}-fuzzy topology (i.e., ordinary topology) and [0,1]-fuzzy topology, the so called Bolzano–Weierstrass property does not hold, i.e., a net with a cluster point in anL-fuzzy topological space is not still necessary to have a subnet converging to the point. In this paper, a necessary and sufficient condition for the Bolzano–Weierstrass property is produced, the result is also used in a satisfactory theory of convergence classes inL-fuzzy topological spaces, and the associated characterization theorem betweenL-fuzzy topologies and convergence classes is established.

θ-Convergence of nets of L-fuzzy sets and its applications

In this paper, the theory of θ-convergence on nets of L-fuzzy sets, which is a generalization of the convergence theory of fuzzy nets introduced by Sarma and Ajmal, is established. Some important and interesting results in L-fuzzy topology are obtained by means of the theory.

Two new L-fuzzy topologies on R ( L)

In this paper, two new topologies, τ′ 1, τ′ 2 on R ( L) are constructed. Using τ′ i , we produced two L-fuzzy topologies η′ i on R( L) other than the L-fuzzy topology T introduced by Hutton. We have proved that η′ 1 is finer than η′ 2 and η′ 2 is finer than T. η′ 2 is the coarsest among the induced L-fuzzy topologies which are finer than T. We have shown that ( L R( L) , η i ), the induced space of ( R ( L), τ′ i , possesses many good properties, such as stronger separation, suitability, etc. We proved that closed interval ([ a, b] ( L), ν 2|[ a, b] ( L)) is closed, connected and N-compact. The addition and multiplication defined on R ( L) by Rodabaugh, are still jointly continuous.

Countability in L-fuzzy topology

Good definitions of L-fuzzy countable and sequential compactness and the Lindelöf property are presented for L a fuzzy lattice. Each of them is defined on arbitrary L-fuzzy sets and their properties studied.

L-fuzzy Smirnov compactifications

In this paper we get the pointwise characterization of the L-fuzzy topology induced by an L-fuzzy proximity space, and study some important properties of L-fuzzy proximity induced spaces. Also, we discuss the N-compactness, separated relations and separation axioms of L-fuzzy proximity spaces. On the base of the method of induced spaces and above conclusions, we establish the L-fuzzy Smirnov compactification, and construct an example for L-fuzzy Smirnov compactifications.

InsertingL-fuzzy Real-valued Functions

A new proof of the Katêtov-Tong Insertion Theorem for L-fuzzy topological spaces is given. All the proofs are performed in such a way that the presented results hold true in the more general case in which L-fuzzy topologies are replaced by σ-additive rings of L-fuzzy subsets.

Theory of topological molecular lattices

The aim of this paper is to bring the theories of point set topology, fuzzy topology and L-fuzzy topology into a unified theory - the theory of Topological Molecular Lattices. For this purpose, we first introduce the concepts of molecule for completely distributive complete lattice (molecular lattice for short), the cotopology on it and the concept of remote-neighborhoods of molecules; therefrom we develop the theory of pointwise topology on molecular lattices.

Frame-fuzzy points and membership

For a frame L the collection of L-fuzzy subsets of a set X forms a locale. The localic definition of points gives rise to L-fuzzy points xλ where x ϵ X and λ is a prime element of L. Membership is defined in terms of ⪕/. A satisfactory local theory of L-fuzzy topology emerges from this, supplemented for some purposes by continuity of L.

Fuzzy proximity spaces redefined

We use the concept ‘L-quasi-coincident’ in fuzzy set theory, to redefine L-fuzzy proximity space, and we study the following problems: proximity of L-fuzzy normal spaces, L-fuzzy filters, proximity neighborhoods and the L-fuzzy topology induced by the L-fuzzy proximity.

On L-fuzzy topological spaces

L-fuzzy sets and, subsequently, L-fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L-fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.