L-fuzzy Topological Spaces Research Articles

A fuzzification of the category of M-valued L-topological spaces

A fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FTOP(L,M) extending the category TOP(L,M) of M-valued L- topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak’s sense. Basic properties of the fuzzy category FTOP(L,M) and its objects are studied.

On L-fuzzy topological spaces

In this paper, we study L-fuzzy topological spaces, where L represents a completely distributive lattice. We shall investigate the level decomposition of L-fuzzy topologies on X and the corresponding L-fuzzy continuous maps. In addition, we shall establish the representation theorems of L-fuzzy topologies on X.

Strong and ultra separation axioms on L-fuzzy topological spaces

Given an L-fuzzy topological space ( X, δ), we introduce new notions of L-fuzzy separation axioms, using the family of its level topology: { i α ( δ): α∈Pr( L)}. We check that these are well-defined L-fuzzy topological concepts and we compare them with the analogous L-fuzzy ones introduced in the literature. We verify that these notations are not equivalent, and we give many examples which illustrate this fact. Furthermore, some results concerning the new concepts have been established.

On characterizations of some covering properties in L-fuzzy topological spaces in Šostak's sense

In this work, various characterizations of some weaker and stronger forms of L-fuzzy compactness are obtained in L-fuzzy topological spaces in Šostak's sense, where L is a fuzzy lattice.

L-filters and LF-topologies

For a complete quasi-monoidal lattice L, with some additional properties, we consider the category of L-filtered sets with fixed basis L, and we find out a functorial relation from this category to that of the L-fuzzy topological spaces. On the other hand, a morphism in the category of complete quasi-monoidal lattices allows us to establish a pair of adjoint functors (change of basis) between the corresponding categories of L-filterd sets.

On the Relationship Between Several Basic Categories in Fuzzy Topology

In this paper, a process, which associates with every fibre-small topo- logical category A and every completely distributive lattice L another topological category called the co-tower extension of A, is introduced to investigate the relationship between several basic categories in fuzzy topology. It is proved that the category of stratified Šostak L-fuzzy topological spaces is the co-tower extension of the category of stratified Chang-Goguen spaces; and the category of L-fuzzifying topological spaces (which is isomorphic the category of fuzzy neighborhood spaces when L = [0, 1]) is the co-tower extension of the category of topological spaces. Thus, in some sense, we can say that properties of the category of stratified Šostak L-fuzzy topological spaces and L-fuzzifying topological spaces are determined by the categories of stratified Chang-Goguen spaces, topological spaces and the lattice L. Therefore it can be said that in fuzzy topology the category of (stratified) Chang-Goguen topological spaces is more basic and more important than the other categories.

Initial L-fuzzy closure spaces

We prove that the category L- FC is a topological category over Set. From this fact, we can define subspaces and products of L-fuzzy closure spaces. In particular, an initial structure of L-fuzzy topological spaces can be obtained by the initial structure of L-fuzzy closure spaces induced by those.

L-FUZZY PS-COMPACTNESS

In this paper, the new compactness which is L-fuzzy PS-compactness in L-fuzzy topological spaces is introduced. It is defined for arbitrary L-fuzzy subsets. And some of its characteristic properties and fundamental properties are studied.

L-Fuzzy PS-Compactness

In this paper, the new compactness which is L-fuzzy PS-compactness in L-fuzzy topological spaces is introduced. It is defined for arbitrary L-fuzzy subsets. And some of its characteristic properties and fundamental properties are studied.

On first countable [formula omitted]-fuzzy topological spaces

The fuzzy continuity of a bijective function defined on a first countable, C 1 , L-fuzzy topological space where L is a spatial frame is characterized by using the sequences of L-fuzzy points. Moreover, a local characterization of fuzzy continuity is given.

Solutions to two problems of G. J. Wang

The problem as to whether the sub-T0 separation and complete regularity are invariant under homeomorphism is answered negatively. And, the problem of multiplicativity of the complete regularity in general L-fuzzy topological spaces is also answered negatively.

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.

Some Categorical Aspects of Fuzzy Topology

Research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties. In this paper we discuss some of these aspects. Since the topological construct L-FTS contains nontrivial both initially and finally closed full subconstructs, and each such construct gives rise to a natural autonomous theory of fuzzy topology, it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories, including the classical topology theory as a special case, with each applying to one such subconstruct. Therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of L-FTS to illustrate this idea. The second topic of this paper is the relationship between the construct of stratified L-fuzzy topological spaces and several other familiar constructs in fuzzy topology, for example, the constructs of Šostak fuzzy topological spaces and L-fuzzifying topological spaces.

Relative compactness in L-fuzzy topological spaces

In this paper, taking fuzziness with respect to a fuzzy lattice, we present good extensions of two of the current definitions of relative compactness in general topology. Fuzzy versions of some properties are obtained.

Formula omitted]-Compactness in [formula omitted]-fuzzy topological spaces

In this paper, we introduce a good definition of fuzzy α-compactness in L-fuzzy topological spaces, where L is a fuzzy lattice. We define fuzzy α-compactness on arbitrary L-fuzzy sets, obtain different characterizations and study some of its properties.

Separation axioms in [formula omitted]-fuzzy topological spaces (I): [formula omitted] and [formula omitted

Using the notion of remote neighborhood, we define the separation axioms T 0 and T 1 in L-fuzzy topological spaces ( L-fts). The relations between our definitions, 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 T 0 L -fts and T 1 L -fts are established, which will be used immediately in our subsequent discussion.

Compactness in fuzzy (fuzzy supra) topological spaces

The notion of Q α -compact fuzzy topological spaces was first introduced by Zhongfu (Kexue Tongbao 29(5) (1984) 582–585). In this paper we extend this notion to fuzzy supra topological spaces and study some of their properties. We also introduce the notion of fuzzy Q α -almost compact ( Q α -almost supracompact). This notion is defined for any fuzzy subset and is a good extension. Finally, we try to extend such a notion to L-fuzzy (supra) topological spaces and discuss some of their properties.

Connectedness in [formula omitted]-fuzzy topological spaces

Based on the consideration to the layer structures of fuzzy lattices and the level topologies of L-fuzzy topological spaces, a connectedness is defined for an arbitrary L-fuzzy set in this paper. This definition reflects 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(λ)-intervals are defined 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.

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.