Fuzzy Separation Axioms Research Articles

Fuzzy SSPO-separation Axioms and Fuzzy α-SSPO Compactness

In this paper we will introduce the concept of new separation axioms named fuzzy SSPO-separation axioms by using the fuzzy strong semi preopen sets and we will also introduce and investigate properties of (alpha)-SSPO compactness. We will define and investigate relation between fuzzy separation axioms, fuzzy pre-separation axioms and different forms of fuzzy continuous mappings. We will also investigate the existence of a countable base of fuzzy strong semi preopen sets, we will define the concept of SSPO separability, the concept of (alpha)-SSPO Lindelof sets and examine their properties. With the concept of fuzzy strong semi pre-continuity, SSPO-irresolute continuous mappings and other forms of fuzzy continuity, we will investigate the new concept of fuzzy compactness and its properties in regards to the mentioned mappings.

A STUDY ON Fuzzy SEPARATION AXIOMS (Ti, i = 0; 1; 2) VIA Fuzzy gp* - OPEN SETS

In this paper we have introduced Fuzzy gp∗ closure, Fuzzy gp*-interior and separation axioms via Fuzzy gp*-open sets. Also we found out the relationship between Fuzzy separation axioms, Fuzzy gp* separation axioms and Fuzzy pre separation axioms.

Fibrewise Fuzzy Separation Axioms

Within that research, we introduce fibrewise fuzzy types of the most important separation axioms in ordinary fuzz topology, namely fibrewise fuzzy (T 0 spaces, T 1 spaces, R 0 spaces, Hausdorff spaces, functionally Hausdorff spaces, regular spaces, completely regular spaces, normal spaces, and normal spaces). Too we add numerous outcomes about it.

Fuzzy roughness via ideals

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK'S SENSE (II)

In this paper, we have use a fuzzy bitopological space (X, <TEX>$\tau_1$</TEX>, <TEX>$\tau_2$</TEX>) to create a family <TEX>$\tau_{ij}^s$</TEX> which is a supra fuzzy topology on X. Also, we introduce and study the concepts of r-(<TEX>$\tau_i$</TEX>, <TEX>$\tau_j$</TEX>)-generalized fuzzy regular closed, r-(<TEX>$\tau_i$</TEX>, <TEX>$\tau_j$</TEX>)-generalized fuzzy strongly semi-closed and r-(<TEX>$\tau_i$</TEX>, <TEX>$\tau_j$</TEX>)-generalized fuzzy regular strongly semi-closed sets in fuzzy bitopological space in the sense of <TEX>$\check{S}$</TEX>ostak. Also, these classes of fuzzy subsets are applied for constructing several type of fuzzy closed mapping and some type of fuzzy separation axioms called fuzzy binormal, fuzzy mildly binormal and fuzzy almost pairwise normal.

On Fuzzy Irresolute Functions

As a generalization of the notions of fuzzy <TEX>$\alpha-irresolute$</TEX>, fuzzy preirresolute, fuzzy irresolute and fuzzy <TEX>$\beta-irresolute$</TEX> functions, we introduce the notion of fuzzy <TEX>$\beta\alpha-continuous$</TEX> functions and investigate the relationships between fuzzy <TEX>$\beta\alpha-continuous$</TEX> functions and fuzzy separation axioms.

The relation between GTi-spaces and fuzzy proximity spaces, G-compact spaces, fuzzy uniform spaces

In this paper, we study the relation between the fuzzy separation axioms, which had been introduced by the authors in 2001, and the fuzzy proximity defined by Katsaras in 1980. We study also the relation between our fuzzy separation axioms and the G-compactness defined by Gähler in 1995. Moreover, we show here the relation between these fuzzy separation axioms and the fuzzy uniform structures introduced and studied by Gähler and the first author in 1998.

Fuzzy topology on fuzzy sets: fuzzy semicontinuity and fuzzy semiseparation axioms

The concept of a fuzzy topology on a fuzzy set was introduced in [Fuzzy Set Syst. 45 (1992) 103]. The aim of this paper is to introduce fuzzy semicontinuity and fuzzy semiseparation axioms in this new situation and validity of some characterization of these concepts have been examined. Also, we define a fuzzy generalized semiopen set and introduce fuzzy separation axioms by using this concept. Fuzzy semiconnected and fuzzy semicompact spaces are introduced and some of their properties are discussed.

Operations and its applications on L-fuzzy bitopological spaces: Part I

In this and a subsequent paper, a new operation on the class of all L-fuzzy subsets of a universe endowed with a fuzzy topological space is introduced to generalize several characterization and properties of some fuzzy bitopological concepts such as a fuzzy interior operators and some new classes of fuzzy separation axioms. Also, the relation between these classes and the weaker and stronger forms in fuzzy bitopological spaces are investigated. Finally, we showed that these axioms are good extension in the sense of Lowen (1978) due to Lowen, in Part II this theory is applied on the mappings defined on L-fuzzy bitopological spaces.

On lower separation axioms

In [3] Mashhour et al. introduced the concept of fuzzy disjointness. Two fuzzy sets A, B in X are said to be fuzzy disjoint if A ⩽ co B, where co B is the complement of B. Using this concept, they introduced FT i separation axioms as a generalization for the basic separation axioms T i ( i = 1, …, 4). In [5] Singal and Rajvanshi used the concepts of regular open fuzzy sets [1] and fuzzy disjointness to introduce some fuzzy almost separation axioms which are stronger than those of Mashhour. Unfortunately, some results in [5] are incorrect. In this paper, we use the concept of the quasi-coincident relation [4] to restate lower fuzzy separation axioms and give a corrected version for some definitions and results in [5].

A fuzzy R1 separation axiom

We introduce a new R 1 separation axiom for fuzzy topological spaces. We identify, from the literature, which lower fuzzy separation axioms are compatible with our R 1, axiom. We provide alternative descriptions for this R 1 axiom in the special setting of fuzzy neighborhood spaces. This special case has standard higher fuzzy separation axioms. In it, we establish aspects of compatibility of our R 1 axiom with those higher separation axioms, as well as with fuzzy compactness.

Fuzzy separation axioms and fuzzy continuity in fuzzy bitopological spaces

Let ( X, τ 1, τ 2) be a bitopological space and ω( τ i ) ( i = 1, 2) be the set of all lower semi-continuous functions defined from X into the closed unit interval [0, 1]. In this paper we have succeeded in showing goodness of extension of separation axioms for fuzzy bitopological spaces, namely we prove the space ( X, τ 1, τ 2) is P-T i , P-T iw , P-regular, and P-normal iff the fuzzy bitopological space ( X, ω( τ 1), ω( τ 2)) is P-T i , P-T iw , P-regular and P-normal.

Regularly open sets in fuzzy topological spaces

This paper is devoted to the study of the role of fuzzy regularly open sets. We prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings. We prove that under a fuzzy almost continuous and fuzzy almost open map, the inverse image of a fuzzy regularly open set is fuzzy regularly open. Further we define a new type of fuzzy separation axioms, fuzzy almost separation axioms. It is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology. For example, in a fuzzy almost T 1 space not every fuzzy singleton is δ-closed. Also a fuzzy space which is fuzzy almost T 2 1 2 as well as fuzzy almost T 0 is fuzzy almost regular. While in general topology we have to take an almost T 2 space in place of almost T 0 space.

Separation axioms in fuzzy topological spaces

In this paper, certain fuzzy separation axioms are studied in terms of the notions of quasi-coincidence, q-neighbourhoods and fuzzy θ-closure operators. Fuzzy T 2 and Urysohn spaces are defined, and fuzzy spaces satisfying these axioms are characterized. A (fuzzy) pointwise formulation of a fuzzy T 2-axiom of Hutton and Reilly (1980) is found. Fuzzy regularity and almost regularity are investigated yielding various characterizing properties of these spaces. Certain relationships among all those fuzzy separation axioms are also discussed.

Fuzzy preopen sets and fuzzy preseparation axioms

In this paper, the concept of preopen sets due to A.S. Mashhour, M.E. Abd El-Monsef and S.N. El-Debb [7] has been generalised to the fuzzy setting. Further fuzzy separation axioms have been introduced and investigated with the help of fuzzy preopen sets.

Fuzzy θ‐closure operator on fuzzy topological spaces

The paper contains a study of fuzzy θ‐closure operator, θ‐closures of fuzzy sets in a fuzzy topological space are characterized and some of their properties along with their relation with fuzzy δ‐closures are investigated. As applications of these concepts, certain functions as well as some spaces satisfying certain fuzzy separation axioms are characterized in terms of fuzzy θ‐closures and δ‐closures.

Separation axioms for fuzzy spaces

Using the concepts of fuzzy points as introduced by Wong [15] and of ‘belonging to’ as introduced by R. Srivastava, S.N. Lal and A.K. Srivastava [13], we introduce several notions of fuzzy separation axioms and examine their productivity. Relations between these axioms have been observed. We also introduce four types of separated fuzzy sets and clarify some relations between them.

On separation axioms and T i-fuzzy continuity

We introduce and generalise concepts of T;-continuous functions in a fuzzy setting and study them in connection with the fuzzy continuity and fuzzy separation axioms. Some fuzzy topological properties are also investigated under such functions.