On generalised fuzzy soft topological spaces

In this paper, union, and intersection of generalised fuzzy soft sets are introduced and some of their basic properties are studied. The objective of this paper is to introduce the generalised fuzzy soft topology over a soft universe with a fixed set of parameters. Generalised fuzzy soft points, generalised fuzzy soft closure, generalised fuzzy soft neighbourhood, generalised fuzzy soft interior, generalised fuzzy soft base are introduced and their basic properties are investigated. Finally, generalised fuzzy soft compact spaces are introduced and a few basic properties are taken up for consideration.


INTRODUCTION
Most of our real life problems in engineering, social and medical science, economics, environment etc. involve imprecise data and their solutions involve the use of mathematical principles based on uncertainty and imprecision.To handle such uncertainties, Zadeh (1965) introduced the concept of fuzzy sets and fuzzy set operations.The analytical part of fuzzy set theory was practically started with the paper of Chang (1968) who introduced the concept of fuzzy topological spaces; however, this theory is associated with an inherent limitation, which is the inadequacy of the parametrization tool associated with this theory as it was mentioned by Molodtsov (1999).
In 1999, Russian researcher Molodtsov introduced the concept of soft set theory which is free from the above problems and started to develop the basics of the corresponding theory as a new approach for modelling uncertainties.Shabir and Naz (2011) studied the topological structures of soft sets.
In recent times, many researches have contributed a lot towards fuzzification of soft set theory.In 2001, Maji et al. introduced the fuzzy soft set which is a combination of fuzzy set and soft set.Tanay and Burc Kandemir (2011) introduced topological structure of fuzzy soft set and gave an introductory theoretical base to carry further study on this concept.The study was pursued by some others (Chakraborty et al., (2014); Gain et al., 2013).
In 2010, Majumdar and Samanta introduced generalised fuzzy soft sets and successfully applied their notion in a decision making problem.Yang (2011) pointed out that some results put forward by Majumdar and Samanta (2010) are not valid in general.Borah et al. (2012) introduced application of generalized fuzzy soft sets in teaching evaluation.
The objective of this paper is divided into three parts.In the first part we introduce the generalised fuzzy soft union, generalised fuzzy soft intersection, and several other properties of generalised fuzzy soft sets are studied.In the second part we introduce "generalised fuzzy soft topological spaces" over the soft universe with a fixed set of parameter.Then we discussed some basic properties of generalised fuzzy soft topological spaces with an example and define generalised fuzzy soft open and closed sets.By this way we define the generalised fuzzy soft closure, generalised fuzzy soft points, generalised fuzzy soft neighbourhood, generalised fuzzy soft interior, generalised fuzzy soft base and also we established then some important theorems related to these spaces.Finally we define generalised fuzzy soft compactness and give some important definitions and theorems.
This paper may be the starting point for the studies on "Generalised fuzzy soft topology" and all results deducted in this paper can be used in the theory of information systems.

PRELIMINARIES
Throughout this paper X denotes initial universe, E denotes the set of all possible parameters for X ,   X P denotes the power set of X , X I denotes the set of all fuzzy sets on X , E I denotes the collection of all fuzzy sets on E ,   E X , denotes the soft universe and (Zadeh, 1965).Zadeh (1965), we have the following:  : (Molodtsov, 1999).In other words, a soft set is a parameterized family of subsets of the set X .For A e  ,   e f may be considered as the set of  e approximate elements of the soft set   A f , .

Definition 3
is called a fuzzy soft set over X , where et al., 2001).

Definition 4
Let X be the universal set of elements and E be the universal set of parameters for X (Majumdar and Samanta, 2010).Let

  
We define a function (Majumdar and Samanta, 2010) In this case we write (Majumdar and Samanta, 2010).Then the complement of and is defined by

, ~
, where , where  is an index set, be a family of GFSSs.The union of these family is denoted by Chakraborty and Mukherjee , where  is an index set, be a family of GFSSs.The intersection of these family is denoted by A GFSS is said to be a generalised null fuzzy soft set, denoted by

Definition 10
A GFSS is said to be a generalised absolute fuzzy soft set, denoted by (Majumdar and Samanta, 2010).
then the following holds: then the following holds: , then the following holds: where , where

GENERALISED FUZZY SOFT TOPOLOGICAL SPACE
Here, we introduce the notion of generalised fuzzy soft topology over a soft universe and study some of its basic properties.

Definition 11
Let T be a collection of generalised fuzzy soft sets over   E X , .Then T is said to be a generalised fuzzy soft topology (GFS topology, in short) over   E X , if the following conditions are satisfied: Follows from the definition of GFST-space and De-Morgan's law for GFSS which is given in Proposition 4.

Example 4
Let T be the collection of all GFSS which can be defined over   E X , .Then T forms a GFS topology over   E X , ; it is called the GFS discrete topology over   E X , .

Chakraborty and Mukherjee 5
Example 5 We consider the following GFSS over   E X , defined as

Remark:
The union of two GFS topologies over   E X , may not be a GFS topology over   E X , , which follows from the following Example.

Example 7
, where Definition 15 Clearly, Let us consider the following GFSS over  
(3) Let  F ~ be a GFS closed set.By (2) we have That is, every GFS closed Hence the intersection of GFS closed super sets of

Chakraborty and Mukherjee 7
Conversely, as being the union of two GFS closed sets.Proof: Obvious.

Definition 17
is called a generalised fuzzy soft neighbourhood of the generalised

Proof
(1) is obvious. ( , then there exist  That is, , be a GFST-space.A subfamily  of T is said to be a GFS base for T if every member of T can be expressed as the union of some members of  .

GENERALISED FUZZY SOFT COMPACT SPACES
The closed and bounded sets of real line were considered an excellent model on which to fashion the generalised version of compactness in topological space.The concept of compactness for a fuzzy topological space has been introduced and studied by many mathematicians in different ways.The first among them was Chang (1968).
In this article, we introduce and study compactness in generalised fuzzy soft topological perspective.

Definition 22
A family  of generalised fuzzy soft sets is a cover of a generalised fuzzy soft set

Definition 24
A family  of generalised fuzzy soft sets has the finite intersection property if the intersection of the members of each finite sub family of  is not the generalised null fuzzy soft set.

Theorem 5
A generalised fuzzy soft topological space is GFS compact if and only if each family of GFS closed sets with the finite intersection property, has a generalised nonempty fuzzy soft intersection.

Conclusion
The proposed work is basically a theoretical one.If these theoretical back-ups are properly nurtured and fruitfully developed, it will definitely usher in various nice applications in the fields of Engineering Science, Medical Science and Social Sciences, by skilfully analyzing and interpreting imprecise data mathematically.
the degree of belongingness of the elements of X in   e F but also the degree of possibility of such belongingness which is represented by   the union of all generalised fuzzy soft open subsets of  N ~.
a GFST-space and  be a GFS base for T over   Then  is a GFS base for T if and only if for any  of GFS closed sets with generalised non-null fuzzy soft intersection and so by hypothesis this collection does not have the FIP.Hence there exists a finite number of GFSS Arbitrary unions of members of T belong to T .(iii) Finite intersections of members of T belong to T .
It is a generalised fuzzy soft open cover iff each member of  is generalised fuzzy soft open set.A sub cover of  is a subfamily of  which is also a cover.