Vol. 8(1), pp. 1-11, January, 2015 DOI: 10.5897/AJMCSR2014.0562 Article Number: E67B4E650091 ISSN 2006-9731 Copyright © 2015 Author(s) retain the copyright of this article http://www.academicjournals.org/AJMCSR
African Journal of Mathematics and Computer Science Research
Full Length Research Paper
On generalised fuzzy soft topological spaces R. P. Chakraborty* and P. Mukherjee Department of Mathematics, Hijli College, Hijli, Kharagpur, Paschim Medinipur-721301, West Bengal, India. Received 7 August, 2014; Accepted 8 September, 2014
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. Key words: Fuzzy soft set, generalised fuzzy soft set, generalised fuzzy soft topology, generalised fuzzy soft open sets, generalised fuzzy soft closed sets, generalised fuzzy soft topological spaces, generalised fuzzy soft compact spaces.
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.
*Corresponding author. Email:
[email protected]. Tel: +91 9143028586. Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Subject Classification Code: 54A40, 54D99.
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Afr. J. Math. Comput. Sci. Res.
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 X , E 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.
f is a mapping from A into P X , that is, f : A P X (Molodtsov, 1999).
where
In other words, a soft set is a parameterized family of subsets of the set X . For e A , f e may be considered as the set of e approximate elements of the soft set f , A .
Definition 3 Let A E . A pair
F, A
is called a fuzzy soft set
over X , where F : A I is a function, that is, for each a A , F a Fa : X 0,1 is a fuzzy set on X X
(Maji 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
PRELIMINARIES
Samanta, 2010). Let F : E I
Throughout this paper X denotes initial universe, E denotes the set of all possible parameters for X , P X denotes the power set of X , I
X
denotes the set of all
E
fuzzy sets on X , I denotes the collection of all fuzzy I sets on E , X , E denotes the soft universe and
stands for 0,1 .
Definition 1 A fuzzy set A in X is defined by a membership function A : X 0,1 whose value A x represents the “grade
subset of E , that is,
X
and
: E I 0,1 .
mapping F : E I
be a fuzzy
Let F be the
I be a function defined as ~ ~ I X and follows: F e F e , e , where F e ~ I E Then F~ is called a generalised fuzzy soft e X
set (GFSS in short) over X , E . Here, for each parameter e E , ~ F e F e , e indicates not only the degree of
belongingness of the elements of X in F e but also the degree of possibility of such belongingness which is represented by e .
of membership” of x in A for x X (Zadeh, 1965). If A, B I X then from Zadeh (1965), we have the following:
Example 1
A B A x B x , for all x X .
Let X x1 , x 2 , x3 be a set of three houses under
A B A x B x , for all x X .
consideration. Let E e1 , e2 , e3 be a set of qualities
D A B D x min A x , B x , for all x X .
surroundings. Let
C A B C x max A x , B x , for all x X .
E A E x 1 A x , for all x X . C
where e1 expensive, e2 beautiful, e3 in the green follows:
: E I 0 ,1 be defined e1 0.4 , e2 0.5 , e3 0.7
We define a function F : E I Definition 2 Let A E . A pair
f , A is called a soft set over
X
X
I as follows:
~ F e1 x1 \ 0.4, x2 \ 0.2, x3 \ 0.5,0.4, F e2 x1 \ 0.6, x2 \ 0.4, x3 \ 0.7,0.5, F F e3 x1 \ 0.7, x2 \ 0.5, x3 \ 0.1,0.7
as
Chakraborty and Mukherjee
Then F is a GFSS over X , E .
~
Definition 5 Let F and G be two GFSS over X , E (Majumdar
~
~
~ and Samanta, 2010). Now F is said to be a GFS subset ~ ~ ~ of G or G is said to be a GFS super set of F if
is a fuzzy subset of
F e is also a fuzzy subset of G e , e E ~ ~ ~ In this case we write F G .
3
~ M : E I X I such that M e M e , e , where M e F e G e and e e e , e E . ~ Let F , , where is an index set, be a
family of GFSSs. The intersection of these family is
,
~ ~ F
denoted by
~
is a GFSS M , defined as
~ M : E I X I such that M e M e , e , where M e F e and e e ,
e E . Example 2
Definition 6
~ Let F be a GFSS over X , E (Majumdar and ~ Samanta, 2010). Then the complement of F , is ~ c ~ c ~ denoted by F and is defined by F G , where
Consider the GFSS
~ F e1 x1 \ 0.4, x2 \ 0.2, x3 \ 0.5,0.4, F e2 x1 \ 0.6, x2 \ 0.4, x3 \ 0.7, 0.5, . F F e3 x1 \ 0.7, x2 \ 0.5, x3 \ 0.1, 0.7
~
e c e and G e F c e , e E .
Let G
~ c Obviously F
follows:
c
~ F .
~ ~ ~ ~ ~ Union of two GFSS F and G , denoted by F G , ~ X is a GFSS H , defined as H : E I I such that ~ H e H e , e , where H e F e G e and e e e , e E . ~ Let F , , where is an index set, be a
family of GFSSs. The union of these family is denoted
~ ~ F
be another GFSS over
X , E
defined as
~ G e1 x1 \ 0.3, x2 \ 0.5, x3 \ 0.2,0.2, G e2 x1 \ 0.5, x2 \ 0.3, x3 \ 0.4, 0.4, . G G e3 x1 \ 0.8, x2 \ 0.6, x3 \ 0, 0.8
Definition 7
by
~ F over X , E given in Example 1
;
is
a
GFSS
~ H ,
defined
H : E I I such that where H e F e and e e ,
~ H e1 x1 \ 0.4, x2 \ 0.5, x3 \ 0.5,0.4, H e2 x1 \ 0.6, x2 \ 0.4, x3 \ 0.7,0.5, . H H e3 x1 \ 0.8, x2 \ 0.6, x3 \ 0.1, 0.8
~ ~ ~ ~ Again F G M M e1 x1 \ 0.3, x2 \ 0.2, x3 \ 0.2,0.2 , M e2 x1 \ 0.5 , x2 \ 0.3, x3 \ 0.4, 0.4 , . ~ M M e3 x1 \ 0.7 , x2 \ 0.5 , x3 \ 0, 0.7
as
~ H e H e , e ,
X
~ ~ ~ ~ Then F G H
c c . ~ c F e1 x1 \ 0.6 , x 2 \ 0.8 , x3 \ 0.5,0.6, F e2 x1 \ 0.4 , x 2 \ 0.6 , x3 \ 0.3, 0.5, F c F e3 x1 \ 0.3 , x 2 \ 0.5 , x3 \ 0.9, 0.3
e E .
Definition 9 A GFSS is said to be a generalised null fuzzy soft set,
~
: E I X I
denoted
Intersection of two GFSS
e F e , e , where F e 0 e E and e 0 e E (Where 0 denotes the null fuzzy set)
~ ~ F and G , denoted by
~ ~ ~ ~ F G , is a GFSS M , defined as
~
by
Definition 8
,
if
(Majumdar and Samanta, 2010).
such
that
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Afr. J. Math. Comput. Sci. Res.
Proof: Straightforward.
Definition 10 A GFSS is said to be a generalised absolute fuzzy soft
~ 1 , if 1 : E I X I , where
set, denoted by
~ 1 e 1e , e is defined by 1e 1 e E and e 1, e E (Where 1 ( x) 1, x X ) (Majumdar
Proposition 4 Let
following holds:
and Samanta, 2010).
~ F be a GFSS over X , E , then the following holds:
~ ~ ~ ~ F F F ~ ~ ~ ~ (ii) F F F ~ ~ ~ ~ (iii) F F ~ ~ ~ ~ (iv) F
(ii) Let
then the following holds:
~ ~ ~ ~ ~ ~ (i) F G G F ~ ~ ~ ~ ~ ~ (ii) F G G F ~ ~ ~ ~ ~ ~ (iii) F G H F ~ ~ ~ ~ ~ ~ (iv) F G H F
~ ~ G ~ ~ G
~ H~ ~ H~
c
c
c
c
c
~ ~ ~ ~ F G M then
~ M e M e , e , e E , M e F e G e and e e e .
~ ~ ~ Thus F G
where
M~ M e, e Fe Ge , e e c
c
c
c
c
c
Here, we introduce the notion of generalised fuzzy soft topology over a soft universe and study some of its basic properties.
Definition 11
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ F G H F G F H ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (ii) F G H F G F H
Let T be a collection of generalised fuzzy soft sets over X , E . Then T is said to be a generalised fuzzy
soft topology (GFS topology, in short) over X , E if the following conditions are satisfied:
then the following holds:
c
GENERALISED FUZZY SOFT TOPOLOGICAL SPACE
~ ~ ~ F , G and H be any three GFSS over X , E ,
H~ H e, e Fe Ge , e e
Proposition 3
(i)
e e .
F c e G c e, c e c e ~c ~ ~c F G
Proof: Straightforward.
Let
~
~
F c e Gc e, c e c e ~c ~ ~c F G
Let F , G and H be any three GFSS over X , E ,
Proof
~ ~~ Thus F G
Proposition 2
c
~ G~ c c ~ ~ c G
where H e F e G e and e
Proof: Follow from the respective definitions.
~
c
~ ~ ~
~ ~ ~ ~ (v) F 1 1 ~ ~ ~ ~ (vi) F 1 F
~
c
(i) Let F G H then H~ e H e , e , e E ,
(i)
F~ F~
~ G~ (i) F~ ~ ~ ~ (ii) F G
Proposition 1 Let
~ ~ F and G are two GFSS over X , E , then the
(i)
~ ~ and 1
are in
T.
Chakraborty and Mukherjee
(ii) Arbitrary unions of members of T belong to T . (iii) Finite intersections of members of T belong to The triplet
X ,T , E
5
Example 5
T. Let X
is called a generalised fuzzy soft
x1 , x 2 , x3 and E e1 , e2 , e3
topological space (GFST-space, in short) over X , E .
We consider the following GFSS over X , E defined as
Definition 12
~ F e1 x1 \ 0.6, x2 \ 0.5, x3 \ 0.4,0.2, F e2 x1 \ 0.2, x2 \ 0.3, x3 \ 0.8, 0.5, F F e3 x1 \ 0.7, x2 \ 0.4, x3 \ 0.3, 0.6
Let
X ,T , E
members of X ,T , E .
be a GFST-space over X , E , then the
T are said to be a GFS open sets in
~ F
IE
~ G e1 x1 \ 0.3, x2 \ 0.2, x3 \ 0.2,0.1, G e2 x1 \ 0.1, x2 \ 0, x3 \ 0.5, 0.3, G G e3 x1 \ 0.5, x2 \ 0.3, x3 \ 0.3, 0.5
Definition 13 Let
where
X , T , E be a GFST-space over X , E . A GFSS over X , E is said to be a GFS closed in X , T , E ,
if its complement
where
IE.
Let us consider
~ ~ ~ ~ T , 1 , F , G . Then T forms
a GFS topology over X , E .
~ c F belongs to T . Definition 14
Proposition 5
Let
Let X , T , E be a GFST-space over X , E . Let the collection of all GFS -closed sets. Then (i)
~ ~ and 1
are in
T be
X ,T , E
and
X , T1 , E
be two GFST-spaces. If
T belongs to T1 , then T1 is called GFS finer (larger) than T or (equivalently) T is GFS coarser than T1 .
each member of
T
(ii) Arbitrary intersections of members of
T
(iii) Finite unions of members of
T belongs to
T belongs to T
Example 6
x1 , x 2 , x3 and E e1 , e2 , e3 . Consider the GFS topology T over X , E given in
Let X
Example 3.7 Let us
Proof Follows from the definition of GFST-space and DeMorgan’s law for GFSS which is given in Proposition 4.
consider ~ ~ ~ ~ ~ topology T1 , 1 , F , G , H , given in Example 5.
Example 3
~ H e1 x1 \ 0.7, x2 \ 0.6, x3 \ 0.4,0.5, H e2 x1 \ 0.3, x2 \ 0.5, x3 \ 0.8,0.6, . H H e3 x1 \ 0.7, x2 \ 0.8, x3 \ 0.6,0.7
another
~ ~ where F , G
GFS are
~ ~ T , 1 forms a GFS topology over X , E , which is said to be the GFS indiscrete topology over X , E .
Then T1 is finer than
Example 4
Proposition 6
Let T be the collection of all GFSS which can be defined over X , E . Then T forms a GFS topology over X , E ;
X , T2 , E be two GFST-spaces ~ over the same universe X , E , then X , T1 T2 , E is also a GFST-space over X , E .
it is called the GFS discrete topology over X , E .
Let
X , T1 , E
and
T.
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Afr. J. Math. Comput. Sci. Res.
Proof
~T . Now H~
~ ~ ~ ~ , 1 T1 T2 . ~ F i : i , being an index set (ii) Let ~ ~ ~ family of GFSS in T T . Then F T
Thus
(i)
1
~ ~ ~T F i T2 , i , so ~ F~ i 1 i
and ~ ~ ~ and F T . i
2
be a
1
i
i
2
~ ~ ~ ~ F i T1 T2 .
Hence
i
(iii)
~
Let F
~ ~ ~ , G T1 T2 . Then
~ ~ ~ F , G T1
and
~ ~ ~ F , G T2 . ~ ~ ~ ~ ~ ~ ~ ~ Since F G T1 and F G T2 , so ~ ~ ~ ~ ~ ~ T forms a GFS F G T1 T2 . Hence T1 2
topology over X , E .
Remark: The union of two GFS topologies over X , E
may not be a GFS topology over X , E , which follows from the following Example.
Example 7
x1 , x 2 , x3 and E e1 , e2 , e3 ~ ~ ~ ~ ~ ~ Let us consider T1 , 1 , F and T2 , 1 , G ~ ~ be two GFS topologies over X , E , where F , G are Let X
GFSS over X , E defined as follows
~ F e1 x1 \ 0.6, x2 \ 0.4, x3 \ 0.3,0.1, F e2 x1 \ 0.2, x2 \ 0.3, x3 \ 0.5, 0.3, , F F e3 x1 \ 0.7, x2 \ 0.6, x3 \ 0.1, 0.5
where
IE
X , T , E be a GFST-space and F~ be a GFSS over X , E . Then the generalised fuzzy soft closure of
Let
~ ~ F , denoted by F is the intersection of all GFS closed ~ supper sets of F . ~ Clearly, F is the smallest GFS closed set ~ over X , E which contains F . Example 8 Let X x 1 , x 2 , x 3 and E e 1 , e 2 Let us consider the following GFSS over X , E .
~ F F e1 x1 \ 0.7, x2 \ 0.3, x3 \ 0.2,0.4 , F e2 x1 \ 0.6, x2 \ 0.1, x3 \ 0.4, 0.5
~ G G e1 x1 \ 0.6, x2 \ 0.4, x3 \ 0.5,0.5 ,G e2 x1 \ 0.5, x2 \ 0.3, x3 \ 0.1, 0.3 ~ H H e1 x1 \ 0.7, x2 \ 0.4, x3 \ 0.5,0.5 , H e2 x1 \ 0.6, x2 \ 0.3, x3 \ 0.4, 0.5 ~ J J e1 x1 \ 0.6, x2 \ 0.3, x3 \ 0.2,0.4 , J e2 x1 \ 0.5, x2 \ 0.1, x3 \ 0.1, 0.3
Let
~
us
consider
~ ~ ~ ~ ~ T , 1 , F , G , H , J
the GFS over X , E .
topology
Now,
~c c c F F e1 x1 \ 0.3, x2 \ 0.7, x3 \ 0.8,0.6, F e2 x1 \ 0.4, x2 \ 0.9, x3 \ 0.6,0.5
~c c c G G e1 x1 \ 0.4, x2 \ 0.6, x3 \ 0.5,0.5,G e2 x1 \ 0.5, x2 \ 0.7, x3 \ 0.9,0.7
IE
~c c c J J e1 x1 \ 0.4, x2 \ 0.7, x3 \ 0.8,0.6, J e2 x1 \ 0.5, x2 \ 0.9, x3 \ 0.9,0.7
~ ~ ~ ~ Clearly, F c , G c , H c , J c are GFS closed sets. Let us
~ ~ ~ T ~ , ~ T T1 2 1 , F , G ~ ~ ~ ~ Let F G H Now,
Definition 15
~c c c H H e1 x1 \ 0.3, x2 \ 0.6, x3 \ 0.5,0.5, H e2 x1 \ 0.4, x2 \ 0.7, x3 \ 0.6,0.5
~ G e1 x1 \ 0.5, x2 \ 0.2, x3 \ 0.4,0.3,G e2 x1 \ 0.1, x2 \ 0.2, x3 \ 0.3,0.2, , G G e3 x1 \ 0.8, x2 \ 0.5, x3 \ 0,0.4
where
T is not a GFS topology over X , E .
Where ~ H e1 x1 \ 0.6, x2 \ 0.4, x3 \ 0.4,0.3, H e2 x1 \ 0.2, x2 \ 0.3, x3 \ 0.5,0.3, H H e3 x1 \ 0.8, x2 \ 0.6, x3 \ 0.1,0.5
consider the following GFSS over X , E .
~ M M e1 x1 \ 0.4, x2 \ 0.5, x3 \ 0.4,0.6, M e2 x1 \ 0.3, x2 \ 0.6, x3 \ 0.7, 0.3
~ , denoted by ~ is the Then the GFS closure of M M intersection of all GFS closed sets containing M~ . ~ c ~ ~ That is, M~ J~ c 1 J .
Chakraborty and Mukherjee
Theorem 1
X , T , E be a GFST-space. Let GFSS over X , E . Then Let
~
~ ~ F and G are
~ ~ ~ G G .
So
~ ~ ~ ~ ~~ ~ ~ ~~ G is a GFS closed F G F G , by (5), F
~ ~~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ F G F G . Thus, F G F G . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (7) Since F G F and F G G , so by (5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ F G F and F G G . Thus ~ ~ ~ ~ ~ ~ ~ F G F G . Then
~ ~ (1) , 1 1 ~ ~ ~ (2) F F ~ ~ ~ (3) F is GFS closed if and only if F F ~ ~ (4) F F ~ ~ ~ ~ ~ ~ (5) F G F G ~ ~ ~ ~ ~ ~ (6) F G F G
and
set over X , E being the union of two GFS closed sets.
~
~ ~ ~ F as F
Conversely,
7
Definition 16
~
Let F
~ ~ ~ ~ ~ ~ ~ (7) F G F G
that
e
be
x
GFSS
over X , E .
We
say
~ F~ read as e , e belongs to the GFSS , e x
~ F
Proof
a
F e x 0 1
if
and
F e y 0 , y X \ x , e .
(1) and (2) are obvious. (3)
~
Let F
be
a
GFS
closed
set.
By
(2)
we
~ ~ ~ ~ have F F . Since F is the smallest GFS closed set over
X , E
~ contains F ,
which
~ ~ ~ then F F .
~ ~ Hence F F .
Proposition 7
~ F can be expressed as the union
Every non-null GFSS
of all the generalised fuzzy soft points which belong
~
to F .
~
~
~
Conversely, let F F . Since F is a GFS closed
~ F is a GFS closed set over X , E . ~ (4) Since F is a GFS closed set therefore, by (3) we ~ ~ have F F . ~ ~ ~ (5) Let F G . Then every GFS closed super set set, then
~ ~ of G will also contain F . That is, every GFS closed ~ ~ super set of G is also a GFS closed super set of F . ~ Hence the intersection of GFS closed super sets of F is
Proof: Obvious.
Definition 17 A GFSS
~ F in a GFST-space X , T , E is called a
generalised fuzzy soft neighbourhood [GFS-nbd, in short]
~
~
of the GFSS G if there exists a GFS open set H such that
~ ~ ~ ~ ~ G H F .
contained in the GFS intersection of GFS closed super Definition 18
~ ~ ~ ~ G . set of G . Thus F
~ ~ ~ ~ ~ ~ F G and G ~ ~ ~ ~ ~ ~ F F G and G
(6) Since F
~ ~~ ~ ~ ~~ F G F G
.
~ ~G ~ F~ , so by (5) ~ ~G ~ F~ Thus, .
A GFSS
~ F in a GFST-space X , T , E is called a
generalised fuzzy soft neighbourhood of the generalised
~~ 1 if there exists a GFS open fuzzy soft point e x , e
~
set G such that
e
x
~ ~ ~ ~G , e F .
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Afr. J. Math. Comput. Sci. Res.
Proposition 8
over X , E . The generalised fuzzy soft interior of F ,
~
Let X , T , E be a GFST-space over X , E . Then
~0 F is the union of all generalised fuzzy soft ~ open subsets of F . ~0 Clearly, F is the largest generalised fuzzy soft open ~ set over X , E which contained in F . denoted by
~~ (1) Each e x , e 1 has a GFS nbd;
~ ~ ~~ (2) If F and G are GFS nbd of some e x , e 1 , ~ ~ ~ then F G is also a GFS nbd of e x , e . ~ ~ ~ ~ ~~ (3) If F is a GFS nbd of e x , e 1 and F G , ~ then G is also a GFS nbd of e x , e .
Example 9
x1 , x 2 , x3 and E e1 , e2 .
Let X
Let us consider the following GFSS over X , E
Proof (1) is obvious.
~ ~ F and G be the GFS neighbourhoods of ex , e ~ ~1 , then there exist H~ , N~ ~ T such that ex , e ~ H~ ~ F~ and ex , e ~ N~ ~ G~ . Now ex , e ~ H~ and ex , e ~ N~ implies that ex , e ~ H~ ~ N~ and H~ ~ N~ ~ T ; so we ~ ~ ~ ~~ ~~ ~~ ~~ have ex , e H N F G . Thus F G (2) Let
is a GFS neighbourhood of
e
x
, e .
~ ~~ 1 and (3) Let F be a GFS neighbourhood of e x , e ~ ~ ~ ~ F G . By definition there exists a GFS open set H such that e , e ~ H~ ~ F~ ~ G~ .
x
~ ~ ~ ~H Thus e x , e G . Hence neighbourhood of
e
x
~ G
is
a
GFS
, e .
~ F F e1 x1 \ 0.7, x2 \ 0.3, x3 \ 0.2,0.4 , F e2 x1 \ 0.6, x2 \ 0.1, x3 \ 0.4, 0.5 ~ G G e1 x1 \ 0.6, x2 \ 0.4, x3 \ 0.5,0.5 , G e2 x1 \ 0.5, x2 \ 0.3, x3 \ 0.1, 0.3
~ H H e1 x1 \ 0.7, x2 \ 0.4, x3 \ 0.5,0.5 , H e2 x1 \ 0.6, x2 \ 0.3, x3 \ 0.4, 0.5 ~ J J e1 x1 \ 0.6, x2 \ 0.3, x3 \ 0.2,0.4 , J e2 x1 \ 0.5, x2 \ 0.1, x3 \ 0.1, 0.3
Let
us
consider
the
GFS
topology
~ ~ ~ ~ ~ ~ T , 1 , F , G , H , J over X , E .
Let us consider the following GFSS over X , E .
~ N N e1 x1 \ 0.6, x2 \ 0.5, x3 \ 0.4,0.5, N e2 x1 \ 0.6, x2 \ 0.3, x3 \ 0.8,0.7
~
Then the generalised fuzzy soft interior of N , denoted
~
by N
0
, is the union of all generalised fuzzy soft open
~
subsets of N . That is,
~ 0 ~ ~ ~ ~ N J J .
Definition 19
~ Let X , T , E be a GFST-space. Let F be a GFSS over X , E and
~ ~ . Then e , e is said to ex , e 1 x
~
be a generalised fuzzy soft interior point of F if there
~ ~ ~ ~ ~G exists a GFS open set G such that e x , e F .
Theorem 2
X , T , E be a GFST-space. Let GFSS over X , E . Then Let
~ ~ , ~1 ~ ~ ~ (2) F F
(1)
0
0
~ ~ F and G are
~ 1
0
Definition 20 Let X , T , E be a GFST-space. Let
~ F be a GFSS
~ ~ (3) F is GFS open if and only if F ~0 0 ~0 (4) F F
0
~ F
Chakraborty and Mukherjee
~ ~ ~ ~0 ~ ~ 0 F G F G ~0 ~ ~ 0 ~ ~ ~ (6) F G F G ~0 ~ ~ 0 ~ ~ ~ ~ (7) F G F G (5)
~ G~ c | G~ ~ c F
c
0
is GFS closed
~ set and G
~ F~ c
c
9
0
Definition 21 Let X , T , E be a GFST-space. A subfamily of
Proof: Straightforward.
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 .
Theorem 3
~ A GFSS F over X , E is a GFS open set if and only if ~ ~ ~ F is a GFS nbd of each GFSS G contained in F .
Lemma: Let X , T , E be a GFST-space and be a GFS base for
T over
X ,T , E .
Then
T equals the
collection of all unions of elements of .
Proof: Obvious.
Proof
~ ~ F is a GFS open and G is any GFSS ~ ~ ~ ~ ~ ~ contained in F . We have G F F , it follows ~ ~ that F is a neighbourhood of G . ~ Conversely, suppose F is a neighbourhood for every ~ ~ ~ GFSS contained it. Since F F there exist a GFS ~ ~ ~ ~ ~ ~ open set H such that F H F . First suppose
Proposition 9
~ T . Then is Let X , T , E be a GFST-space and
~~ a GFS base for T if and only if for any e x , e 1 and
~ any GFS open set G containing e x , e , there exists
~ ~ ~ R~ and R~ ~ G~ . R such that e x , e
~ ~ ~ Hence F H and F is GFS open.
Proof: Straightforward.
GENERALISED FUZZY SOFT COMPACT SPACES Theorem 4
~ Let X , T , E be a GFST-space and F be a GFSS over X , E . Then
~ F~ F~ F c
(1) F (2)
0
c
c
0
c
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
Proof
~ G~
(1) F~
c
=
~ G~ ~ = F c
c
~ ~ ~ ~ | G is GFS closed set and F G
~ c ~ c ~ ~c | G is GFS open set and G F
c
~ ~ ~ ~ ~ ~ G | G is GFS open set and G F
A family of generalised fuzzy soft sets is a cover of a ~ ~ generalised fuzzy soft set F iff F is contained in the union
of
c
members
~ ~~ ~ F G
0
~0 (2) F
c
~ : G
of
,
.
That
is,
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.
10
Afr. J. Math. Comput. Sci. Res.
Definition 23 Let ( X , T , E ) be a generalised fuzzy soft topological ~ space and F be a generalised fuzzy soft set, over ( X , E ) is called generalised fuzzy soft compact ( henceforth GFS compact) if each generalised fuzzy soft ~ open cover of F has a finite sub cover. Also,
~ ~ ~ G 1 : ~ ~ ~ G : Thus,
G~ c
c
:
~ c ~ G
: is a collection of GFS closed
generalised fuzzy soft topological space ( X , T , E ) is called generalised fuzzy soft compact if each generalised fuzzy
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 ~ c such that G : 1,2,......n
soft open cover of
~ G c : 1,2,......n ~ G : 1,2,....n
~ 1 has a finite sub cover.
~
~
~
~
: 1,2,.....n.
~ ~
This implies that 1 G
Hence ( X , T , E ) is GFS compact.
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.
Conclusion
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.
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.
Conflict of Interest The author(s) have not declared any conflict of interest.
Proof Let
c
Definition 24
Theorem 5
.
( X , T , E ) be a GFS compact space and let ~ G : be a collection of GFS closed
ACKNOWLEDGEMENT
subsets of ( X , T , E ) with the FIP and suppose, if possible, ~ ~ ~ ~ ~ G ~ G : . Then : c 1 or
The first author is thankful to University Grants Commission, New Delhi, for sponsoring this work under grants of Minor Research Project.
~ ~ G
~ : 1 by De-Morgan law. This ~ ~ means that G : is open cover of 1 , ~ since G : are GFS closed. Since ( X , T , E ) is
c
c
~ : ~1
n
~ G c GFS compact, we have that
:
~ ~n G so by De-Morgan law implies that contradicts
~ ~ G
~ ~ G n
1
the
~
Hence
: .
~ Conversely, let G
~ cover of 1 so that
~ : .
FIP.
1
1
:
~ 1 which
However, this
we
c
and
must
have
be a GFS open
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