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 Total Downloads : 593
 Authors : C. Janaki, Jeyanthi. V
 Paper ID : IJERTV3IS20018
 Volume & Issue : Volume 03, Issue 02 (February 2014)
 Published (First Online): 20022014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
on Soft GrContinuous Functions in Soft Topological Spaces
1.C. Janaki, 2. Jeyanthi. V
1. Asst Professor, Department of Mathematics, L.R.G. Government College for Women, Tirupur4.
2. Asst Professor, Department of Mathematics, Sree Narayana Guru College, Coimbatore105.
Abstract: The purpose of this paper is to study soft gr continuous function and soft grirresolute function in soft topological spaces. Also,we introduce the concepts such as soft
grclosure,soft grinterior and exhibit some results related to soft gr continuity,soft gropen map and soft grclosed map
. Further, we study the relation between soft grcontinuous function and other soft continuous functions.
Key Words: Soft Set , Soft Topology, Soft Continuity , Soft Point, Soft grclosure, Soft grinterior, Soft Function, Soft
grcontinuity, Soft grirresolute, soft gr neighborhood.
Mathematics Subject Classification: 06D72, 54A40.

INTRODUCTION.
Molodtsov[16] introduced the concept of soft set as a new mathematical tool.
Furthermore, D.Pei and D.Miao [19] showed that soft sets are a class of special information systems. M.Shabir and M.Naz [20] introduced soft topological spaces. D.N.Georgiou and A.C. Megaritis[6], Soft Set Theory and Topology, Applied General Topology, 14, (2013) for the soft set theory, new definitions , examples , new classes of soft sets and properties for mappings between different classes of soft sets are introduced and studied. Levine[13] Introduced gclosed sets in general topology. Kannan [11] introduced soft gclosed sets in soft topological spaces, The concept of closed sets in topological spaces was initiated by Zaitsav[23] and the concept of gclosed set was introduced by Noiri and Dontchev[4]. N.Palaniappan[17]

PRELIMINARIES.
Definition 2.1 ([15],[16],[22])
Let U be the initial universe and P(U) denote the power set of U. Let E denote the set of all parameters. Let A be a non empty subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given by F: AP(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For A, F() may be considered as the set – approximate elements of the soft set (F,A).Clearly, a soft set is not a set.
For two soft sets (F,A) and (G,B) over the common universe U, we say that (F,A) is a soft subset of (G,B) if (i)AB and (ii) for all eA, F(e) and G(e) are identical
approximations. We write (F,A) ~ (G,B). (F,A) is said to be
a soft superset of (G,B) , if (G,B) is a soft subset of (F,A). Two soft sets (F,A) and (G,B) over a common universe U are said to be soft equal if (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A).
Definition:2.2([15],[16],[22])
For a soft set (F,A) over the universe U, the relative complement of (F,A) is denoted by (F,A) and is defined by (F,A)=(F,A), where F: A P(U) is a mapping defined by F(e)= U F(e) for all eA.
Definition :2.3([15],[16],[22])
A Soft set (F,A) over X is said to be a Null soft set denoted by or A if for all eA, F(e)=(null set).
Definition:2.4([15],[16],[22])
A soft set (F,A) over X is said to be absolute soft set denoted by or XA if for all eA, F(e) =X. Clearly, we have
= Aand =XA.
studied and introduced regular closed sets in topological
spaces. Jeyanthi.V and Janaki.C [8]introduced grclosed sets in topological spaces. Soft semiopen sets and its properties were introduced and studied by Bin Chen[2]. Kharal et al.[12]introduced soft function over classes of soft sets. Cigdem Gunduz Aras et al., [3] in 2013 Studied and discussed the properties of Soft Continuous mappings
.Recently, we introduced Soft grclosed sets [9]in soft topological spaces. The concept of gclosed,gp closed,gsclosed,*gclosed,gbclosed sets in topological spaces was introduced by C.Janaki[7], Park. J.H[18], Aslim [1], Ganes M Pandya [5] and Sreeja and Janaki[21].The purpose of this paper is to study soft grcontinuity on soft topological spaces and investigate its properties.
Definition :2.5 ([15],[16],[22])
The union of two soft sets of (F,A) and (G,B) over the common universe U is soft set (H,C), where C = B and for all eC, H(e) = F(e) if e AB, H(e) = G(e) if e BA and H(e)=F(e) G(e) if e B. We write (F,A) (G,B)= (H,C).
Definition:2.6([15],[16],[22])
The Intersection (H,C) of two soft sets (F,A) and (G,B) over a common universe U denoted (G,B) is defined as C= B and H(e) = F(e) G(e) for all eC.
Definition:2.7 ([2],[11],[15])
Let be the collection of soft sets over X, then is called a soft topology on X if satisfies the following axioms:
X

, ~ belong to .

The union of any number of soft sets in belongs to .

The intersection of any two soft sets in belongs to
.
The triplet (X, , E) is called a soft topological space over

For simplicity, we can take the soft topological spaces (X,1,E) as X and (Y,2,E) as Y throughout this work respectively. Let us denote the collection of soft sets over the universe X and Y as SS(X) and SS(Y) respectively. Definition:2.8 ([2],[11],[15])
Let (X, , E) be soft topological space over X. A soft set (F,E) over X is said to be soft closed in X, if its relative complement (F,E) belongs to . The relative complement is a mapping F:EP(X) defined by F(e)= XF(e) for all eA.
Definition:2.9 ([2],[11],[15])
Let (X,, E) be a soft topological space over X and the soft interior of (F,E) denoted by sint(F,E) is the union of all soft open subsets of (F,E). Clearly, (F,E) is the largest soft open set over X which is contained in (F,E).The soft closure of (F,E) denoted by scl(F,E) is the intersection of all closed sets containing (F,E). Clearly, (F,E) is smallest soft closed
The soft regular open set of X is denoted by SRO(X) or SRO(X,,E).
Definition:2.13([11])
A soft topological space X is called a soft T1/2space if every soft gclosed set is soft closed in X.
Definition:2.14([9])
The soft regular closure of (A,E) is the intersection of all soft regular closed sets containing (A,E). (i.e)The smallest soft regular closed set containing (A, E) and is denoted by srcl(A,E).The soft regular interior of (A,E ) is the union of all soft regular open sets contained in (A,E) and is denoted by srint(A,E).
Similarly , we define soft closure, soft preclosure, soft semi closure and soft bclosure of the soft set (A,E) of a topological space X and are denoted by scl(A,E), spcl(A,E), sscl(A,E) and sbcl(A,E) respectively.
Definition:2.15 ([10],[14])
Let (F,E) be a soft set over X. The soft set (F,E) is called a soft point denoted by eF if for the element eE , F(e) and F(e)= for all eE{e}.
Definition:2.16 ([3])
A soft set (G,E) in a soft topological space (X,,E) is called a soft neighborhood of the soft point eF if there exists a soft
open set (H,E) such that eF(H,E) ~ (G,E).
set containing (F,E).
~ A soft set (G,E) in a soft topological space (X,,E) is called
sint (F,E) { (O,E): (O,E) is soft open and (O,E)
(F,E)}.
scl(F,E) = { (O,E): (O,E) is soft closed and (F,E) ~
(O,E)}.
Definition:2.10[3]
Let (X,,E) be a soft topological space over X, (G,E) be a soft set over X and xX. Then (G,E) is said to be a soft neighborhood of x, if there exists a soft open set (F,E) such
that x(F,E) ~ (G,E).
Definition:211 [24]
Let (X,,E) be a soft topological space over X, (G,E) be a soft set over X and xX. Then (G,E) is said to be a soft interior point of (G,E), if there exists a soft open set (F,E)
such that x(F,E) ~ (G,E).
Definition:2.12
A soft subset (A,E) of X is called

a soft generalized closed (Soft gclosed)[11] in a soft
topological space (X,,E) if scl(A,E) ~ (U,E) whenever
a soft neighborhood of the soft set (F,E) if there exists an soft open set (H,E) such that (F,E)(H,E) ~ (G,E).
The neighborhood system of the soft point eF denoted by
N(eF) , is the family of all its neighborhoods.
Definition:2.17([3])
Let (X,,E) be a soft topological space over X, (G,E) be a soft set over X and xX. Then (G,E) is said to be a soft interior point of (G,E), if there exists a soft open set (F,E)
such that x(F,E) ~ (G,E).
Definition:2.18 ([10],[14] )
Let SS(X)A and SS(Y)B be two soft classes. Then u: XY and p:AB be two functions. Then the function fpu: SS(X)E
SS(Y)B and its inverse are defined as

Let (F,A) be a soft set in SS(X)A. The image of (F,A) under fpu, written as fpu((F,A)) = (fpu(F),p(A))is a soft set in SS(Y)B such that
1
fpu(F)(y) = u(F (x)), p ( y) A
(A,E) ~ (U,E) and (U,E) is soft open in X.

a soft semi open [ 3]if (A,E) ~ sint(scl(A,E))

(iii))a soft regular open if (A,E)= sint(scl(A,E)).

a soft open if (A,E) ~ sint(scl(sint(A,E)))
xp1 ( y ) A
, otherwise. for all yB.

a soft bopen if (A,E) ~ scl(sint(A,E)) sint(s
(ii) Let (G,B) be a soft set in SS(Y)B. Then the inverse image of (G,B) under fpu, written as 1 ((G,B))=
cl(A,E))

a soft preopen set if (A,E)
~ sint(scl(A,E)).
(1 (G),p1(B)) is a soft set in SS(X)A such that
1 (G)(x)= 1 (G(p(x))), p(x)B,

a soft clopen if (A,E) is both soft open and soft closed. The complement of the soft semi open , soft regular open , soft open, soft bopen , soft preopen sets are their respective soft semi closed , soft regular closed , soft – closed , soft bclosed and soft pre closed sets.
The finite union of soft regular open sets is called soft – open set and its complement is soft closed set.
, otherwise. for all xA.
Definition:2.19([10],[14])
Let (X,,A) and (Y,*,B) be soft topological spaces and fpu : SS(X)ASS(Y)B be a function. Then function fpu is called soft continuous if 1 ((G,B)) for all (G,B)*.
Definition:2.20 [10]
Let (X,,A) and (Y,*,B) be soft topological spaces and fpu:SS(X)ASS(Y)B be a function. Then the function fpu is called soft open mapping if fpu((G,A)) * for all (G,A). Similarly, a function fpu:SS(X)ASS(Y)B is called a soft closed map if for a closed set (F,A) in , the image fpu((G,B)) is soft closed in *.
Definition:2.21([9])
A soft subset (G,A) of a soft topological space X is called a soft grclosed set in X if srcl(G,A) ~ (X,A)) whenever (G,A) ~ (X,A), where (X,A) is soft – open in X. We
denote the soft grclosed set of X by SGRC(X) . The
complement of soft grclosed is soft gropen set and is denoted by SGRO(X).
Lemma:3.5
Let (F,A) and (G,A) be soft subsets of a soft topological space X. Then
(i)sgrint(X) =X, sgrint(A) = A (ii)sgrint((F,A)) ~ (F,A)

If (F,A) is any soft gropen set contained in (G,A), then ((F,A) ~ sgrint(G,A)

If (F,A) ~ (G,A) , then sgrint((F,A)) ~ sgr
int(G,A)

sgrint(sgrint((F,A)))= sgrint((F,A)).
Proof: Straight Forward.
Theorem:3.6
If a subset (F,A) of a soft topological space X is soft gr open, then sgrint((F,A)) =(F,A).
Proof: Obvious.
Theorem:3.7
If (F,A) and (G,A) are soft subsets of a soft topological
space X, then sgrint((F,A)) sgrint((G,A)) ~ sgr
III. SOFT GRCLOSURE AND SOFT GR INTERIOR.
int ((F,A) (G,A))
Proof: We know that (F,A) ~
~ (F,A) (G,A)
(F,A) (G,A) and (G,A)
Let us introduce the following definitions.
Definition:3.1
Let (X,,A) be a soft topological space over X, (G,A) be a soft set over X and xX. Then (G,A) is said to be a soft gr
neighborhood of x, if there exists a soft gropen set (F,A) such that x(F,A) ~ (G,A).The soft grneighborhood of a
point x is denoted by sgrnbd(x).
Definition:3.2
The Soft grClosure of a soft set (G,A) is defined to be the intersection of all soft grclosed sets containing the soft set (G,A) and is denoted by sgrcl(G,A).
The Soft grInterior of a soft set (G,A) is defined to be the union of all soft gropen sets contained the soft set (G,A) and is denoted by sgrint(G,A).
Lemma :3.3
Let (F1,A) and (F2,A) be subsets of (X,,A) . Then

sgrcl (A) = A and sgrcl(X) =X.

If (F1,A) ~ (F2,A), then sgrcl(F1,A) ~
sgrcl(F2,B)

(F1,A) ~ sgrcl(F1,A)
Then sgrint((F,A)) ~ sgrint ((F,A) (G,A)), sgr
int((G,A)) ~ sgrint ((F,A) (G,A)
Hence sgrint ((F,A)) sgrint((G,A)) ~ sgrint ((F,A) (G,A)).
Theorem:3.8
If (F,A) and (G,A) are soft subsets of a space X, then sgr int ((F,A)(G,A))= sgrint ((F,A) sgrint ((G,A))
Proof: We know that (F,A)(G,A ) ~ (F,A),
(F,A)(G,A) ~ (G,A).Then sgrint ((F,A) (G,A)) ~
sgrint((F,A)) and sgrint ((F,A)(G,A)) ~ sgr
int((G,A)). Hence sgrint ((F,A)(G,A))= sgrint
((F,A)) sgrint ((G,A))—————(*)
Again , let x sgrint ((F,A)) sgrint ((G,A)). Then x sgrint ((F,A)) and x sgrint
((G,A)).Hence x is a soft grinterior point of each of sets (F,A) and (G,A). It follows that ((F,A) and (G,A) are sgr nbds of x, so that their intersection (F,A)(G,A) is also a
sgrnbd of x . Hence x sgrint ((F,A)(G,A))
Thus, xsgrint ((F,A)) sgrint ((G,A)) x sgr
int ((F,A)(G,A))
~

sgrcl(F1,A)= sgrcl( sgrcl(F1,A))

sgrcl((F1,A) (F2,B))= sgrcl(F1,A)

sgrcl(F2,B)
Therefore, sgrint ((F,A)) sgrint ((G,A))
((F,A)(G,A))—(**)
From (*) and (**) ,
sgrint
Proof:Obvious. Lemma: 3.4
Let (F,A) and (G,A) be soft subsets of the soft topological space X. Then sgrcl ((F,A) (G,A)) ~ sgrcl((F,A))
sgrcl((G,A))
Proof: Since (F,A)(G,A) ~ (F,A), (G,A)
sgrcl((F,A)(G,A)) ~ sgrcl((F,A)) and sgr
cl((F,A)(G,A)) ~ sgrcl((G,A))
sgrcl((F,A)(G,A)) ~ sgrcl(F,A) sgrcl(G,A)
sgrint ((F,A)(G,A))= sgrint ((F,A)) sgrint ((G,A)).
Lemma:3.9
If (F,A) be a soft subset of a soft topological space X, then (X sgrint((F,A)) = sgrcl(X(F,A)).
Proof: Let xX (sgrint((F,A))). Then x sgr
int((F,A)). That is every soft gropen set (G,A) containing x is such that (G,A) ~ (F,A).Every soft gropen set (G,A)
containing x intersects X(F,A).
x sgrcl(X(F,A))
Hence (X sgrint((F,A))) ~ sgrcl(X(F,A)). Conversely, let xsgrcl(X(F,A)). Thus every soft gr
open set (H,A) containing x intersects X(F,A) (i.e) is every
soft gropen set (H,A) containing x is such that (H,A) ~ (F,A).
x sgrint((F,A)). Thus sgrcl(X(F,A)) ~ (X
sgrint((F,A))) and hence
(X sgrint((F,A))) = sgrcl(X(F,A)).
Remark:3.10
If (F,A) be a soft subset of a soft topological space X, then (X sgrcl((F,A))) = sgrint(X(F,A)).
IV. SOFT GRCONTINUOUS FUNCTIONS.
Definition:4.1
Let (X,,A) and (Y,*,B) be soft topological spaces and fpu:SS(X)ASS(Y)B be a function. Thn the function fpu is called

soft grcontinuous if 1((G,B)) is soft gr closed in (X,,A) for every soft closed set (G,B) in (Y,*,B).

soft grirresolute if 1 ((G,B)) is soft grclosed in (X,,A) for every soft gr closed set (G,B) in (Y,*,B).

soft regular continuous if 1 ((G,B)) is soft regular closed in (X,,A) for every soft closed set
(G,B) in (Y,*,B).
G1(e1) =, G1(e2) = {a}
G2(e1)= {a}, G2(e2)=Y.
Then 2={, Y, (G1,E), (G2,E)} be a soft topology on Y. Let f: XY be a function defined as f(p)=f(p)={a} and f(p)={b}.
Here the inverse image of the soft open sets (G1,E) and (G2,E) in Y are soft gropen in X but not soft open in X. Hence soft grcontinuity need not be soft continuity.
(ii) Let X={a,b,c,d}=Y, E={e1,e2}. Let F1,F2,, F6 are functions from E to P(X) and are defined as follows:
F1(e1)={c}, F1(e2)={a},
F2(e1)={d}, F2(e2)={b},
F3(e1)={c,d}, F3(e2)={a,b},
F4(e1)={a,d}, F4(e2)={b,d},
F5(e1)={b,c,d}, F5(e2)={a,b,c},
F6(e1)={a,c,d}, F6(e2)={a,b,d}.

soft R map if for a soft regular closed set (G,B) in
Then
~
={, ,(F ,E),,(F ,E)} is a soft topology and
1 X 1 6
* , 1 ((G,B)) is soft regular closed in .
Definition:4.2
Let (X,,A) and (Y,*,B) be soft topological spaces and fpu:SS(X)ASS(Y)B be a function. Then

The function fpu is called soft gropen if fpu((G,A)) is soft gropen in (Y,*,B) for every soft open set (G,A) in (X,,A).

The function fpu is called soft grclosed if fpu((G,A)) is soft grclosed in (Y,*,B) for every soft closed set (G,A) in (X,,A).
Remark:4.3
Soft grcontinuity and soft continuity are independent concepts.
Example:4.4
(i)Let X={p, p,p}, Y= {a,b}, E={e1,e2}. Let F1, F2, F3 are
functions defined from E to P(X) as follows: F1(e1)={p,p}, F2(e2={p}
F2(e1)=X, F2(e2)= {p,p}
F3(e1) ={p}, F3(e2)={p}.
Then 1= {, X, (F1,E),(F2,E),(F3,E)} is soft topology on X.
Let G1, G2 are functions from E to P(Y) and are defined as follows:
elements in are soft open sets.
Let G1, G2 ,G3 are functions from E to P(Y) and are defined as follows:
G1(e1) ={a}, G1(e2) = {d},
G2(e1)= {b}, G2(e2)={c},
G3(e1)={a,b},G3(e2)={c,d},
G4(e1)={b,c,d},G4(e2)={a,b,c}.
Then 2={, Y, (G1,E), (G2,E),(G3,E),(G4,E)} be a soft
topology on Y. Let f: XY be an identity map. Here the inverse image the soft open set (G4,E)={{b,c,d},{a,b,c}} in Y is not soft gropen in X but soft open in X. Hence soft continuous function need not be soft grcontinuous.
Theorem:4.5
Every soft regular continuous function is soft gr continuous but not conversely.
Proof: Straight forward.
Example :4.6
In example 4.4(i), The inverse image of the soft open sets (G1,E) and (G2,E) in Y are soft gropen in X but not soft regular open in X. Hence soft grcontinuity need not be soft regular continuous.
Theorem:4.7
Every soft grcontinuous function is soft rgcontinuous, soft gcontinuous, soft *gcontinuous, soft g continuous, soft gpcontinuous, soft gscontinuous and soft gbcontinuous .
Proof: Follows from the definitions.
Remark: 4.8
The converse of the above need not be true and is shown in the following example.
Example:4.9
In example 4.4(ii), the inverse image the soft open set (G4,E)={{b,c,d},{a,b,c}} in Y is soft not gropen in X but soft rgopen, soft gopen, soft gsopen, soft gopen, soft *gopen, soft gpopen and soft gb open in X. Hence soft rgcontinuous, soft gcontinuous, soft gs continuous, soft gcontinuous, soft *gcontinuous, soft
gpcontinuous and soft gbcontinuous but not soft gr continuous.
Remark:4.10
Soft grcontinuity and soft grirresolute are independent concepts.
Example:4.11

Let X={a,b,c,d}=Y, E={e1,e2}. Let F1,F2,, F6 are functions from E to P(X) and are defined as follows:
F1(e1)={c}, F1(e2)={a},
F2(e1)={d}, F2(e2)={b},
F3(e1)={c,d}, F3(e2)={a,b},
F4(e1)={a,d}, F4(e2)={b,d},
F5(e1)={b,c,d}, F5(e2)={a,b,c},
F6(e1)={a,c,d}, F6(e2)={a,b,d}.
F2(e1)={b}, F2(e2)={a},
F3(e1)= X, F3(e2)={a}
~
Then 1 ={, X ,(F1,E),(F2,E),(F3,E)} is a soft topology on
X and elements in are soft open sets of X.
Let G1, G2 ,G3 ,G4,G5 and G6 are functions from E to P(Y) and are defined as follows:
G1(e1) ={a}, G1(e2) = {b},
G2(e1)={b,c},G2(e2)={a},
G3(e1)=X ,G3(e2)={a,b},
G4(e1)={a,b},G4(e2)={b,c}, G5(e1)={b},G5(e2)=,
G6(e1)={a,b},G6(e2)={b}.
Then 2={, Y, (G1,E), (G2,E),(G3,E),(G4,E),(G5,E),(G6,E)}
be a soft topology on Y. Let f: XY be an identity map. Here the inverse image of the soft gropen sets in Y are soft gropen in X, but the inverse image of soft open set (G4,E)={{a,b},{b,c}} is not soft gropen in X. Hence soft
grirresoluteness need not be soft grcontinuous.
Remark:4.12
The above discussions are represented diagrammatically as follows:
2
1
5 7 9
~
Then 1 ={, X ,(F1,E),,(F6,E)} is a soft topology and
elements in are soft open sets.
4 10
Let G1, G2 ,G3 are functions from E to P(Y) and are defined as follows:
6
3

Soft continuous
8 11
G1(e1) ={a}, G1(e2) = {d},
G2(e1)={b,c,d},G2(e2)={a,b,c}.
Then 2={, Y, (G1,E), (G2,E)} be a soft topology on Y. Let f: XY be an identity map. Here the inverse image of the soft open set in Y is soft gropen in X, but the inverse image of soft gropen sets in Y are not soft gropen in X. Hence soft grcontinuous function need not be soft gr irresolute.


Let X={a,b,c}=Y, E={e1,e2}. Let F1, F2, F3 are functions from E to P(X) and are defined as follows:
F1(e1)={a,c}, F1(e2)={},

soft regular continuous 3soft grirresolute
4soft grcontinuous 5soft gcontinuous 6soft rgcontinuous 7soft *gcontinuous 8soft gpcontinuous 9soft gcontinuous
10soft gbcontinuous 11soft gscontinuous.
Theorem:4.13
A soft function fpu: SS(X)ASS(Y)B is soft grcontinuous
when f (sgrcl(F,A)) ~ scl(fpu((F,A))) for every soft set
Proof: Let fpu: SS(X)ASS(Y)B is soft gropen map and (F,A) be any soft set of (X,,A).Then sint (F,A) is soft open in (X,,A). Since f is soft gropen map, fpu(sint(F,A))
is soft gropen in (Y,*,B). We have f (sint (F,A) ) ~
pu
(F,A) of a soft topological space (X,,A).
Proof: Let fpu: SS(X)ASS(Y)B be a soft grcontinuous
pu
fpu((F,A)) for every soft set (F,A) of a soft topological
space (X,,A). Also, fpu(sint(F,A)) = sgrint(fpu(sint(F,A
)) Hence f (sint(F,A)) ~ sgrint(f ((F,A))).
function. Now, scl(f ((F,A))) is a soft closed set of pu pu
pu
(Y,*,B). By the soft grcontinuity of fpu,
1
(s
Theorem:4.17
cl(fpu((F,A))) is soft grclosed and (F,A)
~
(s
pu pu
cl(fpu((F,A))).But sgrcl (F,A) is the smallest soft gr closed set containing (F,A), hence sgrcl(F,A) ~ 1 (s cl(f ((F,A))). The above implies f (sgrcl(F,A)) ~ s
cl(fpu((F,A))).
Theorem:4.14
A soft function fpu: SS(X)ASS(Y)B is soft grcontinuous
Let (X,,A) be a soft grT1/2space fpu: SS(X)ASS(Y)B be a soft function. Then fpu is soft grcontinuous iff fpu is soft regular continuous.
Proof: Let fpu be a soft grcontinuous function. Then
1((G,B)) is soft grclosed in (X,,A) for every soft closed set (G,B) of (Y,*,B). Since (X,,A) is a soft gr T1/2space, every soft grclosed set is soft reglar closed .
Hence 1 ((G,B)) is soft regular closed in (X,,A) for
when
1 ~
(sint(G,B))
sgrint(1
((G,B)) for every
every soft closed set (G,B) in (Y,*,B) and hence fpu is soft
soft set (G,B) of a soft topological space (Y,*,B).
Proof :Let fpu: SS(X)ASS(Y)B be a soft grcontinuous function. Now, sint(f ((G,B))) is a soft open set of
regular continuous.
Let fpu be a soft regular continuous function in (X,,A) . Then 1((G,B)) is soft regular closed in (X,,A) for every
pu
(Y,*,B), so by soft grcontinuity of fpu ,
1(s
1
soft closed set (G,B) in (Y,*,B). Since every soft regular closed set is soft grclosed. Then 1((G,B)) is soft gr
int(fpu((G,B))) is soft gropen in (X,,A) and (s
int((G,B)))) ~ (G,B). As sgrint(G,B) is the largest soft
gropen set contained in (G,B), 1 (sint(G,B)) ~ sgr
int(1 ((G,B)) for every soft set (G,B) of a soft topological space (Y,*,B).
Theorem:4.15
closed in (X,,A) for every soft closed set (G,B) of (Y,*,B) and hence fpu is soft grcontinuous.
Theorem:4.18
A soft function fpu: SS(X)ASS(Y)B is soft grirresolute, then
A soft function fpu: SS(X)ASS(Y)B is soft grclosed if

f
(sgrcl(F,A)) ~ sgrcl(fpu((F,A))) for
s ~ f
pu
(scl(F,A)) for every soft set (F,A)
grcl(fpu((F,A))) pu
every soft set (F,A) of (X,,A).
of a soft topological space (X,,A).
(ii) sgrcl( 1(G,B))) ~ 1(sgrcl(G,B)) for
~
pu pu
Proof: Suppose that f is soft grclosed and (F,A) is any soft set of (X,,A). We have, f ((F,A)) ~ f (scl(F,A)).
Now,sgrcl(fpu((F,A))) sgrcl(fpu(scl(F,A) ) ).
every soft set (G,B) of (Y,*,B).
Proof :(i) For every soft set (F,A) of (X,,A), sgr cl(fpu((F,A))) is soft grclosed in (Y, *,B). By hypothesis ,
1(
Since fpu(scl(F,A)) is soft grclosed in (Y,*,B), sgr
sgrcl(fpu((F,A)))) is soft grclosed in (X,,A).Also ,
(F,A) = 1 (fpu((F,A))) ~ 1 (sgrcl(fpu(F,A)))). By the
cl(fpu(scl(F,A)))=fpu(scl(F,A)) for every soft set (F,A) of
~
(X,,A).Hence sgrcl(f ((F,A))) ~ fpu(scl(F,A)) for every
definition of soft grclosure, we have sgrcl(F,A)
pu
soft set (F,A) of a soft topological space (X,,A).
Theorem:4.16
1(sgrcl(F,A)). Hence, we get fpu(sgrcl(F,A)) ~
sgrcl(fpu((F,A))).
A soft function fpu: SS(X)ASS(Y)B is soft gropen if

Since sgrcl(G,B) is soft grclosed in Y and so by
hypothesis, 1(
~
sgrcl(G,B)) is soft grclosed in (X,,A).
fpu(sint((F,A))) sgrint(fpu((F,A))) for every soft set (F,A) of a soft topological space (X,,A).
Since 1((G,B)) ~ 1 (sgrcl(G,B)), it follows that sgrcl( 1((G,B))) ~ 1 (sgrcl(G,B)).
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