on Soft πGr-Continuous Functions in Soft Topological Spaces

on Soft Gr-Continuous Functions in Soft Topological Spaces

1.C. Janaki, 2. Jeyanthi. V

1. Asst Professor, Department of Mathematics, L.R.G. Government College for Women, Tirupur-4.

2. Asst Professor, Department of Mathematics, Sree Narayana Guru College, Coimbatore-105.

Abstract: The purpose of this paper is to study soft gr- continuous function and soft gr-irresolute function in soft topological spaces. Also,we introduce the concepts such as soft

gr-closure,soft gr-interior and exhibit some results related to soft gr- continuity,soft gr-open map and soft gr-closed map

. Further, we study the relation between soft gr-continuous function and other soft continuous functions.

Key Words: Soft Set , Soft Topology, Soft Continuity , Soft Point, Soft gr-closure, Soft gr-interior, Soft Function, Soft

gr-continuity, Soft gr-irresolute, soft gr- neighborhood.

Mathematics Subject Classification: 06D72, 54A40.

1. 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 g-closed sets in general topology. Kannan [11] introduced soft g-closed sets in soft topological spaces, The concept of -closed sets in topological spaces was initiated by Zaitsav[23] and the concept of g-closed set was introduced by Noiri and Dontchev[4]. N.Palaniappan[17]

2. 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 gr-closed sets in topological spaces. Soft semi-open 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 gr-closed sets [9]in soft topological spaces. The concept of g-closed,gp- closed,gs-closed,*g-closed,gb-closed 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 gr-continuity 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

1. , ~ belong to .

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

3. The intersection of any two soft sets in belongs to

.

The triplet (X, , E) is called a soft topological space over

1. 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)= X-F(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 s-int(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 s-cl(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 T1/2-space if every soft g-closed 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 pre-closure, soft semi closure and soft b-closure 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

   s-int (F,E) { (O,E): (O,E) is soft open and (O,E)

   (F,E)}.

   s-cl(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

   1. a soft generalized closed (Soft g-closed)[11] in a soft

      topological space (X,,E) if s-cl(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

      1. 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.

      2. a soft semi open [ 3]if (A,E) ~ s-int(s-cl(A,E))

   (iii))a soft regular open if (A,E)= s-int(s-cl(A,E)).

   1. a soft -open if (A,E) ~ s-int(s-cl(s-int(A,E)))

      xp1 ( y ) A

      , otherwise. for all yB.

   2. a soft b-open if (A,E) ~ s-cl(s-int(A,E)) s-int(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))

   1. a soft pre-open set if (A,E)

      ~ s-int(s-cl(A,E)).

      (1 (G),p-1(B)) is a soft set in SS(X)A such that

      1 (G)(x)= 1 (G(p(x))), p(x)B,

   2. 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 b-open , soft pre-open sets are their respective soft semi closed , soft regular closed , soft – closed , soft b-closed 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 gr-closed 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 gr-closed set of X by SGRC(X) . The

   complement of soft gr-closed is soft gr-open 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)sgr-int(X) =X, sgr-int(A) = A (ii)sgr-int((F,A)) ~ (F,A)

   1. If (F,A) is any soft gr-open set contained in (G,A), then ((F,A) ~ sgr-int(G,A)

   2. If (F,A) ~ (G,A) , then sgr-int((F,A)) ~ sgr-

      int(G,A)

   3. sgr-int(sgr-int((F,A)))= sgr-int((F,A)).

   Proof: Straight Forward.

   Theorem:3.6

   If a subset (F,A) of a soft topological space X is soft gr- open, then sgr-int((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 sgr-int((F,A)) sgr-int((G,A)) ~ sgr-

   III. SOFT GR-CLOSURE 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 gr-open set (F,A) such that x(F,A) ~ (G,A).The soft gr-neighborhood of a

   point x is denoted by sgr-nbd(x).

   Definition:3.2

   The Soft gr-Closure of a soft set (G,A) is defined to be the intersection of all soft gr-closed sets containing the soft set (G,A) and is denoted by sgr-cl(G,A).

   The Soft gr-Interior of a soft set (G,A) is defined to be the union of all soft gr-open sets contained the soft set (G,A) and is denoted by sgr-int(G,A).

   Lemma :3.3

   Let (F1,A) and (F2,A) be subsets of (X,,A) . Then

   1. sgr-cl (A) = A and sgr-cl(X) =X.

   2. If (F1,A) ~ (F2,A), then sgr-cl(F1,A) ~

      sgr-cl(F2,B)

   3. (F1,A) ~ sgr-cl(F1,A)

      Then sgr-int((F,A)) ~ sgr-int ((F,A) (G,A)), sgr-

      int((G,A)) ~ sgr-int ((F,A) (G,A)

      Hence sgr-int ((F,A)) sgr-int((G,A)) ~ sgr-int ((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))= sgr-int ((F,A) sgr-int ((G,A))

      Proof: We know that (F,A)(G,A ) ~ (F,A),

      (F,A)(G,A) ~ (G,A).Then sgr-int ((F,A) (G,A)) ~

      sgr-int((F,A)) and sgr-int ((F,A)(G,A)) ~ sgr-

      int((G,A)). Hence sgr-int ((F,A)(G,A))= sgr-int

      ((F,A)) sgr-int ((G,A))—————(*)

      Again , let x sgr-int ((F,A)) sgr-int ((G,A)). Then x sgr-int ((F,A)) and x sgr-int

      ((G,A)).Hence x is a soft gr-interior 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

      sgr-nbd of x . Hence x sgr-int ((F,A)(G,A))

      Thus, xsgr-int ((F,A)) sgr-int ((G,A)) x sgr-

      int ((F,A)(G,A))

      ~

   4. sgr-cl(F1,A)= sgr-cl( sgr-cl(F1,A))

   5. sgr-cl((F1,A) (F2,B))= sgr-cl(F1,A)

sgr-cl(F2,B)

Therefore, sgr-int ((F,A)) sgr-int ((G,A))

((F,A)(G,A))—(**)

From (*) and (**) ,

sgr-int

Proof:Obvious. Lemma: 3.4

Let (F,A) and (G,A) be soft subsets of the soft topological space X. Then sgr-cl ((F,A) (G,A)) ~ sgr-cl((F,A))

sgr-cl((G,A))

Proof: Since (F,A)(G,A) ~ (F,A), (G,A)

sgr-cl((F,A)(G,A)) ~ sgr-cl((F,A)) and sgr-

cl((F,A)(G,A)) ~ sgr-cl((G,A))

sgr-cl((F,A)(G,A)) ~ sgr-cl(F,A) sgr-cl(G,A)

sgr-int ((F,A)(G,A))= sgr-int ((F,A)) sgr-int ((G,A)).

Lemma:3.9

If (F,A) be a soft subset of a soft topological space X, then (X sgr-int((F,A)) = sgr-cl(X(F,A)).

Proof: Let xX (sgr-int((F,A))). Then x sgr-

int((F,A)). That is every soft gr-open set (G,A) containing x is such that (G,A) ~ (F,A).Every soft gr-open set (G,A)

containing x intersects X(F,A).

x sgr-cl(X(F,A))

Hence (X sgr-int((F,A))) ~ sgr-cl(X(F,A)). Conversely, let xsgr-cl(X(F,A)). Thus every soft gr-

open set (H,A) containing x intersects X-(F,A) (i.e) is every

soft gr-open set (H,A) containing x is such that (H,A) ~ (F,A).

x sgr-int((F,A)). Thus sgr-cl(X(F,A)) ~ (X

sgr-int((F,A))) and hence

(X sgr-int((F,A))) = sgr-cl(X(F,A)).

Remark:3.10

If (F,A) be a soft subset of a soft topological space X, then (X sgr-cl((F,A))) = sgr-int(X(F,A)).

IV. SOFT GR-CONTINUOUS 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

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

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

3. 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 gr-open in X but not soft open in X. Hence soft gr-continuity 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}.

4. 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

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

2. The function fpu is called soft gr-closed if fpu((G,A)) is soft gr-closed in (Y,*,B) for every soft closed set (G,A) in (X,,A).

Remark:4.3

Soft gr-continuity 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 gr-open in X but soft open in X. Hence soft continuous function need not be soft gr-continuous.

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 gr-open in X but not soft regular open in X. Hence soft gr-continuity need not be soft regular continuous.

Theorem:4.7

Every soft gr-continuous function is soft rg-continuous, soft g-continuous, soft *g-continuous, soft g- continuous, soft gp-continuous, soft gs-continuous and soft gb-continuous .

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 gr-open in X but soft rg-open, soft g-open, soft gs-open, soft g-open, soft *g-open, soft gp-open and soft gb -open in X. Hence soft rg-continuous, soft g-continuous, soft gs- continuous, soft g-continuous, soft *g-continuous, soft

gp-continuous and soft gb-continuous but not soft gr- continuous.

Remark:4.10

Soft gr-continuity and soft gr-irresolute are independent concepts.

Example:4.11

1. 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 gr-open sets in Y are soft gr-open in X, but the inverse image of soft open set (G4,E)={{a,b},{b,c}} is not soft gr-open in X. Hence soft

   gr-irresoluteness need not be soft gr-continuous.

   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

   1. 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 gr-open in X, but the inverse image of soft gr-open sets in Y are not soft gr-open in X. Hence soft gr-continuous function need not be soft gr- irresolute.

2. 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)={},

1. soft regular continuous 3-soft gr-irresolute

4-soft gr-continuous 5-soft g-continuous 6-soft rg-continuous 7-soft *g-continuous 8-soft gp-continuous 9-soft g-continuous

10-soft gb-continuous 11-soft gs-continuous.

Theorem:4.13

A soft function fpu: SS(X)ASS(Y)B is soft gr-continuous

when f (sgr-cl(F,A)) ~ s-cl(fpu((F,A))) for every soft set

Proof: Let fpu: SS(X)ASS(Y)B is soft gr-open map and (F,A) be any soft set of (X,,A).Then s-int (F,A) is soft open in (X,,A). Since f is soft gr-open map, fpu(s-int(F,A))

is soft gr-open in (Y,*,B). We have f (s-int (F,A) ) ~

pu

(F,A) of a soft topological space (X,,A).

Proof: Let fpu: SS(X)ASS(Y)B be a soft gr-continuous

pu

fpu((F,A)) for every soft set (F,A) of a soft topological

space (X,,A). Also, fpu(s-int(F,A)) = sgr-int(fpu(s-int(F,A

)) Hence f (s-int(F,A)) ~ sgr-int(f ((F,A))).

function. Now, s-cl(f ((F,A))) is a soft closed set of pu pu

pu

(Y,*,B). By the soft gr-continuity of fpu,

1

(s-

Theorem:4.17

cl(fpu((F,A))) is soft gr-closed and (F,A)

~

(s-

pu pu

cl(fpu((F,A))).But sgr-cl (F,A) is the smallest soft gr- closed set containing (F,A), hence sgr-cl(F,A) ~ 1 (s- cl(f ((F,A))). The above implies f (sgr-cl(F,A)) ~ s-

cl(fpu((F,A))).

Theorem:4.14

A soft function fpu: SS(X)ASS(Y)B is soft gr-continuous

Let (X,,A) be a soft gr-T1/2-space fpu: SS(X)ASS(Y)B be a soft function. Then fpu is soft gr-continuous iff fpu is soft regular continuous.

Proof: Let fpu be a soft gr-continuous function. Then

1((G,B)) is soft gr-closed in (X,,A) for every soft closed set (G,B) of (Y,*,B). Since (X,,A) is a soft gr- T1/2-space, every soft gr-closed set is soft reglar closed .

Hence 1 ((G,B)) is soft regular closed in (X,,A) for

when

1 ~

(s-int(G,B))

sgr-int(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 gr-continuous function. Now, s-int(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 gr-continuity of fpu ,

1(s-

1

soft closed set (G,B) in (Y,*,B). Since every soft regular closed set is soft gr-closed. Then 1((G,B)) is soft gr-

int(fpu((G,B))) is soft gr-open in (X,,A) and (s-

int((G,B)))) ~ (G,B). As sgr-int(G,B) is the largest soft

gr-open set contained in (G,B), 1 (s-int(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 gr-continuous.

Theorem:4.18

A soft function fpu: SS(X)ASS(Y)B is soft gr-irresolute, then

A soft function fpu: SS(X)ASS(Y)B is soft gr-closed if

1. f

   (sgr-cl(F,A)) ~ sgr-cl(fpu((F,A))) for

   s ~ f

   pu

   (s-cl(F,A)) for every soft set (F,A)

   gr-cl(fpu((F,A))) pu

   every soft set (F,A) of (X,,A).

   of a soft topological space (X,,A).

   (ii) sgr-cl( 1(G,B))) ~ 1(sgr-cl(G,B)) for

   ~

   pu pu

   Proof: Suppose that f is soft gr-closed and (F,A) is any soft set of (X,,A). We have, f ((F,A)) ~ f (s-cl(F,A)).

   Now,sgr-cl(fpu((F,A))) sgr-cl(fpu(s-cl(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 gr-closed in (Y, *,B). By hypothesis ,

   1(

   Since fpu(s-cl(F,A)) is soft gr-closed in (Y,*,B), sgr-

   sgr-cl(fpu((F,A)))) is soft gr-closed in (X,,A).Also ,

   (F,A) = 1 (fpu((F,A))) ~ 1 (sgr-cl(fpu(F,A)))). By the

   cl(fpu(s-cl(F,A)))=fpu(s-cl(F,A)) for every soft set (F,A) of

   ~

   (X,,A).Hence sgr-cl(f ((F,A))) ~ fpu(s-cl(F,A)) for every

   definition of soft gr-closure, we have sgr-cl(F,A)

   pu

   soft set (F,A) of a soft topological space (X,,A).

   Theorem:4.16

   1(sgr-cl(F,A)). Hence, we get fpu(sgr-cl(F,A)) ~

   sgr-cl(fpu((F,A))).

   A soft function fpu: SS(X)ASS(Y)B is soft gr-open if

2. Since sgr-cl(G,B) is soft gr-closed in Y and so by

hypothesis, 1(

~

sgr-cl(G,B)) is soft gr-closed in (X,,A).

fpu(s-int((F,A))) sgr-int(fpu((F,A))) for every soft set (F,A) of a soft topological space (X,,A).

Since 1((G,B)) ~ 1 (sgr-cl(G,B)), it follows that sgr-cl( 1((G,B))) ~ 1 (sgr-cl(G,B)).

REFERENCES

1. Aslim.G.Caksu Guler.A and Noiri.T, On gs-closed sets in topological spaces, Acta Math. Hungar., 112(4) (2006),275-283.

2. Bin Chen, Soft semi-open sets and related properties in soft topological spaces,Appl.Math.Inf.Sci.7,No.1, 287

   -294(2013).

3. Cigdem Gunduz Aras, Ayse Sonmez and Huseyin Cakalli, On Soft mappings, Xiv: 1305. 4545 ,v1

   [math. GM], May 2013.

4. Dontchev.J, Noiri.T, Quasi normal spaces and g-closed sets, Acta Math. Hungar, 89 (3) , 2000,211-219.

5. Ganes M Pandya, Janaki.C and Arokiarani.I, On *g-closed sets in Topological spaces, Antarctica J. Antarctica J.Math.,(^)(2010),597-604.

6. D.N.Georgiou and A.C.Megaritis, Soft set Theory and Topology, Applied General Topology, 14,(2013).

7. C.Janaki, Studies on g- closed sets in Topology, Ph.D Thesis, Bharathiar University ,2009.

8. Jeyanthi.V and Janaki.C, gr-closed sets in topological spaces ,Asian Journal of Current Engg. and Maths 1:5 , Sep 2012, 241-246.

9. Janaki.C and Jeyanthi.V, On Soft

   gr-closed sets in soft topological spaces,JAM, Vol 4, No 3, 478-485.

10. A.Kandil, O.A.E.Tantawy, S.A.El-

    Sheikh, A.M.Abd El-Latif-operation and decomposition of some forms of soft continuity in soft topological

    spaces,Annals of Fuzzy Mathematics and Informatics, Vol x,No.x,(Month 201y),pp.1-xx.

11. K.Kannan , Soft generalized closed sets in topological spaces, Journal of Theoretical and Applied Information

    Technology, Mar 2012, Vol.37, No

    1.17-21.

12. Kharal. A,Ahmad.B,Mappings of Soft classes, New Math .Nat Comput.,7(3),(2011), 471- 481.

13. Levine.N,Generalized closed sets in topological spaces, Rend.Circ. Mat

    .Palermo , Vol 19, No.2, 1970, pp

    89-96.

14. J.Mahanta,A study on some topological aspects of fuzzy sets, rough sets and soft sets , Ph.D. Thesis , North Eastern Regional

    Institute of Science and

    technology,(deemed university), Nirjuli , Arunachal Pradesh, April, 2013.

15. P.K. Maji, R.Biswas , R.Roy, An application of soft sets in a decision making problem, Comput. Math. Appl., Vol.45, 2003, pp.1077-1083.

16. Molodtsov.D, Soft set theory first results,Comput. Math. Appl., Vol.45, 2003, pp.555- 562.

17. Palaniappan .N and Rao.K.C, Regular closed sets in topological spaces,Kyungpok Math .J.

    33(1993), 211-219.

18. Park.J.H , On gp-closed sets in topological spaces, Acta Mathematica Hungarica, 12, (4),257-283.

19. D.Pei and D.Miao, From Soft Sets to information systems, In:X.Hu,Q.Liu, A.Skowron,T.Y.Lin,R.R.Yager, B.Zhang(Eds.),Proceedings of

    Granular computing,

    in:IEEE,2(2005)617-621.

20. M.Shabir and M.Naz ,On Soft topological spaces, Comp. And Math. with applications Vol.61,no.7,pp 1786-1799,2011.

21. D. Sreeja and C. Janaki ,On gb- Closed Sets in Topological Spaces,

    International journal of Mathematical

    Archive, Vol 2, 8,2011, 1314-1320.

22. Tridiv Jyoti Neog and Dusmanta Kumar Sut, A New Approach to the theory of Soft sets Ijca, (0975-8887), Vol.32, No.2, October-2011.

23. Zaitsev, On Certain classes of topological spaces and their bicompactifications , Dokl. Akad ,

    Nauk.SSSR., 178(1968), 778-779.

24. .Zorlutuna, M.Akdag, W.K.Min and S.Atmaca, Remarks on soft topological spaces, Annals of

Fuzzy Mathematics and Informatics,

Vol 3, No 2, (April 2012), pp.171-

185.