Download Soft simply open set in soft topological space

Soft simply open set in soft topological space
M. El Sayeda , M. K. El-Bablyb and I. A. Noamanc
a
Department of Mathematics, Faculty of Science and Arts Najran University,K.S.A.
b
Department of Mathematics, Faculty of Science and Arts, Taibah University
c
Department of Mathematics, Faculty of Science and Arts,
EL Moundq, Al-Baha University , K.S.A.
E. mail: [email protected]
Abstract
Soft sets have many applications in various real time problems in the …eld
of engineering, social science, medical science etc. Recently, the concept of soft
topological space has been developed with the help of soft sets. In the present
paper, we introduce new class of soft open set called soft simply open set, delta
sets in soft topological space , we shall also introduce soft simply continuous
functions, strongly soft simply-continuous and soft simply irresoluteness based
on this set, We further investigate some relationship among various soft open
sets , and …nally we study some of their properties.
Keywords: soft set ,soft topology ,soft open set , soft b-open set , soft beta
open set , soft semi open set and soft preopen
1
Introduction
In 1999, Molodtsov [2] introduced the concept of soft set theory as a mathematical
tool for dealing with uncertainties. He [2] established the fundamental results of this
new theory and successfully applied the soft set theory into several directions, such
as smoothness of functions, operations research, Riemann integration, game theory,
theory of probability and so on. Maji et al. [4] de…ned and studied several basic
notions of soft set theory. Shabir and Naz [5] de…ned soft topology by using soft sets
and studied some basic notions of soft topological spaces such as soft open and closed
sets, soft subspace, soft closure, soft neighborhood of a point, soft separation axioms.
After then many authors [5, 6, 7, 8, 9, 10] studied some of basic concepts and properties of soft topological spaces. In 1975 [11] Neubrunnova introduced the concept
of simply-open sets. Chen [12] introduced soft semi open sets and related properties. Gunduz Aras et al. [13] introduced soft continuous mappings which are de…ned
over an initial universe set with a …xed set of parameters. Mahanta and Das [14]
introduced and characterized various forms of soft functions, like semi continuous,
irresolute, semi open soft functions. In1963 [18] N. levine introduced the concept of
semi open sets. Recall that a set A is called semi-open if there exists an open set U
such that U
A Cl(A). Complements of semi-open sets are called semi-closed.
It is well-known that a set A is semi-closed if and only if Int(Cl(A))
A. In
1965 [17] Najasted introduced the concept of
-open sets Given a topological space
(X; ), the -topology
on X is the collection of all subsets of (X; ) satisfying
1
A Int(Cl(Int(A))). In 1991 Julian Dontchev [19], and Maximilian Ganster introduce the concept of B-sets, -set , NDB-set if the boundary of A is nowhere dense.
And in 1987 P. Bhattacharya and B. K. Lahiri [1] introduce the concept semi- generalized closed (brie‡y sg-closed) if semi closure (A) U ( brie‡y scl(A) U ),
whenever
A
U and U 2 SO(X). In the present paper, we introduce new class of soft
open set called soft simply open set, delta sets in soft topological space , we shall
also introduce soft simply continuous functions, strongly soft simply-continuous and
soft simply irresoluteness based on this set, We further investigate some relationship
among various soft open sets , and …nally we study some of their properties.
De…nition 1.1 [22] Let (UA , ) and (UB , ) be two soft topological spaces. A soft
function f : UA ! UB is said to be
1. soft semi continuous if for each soft open set GB of UB , the inverse imagef 1 (GB ) is soft semi open set of UA ;
2. soft irresolute if for each soft semi open set GB of UB , the inverse image
f 1 (GB ) is soft semi open set of UA ;
De…nition 1.2 [25] Let (X; ) be a soft topological space over X and (F; E)
be a soft set is called soft regular open (soft regular closed) in X if (F; E) =
int(cl((F; E)); ((F; E) = cl(int((F; E)))
De…nition 1.3 [24] A soft set (A; E) is called a soft semi generalized open (soft
semi g- open) in a soft topological space(X; ; E) if the relative complement (A; E)0is
soft semi g- closed in X: Equivalently, a soft set (A; E) is called a soft semi generalized open (soft semi g- open) in a soft topological space (X; ; E) if and only if
(F; E) v so sin t(A; E) whenever(F; E) v (A; E) and (F; E) is soft semi closed in
X.
De…nition 1.4 [24] A soft set (A; E) is called a soft semi generalized closed (soft
semi g- closed) in a soft topological space(X; ; E) if the relative complement (A; E)0is
soft g- open in X: Equivalently, a soft set (A; E) is called a soft semi generalized
closed (soft semi g- closed ) in a soft topological space (X; ; E) if and only if
soscl(A; E) v (U; E) whenever (A; E) v (U; E) and (F; E) is soft semi open in X.
De…nition 1.5 (see [2])
1. Let X be an initial universe set, P (X) the power set of X, that is the set of
all subsets of X, and A a set of parameters. A pair (F; A), where F is a map
from A to P (X), is called a soft set over X.
2. Let (F; A), (G; A) 2 SS(X; A).We say that the pair (F; A) is a soft subset
of (G; A) if F (p) G(p), for every p 2 A. Symbolically, we write (F; A) v
(G; A). Also, we say that the pairs(F; A) and (G; A) are soft , and the equal if
(F; A) v (G; A) and (G; A) v (F; A). Symbolically, we write (F; A) = (G; A).
In what follows by SS(X; A) we denote the family of all soft sets (F; A) over
X:
2
De…nition 1.6 (see, [2] and [3]) Let I be an arbitrary index set and f(Fi ; A) : i 2
Ig SS(X; A). The soft union of these soft sets is the soft set (F; A) 2 SS(X; A),
where the map F : A ! P (X) de…ned as follows: F (p) =[fFi (p) : i 2 Ig, for every
p 2 A. Symbolically, we write (F; A) = tf(Fi ; A) : i 2 Ig.
De…nition 1.7 [3] Let X be an initial universe set, A a set of parameters, and
SS(X; A). We say that the family
1. 0A ; 1A 2 .
2. If (G; A); (H; A) 2
, then (G; A) u (H; A) 2
.
3. If (Gi ; A) 2 for every i 2 I, then tf(Gi ; A) : i 2 Ig 2 . The triplet (X; ; A) is
called a soft topological space or soft space. The members of
De…nition 1.8 ( [2] and [3]) Let I be an arbitrary index set and f(Fi ; A) : i 2 Ig
SS(X; A). The soft intersection of these soft sets is the soft set (F; A) 2 SS(X; A),
where the map F : A ! P (X) de…ned as follows: F (p) = \fFi (p) : i 2 Ig, for every
p 2 A. Symbolically, we write (F; A) = uf(Fi ; A) : i 2 Ig:
De…nition 1.9 ( [2] and [3]) Let (F; A) 2 SS(X; A). The complement of soft set
(F; A) is the soft set (H; A) 2 SS(X; A), where the map H : A ! P (X) de…ned as
follows: H(p) = XnF (p), for every p 2 A. Symbolically, we write (H; A) = (F; A)c .
De…nition 1.10 For two soft sets (F,A) and (G; B) over a common universe U,
1. ([4]) union of two soft sets of (F; A) and (G; B) is the soft set (H; C), where
C = A [ B, and 8e 2 C; then H(e) =fF (e);if e 2 A B(e) or G(e) if
e 2 A \ B or F (e) [ G(e), if e 2 A \ B We write (F; A)e [ (G; B) = (H; C).
2. [17] intersection of (F; A) and (G; B) is the soft set (H; C), where C = A\B,
and 8e 2 C; H(e) = F (e) \ G(e). We write(F; A)e \ (G; B) = (H; C).
De…nition 1.11 [20] Let (X;
a soft set over X.
) be a soft topological space over X and (F; E) be
1. The soft closure of (F; E) is the soft set scl(F; E) = \ff (G; E) : (G; E) is
soft closed and (F ; E) v (G; E)g.
2. The soft interior of (F;E) is the soft set sint(F ; E) = [[ f (H; E) : (H; E) is
soft open and (H; E) v (F ; E)g. Clearly, (F; E) is the smallest soft closed
set over X which contains (F; E) and (F; E) is the largest soft open set over
X which is contained in (F; E).
Theorem 1.1 [21] Let (X; ) be a soft topological space overX; (F; E) and (G; E)
soft sets over X: Then
1. ((F; E) \ (G; E)) = (F; E) \ (G; E)
2. ((F; E) [ (G; E))
(F; E) [ (G; E) :
Theorem 1.2 [20] Let (X; ) be a soft topological space over X; (F; E) and (G; E)
soft sets over X. Then
3
1. ((F; E) [ (G; E)) = (F; E) [ (G; E)
2. (((F; E) \ (G; E)) = (F; E) \ (G; E) .
De…nition 1.12 A soft set (F; E) in a soft topological space (X; ; E) is said to be:
1. [12] soft semi-open if (F; E)
2. [15]soft pre-open if (F; E)
3. [15] soft
4. soft
2
-open if
(F; E)
open [11] if (F; E)
cl(int(F; E)).
int(cl(F; E)).
int(cl(int(F; E))).
cl(int(cl(F; E))):
Soft simply-open sets
De…nition 2.1 A soft subset ( F,A) of soft topological space (X; ; B) is called:
soft simply-open set if sint(scl((F; A))) v scl(sint((F; A))):
Proposition 2.1 For a soft subset (V; E)
equivalent:
(X; ; E) the following conditions are
1. (V; E) is soft simply-open.
2. (V; E) is soft semi-locally closed.
3. (V; E) is a soft -set.
4. (V; E) is Null soft -set.
Proof. (1) , (2) obvious
(2) , (3) let (V; E) be soft semi locally closed. Then sin t(scl((V; E))) v
sin t(scl((V; E)))\scl(sin t((V; E))); sin t(scl((V; E))) v scl(sin t((V; E))), then (V; E)
is a soft -set.
(3) , (4) since sin t(scl((V; E))) = sin t(scl((V; E))) \ sin t(scl(X n (V; E))) =
sin t(scl((V; E))) \ (X n scl(sin t((V; E)))) = sin t(scl((V; E))) n scl(sin t((V; E))).
Theorem 2.1 In soft topological space (X; ; E) . Then
1. The union of two soft simply open set is soft simply open set .
2. The …nite intersection of soft simply open set is soft simply open set.
Proof.
1. Let (V; E) and (G; E) be two soft simply simply sets , since (V; E) is soft
simply open set then int(cl((V; E)))v cl(int((V; E))); .........(1) and (G; E) is
soft simply simply set then int(cl((G; E)))v cl(int((G; E))).........(2) , then
from (1),(2) we get int(cl((V; E))) [ int(cl((G; E))) v cl(int((G; E))) [
cl(int((V; E))) this implies that int(cl(((V; E))) [ ((G; E)))) v cl(int(((G; E)))[
((V; E)))) if we put (G; E))) [ ((V; E) = (H; E) , thus int(cl((H; E)) v
cl(int(H; E)); hence(H; E)soft simply open set .
4
2. Let,f(F; A)gni=1 be a collection of soft simply open sets then sint(scl(f(F; A)gni=1 ) v
scl(sin t(f(F; A)gni=1 )) for each i= 1,2,......,n ,then sint(scl(f(F; A)gi=1 ) v
n
n
scl(sin t(f(F; A)gi=1 )) this tends to \ sint(scl(f(F; A)gi=1 ) v \ scl(sin t(f(F; A)gi=1 ))
i=1
n
i=1
n
n
, hence sint(scl( \ f(F; A)gi=1 ) v scl(sin t( \ f(F; A)gi=1 ));therefore \ f(F; A)gi=1
i=1
i=1
i=1
is soft simply open set .
The following diagram gives the relationship between soft simply open set and
some other types of soft near open sets.
soft regular open ! soft open set !soft alpha -open set
.
&
Soft simply open set
soft semi open set
soft pre open set
&
.
soft beta -open set
soft b- open set
Diagram 1.1
The following example show that ( the implication 1) is not reversible.
Example 2.1 . Let X = fa; b; cg; E = fe1 ; e2 g
= fX; f(e1 ; fag); (e2 ; fbg)g; f
(e2 ; fa; bg)gg since (V; E) = {(e2 ; fcg)g is soft simply open set but it is not semi
open set;and (G; K) = f(e1 ; fcg)g is not soft semi open in X.
Remark 2.1 It is up to the reader, which term he wants to utilize. Since soft
simply-continuity was de…ned in terms of soft simply-open sets .
Remark 2.2 Clearly every soft semi-open and every soft semi-closed set is soft
simply-open. Conversely, not every soft simply-open set is soft semi open an soft
semi closed sets. The following example show this remark.
Example 2.2 Let X = fa; b; c; dg; E = fe1 ; e2 g , = fX; f(e1 ; fag); (e2 ; fa; bg)g; f
(e1 fbg); (e2 ; fb; c:dg); (e1 ; ); (e2 ; fbg)gg since (V; E) = {(e1 ; fa; cg; (e2 ; fa; bg)g soft
set and it is soft semi open and it is also soft simply open set;and if (G; E) =
f(e1 ; fa; cg)g is soft set ,then it is soft simply open set but it is not not soft semi
open set , and (V; E) = f(e1 ; fb; dg; (e2 ; fc; dg)g is soft semi closed and it is soft
simply open set , since (G; E) = f(e2 ; fb; cg)g is soft set, then it is soft simply open
set but it is not soft semi closed set
5
De…nition 2.2 A soft topological space (X; ; E) is called soft locally indisceret
space if ever soft open subset is soft closed set .
Theorem 2.2 In soft topological space (X; ; E) if (G; A) is a soft simply open set
and soft regular open. Then (G; A) is soft semi open set .
Proof. Obvious
Theorem 2.3 In soft topological space (X; ; E) if (G; A) is a soft simply open set
and soft regular closed. Then (G; A) is soft semi open closed set .
Proof. Obvious
De…nition 2.3 A function f : (X; ; E) !(Y; ; E) is called:
1. Soft simply-continuous if f 1 (V; E) is soft simply-open in (X; ; E) for every
soft open set (V; E) of (Y; ; E),
2. Soft simply-irresolute if f 1 (V; E) is soft simply-open in (X; ; E) for every
soft simply - open set (V; E) of (Y; ; E)
3. Soft generalized -continuous if f 1 (V; E) is soft generalized open in (X; ; E)
for every soft open set (V; E) of (Y; ; E)
Remark 2.3 If (X; ; E) is soft topological space then from the a above diagram :
1. Soft simply open set and soft
open set are not comparable .
2. Soft simply open set and soft b open set are not comparable
the following example shows this remark
Example 2.3 Let X = fa; b; c; dg; E = fe1 ; e2 g , = fX; f(e1 ; fag); (e2 ; fa; bg)g; f
(e1 fbg); (e2 ; fb; c:dg); (e1 ; ); (e2 ; fbg)gg, since (V; E) = {(e1 ; fb; c; dg; (e2 ; fcg)g it is
soft simply open but not b-open set and if (G; E) = {(e1 ; fc; dg; (e2 ; fb; dg)g; soft
set , since it is not soft simply open set but it is soft b- open set, and (H; E) =
{(e1 ; fa; b; cgg is soft simply open set but it is not soft
open set and G; E) =
{(e1 ; fc; dg; (e2 ; fb; dg)g is soft
open set but is not soft simply open set .
Proposition 2.2 The family of all soft simply-open sets in a soft topological space
(X; ; E) is an algebra of sets, i.e., it contains the complement of each member as well
as the union of each two members. Moreover, …nite intersection of soft simply-open
sets is also soft simply-open.
Proposition 2.3 For a soft subset (F; E)
equivalent:
(X; ; E) the following conditions are
1. (F; E) is soft semi-closed.
2. (F; E) is soft sg- closed and soft simply-open.
6
Proof. . (1) ) (2) let (F; E) be soft semi closed set and let (G; E) be soft semi
open set since sscl((F; E)) v (F; E) v (G; E). Hence (F; E) is soft semi generalized
closed set.
(2) ) (1) Let sscl((F; E)) denote the soft semi-closure of (F; E), i.e. the intersection of all soft semi-closed soft super sets of (F; E). Since (F; E) is soft
simply-open, then (F; E) can be written as the intersection of soft semi-open set
(G; E) and soft semi-closed set (V; E). Since (F; E) is sg- closed, we have that
sscl((F; E)) is contained in (G; E). Since (V; E) is soft semi-closed, sscl((F; E))
is contained in T . Therefore sscl((F; E)) = (F; E), i.e. (F; E) is soft semi-closed.
Proposition 2.4 For a soft topological space (X; ; E) the following conditions are
equivalent:
1. Every soft simply-open set is soft semi-closed.
2. Every soft open set is soft regular open.
3. X is soft locally indiscrete.
4. Every soft simply-open set is soft
-closed.
Proof. . Let (1) ) (2) and (A; E) be soft open. Then (A; E) is also soft semiclosed and thus soft regular open, and so (2) holds.
(2) ) (3) obvious.
(3) ) (4) Let (A; E) be soft simply-open, i.e. sint(scl(A; E))v scl(sin t(scl(A; E))) v
scl(sin t((A; E))) v (A; E). By (3), then (A; E) is soft -closed .
(4) ) (1) let (A; E) be soft soft - closed set and soft simply open set. Then
scl(sin t(scl(A; E) v (A; E) but sint(scl(A; E))v scl(sin t(scl(A; E))) v scl(sin t((A; E)) v
(A; E); this means that (A; E) is soft semi closed set.
Strongly soft simply-continuous functions
3
De…nition 3.1 A soft function f : (X; ; E) !(Y; ; E) is called soft strongly
simply-continuous if for every soft semi-open set (A; E) (resp. soft semi closed
(F; E)) of Y , f 1 ((A; E)) ( resp. f 1 (F; E)) is soft simply-open in X.
Proposition 3.1 For a function the following conditions are equivalent:
1. f is strongly soft simply-continuous.
2. For every soft semi-closed set (F; E) of Y , f
in X.
1
((F; E)) is soft simply-open
Proof. .Let 1 ! 2 and (F; E) 2 SC(Y );since f is strongly soft continuuos . Then
f
1
M
((F; E)) 2 s S o(X):
2 ! 1 Let (V; E) 2 SSC(Y ) since f
soft simply continuous.
Proposition 3.2
continuous.
1.
Every soft
1
M
((V; E)) 2 S S o(X):Therefore f is strongly
irresolute function is strongly soft simply-
7
2. Every strongly soft simply-continuous is soft simply-continuous.
1. Let (V; E) 2 SSO(Y ) and since f is soft irresolute, then f 1 ((V; E)) 2
SSo(X) since soft every soft semi open set is simply open set, thus f is
strongly soft simply continuous f
1
M
((V; E)) 2 S S o(X).
2. Let (V; E) 2 since every soft open set is soft semi open set since f is soft
strongly continuous .
The following diagram shows the relation between the new notions and the other
notion .
soft strongly semi continuous !soft irresolute !strongly soft simply continu
#
soft continuous !soft alpha-continuous ! soft simply continu
Diagram 1.2
Our next two examples show that none of the implications in Proposition are
reversible.
Example 3.1 . Let X = fa; b; cg; E = K = fe1 ; e2 g = fX; f(e1 ; fag); (e2 ; fbg)g; f
(e2 ; fa; bg)ggthe let soft f identity function f : (X; ; E) !(X; ; E) de…ned and
=f (a) = f (b) = a and f (c) = c and let p : E ! K and is de…ned by p(e1 ) = e2 ;
p(e2 ) = e1 , since (V; K) = {(e2 ; fb; cg)g is soft semi open set but fp 1 (V; K) =
f(e1 ; fcg)g is not soft semi open in X thus f is not soft irresolute. But f is strongly
soft simply-continuous.
Example 3.2 Let X = fa; b; cg; E = K = fe1 ; e2 g ={X; ; f(e1 ; fag)g; f(e2 ; fb; cg)g}
, ={X; ; f(e1 ; fag)g} the let soft f identity function f : (X; ; E) !(X; ; E) de…ned and =f (a) = f (b) = a and f (a) = c and let p : E ! K and is de…ned by p(e1 ) = e2 ; p(e2 ) = e1 . Then f is soft simply-continuous but not soft
strongly simply-continuous. Let (V; K) = {(e2 ; fa; bg)g.is soft semi-open in but
fp 1 (V; K) = f(e1 ; fa; bg)g is not soft simply-open in X .
Proposition 3.3 A function f : (X; ; E) !(Y; ; E) is soft simply irresolute if and
only if it is soft simply continuous.
Proof. let (A; E) be soft open set in (Y; ; E); hence (A; E) is soft semi open set
since f is soft irresolute then f 1 (A; E) is soft semi open set in X since every soft
semi open set is soft simply open thus f
continuous .
1
M
(A; E) 2 S S O(X) then f is soft simply
Proposition 3.4 If f : (X; ; E) !(Y; ; E) and g : (Y; ; E) !(Z; ; E) are two
soft functions
1. If f : (X; ; E) !(Y; ; E) is soft irresolute and g : (Y; ; E) !(Z; ; E) is
soft simply continuous . Then g f is soft simply continuous.
2. if g f is soft simply continuous and g is soft simply continuous. Then f is
soft open map
8
3. If f : (X; ; E) ! (Y; ; E) is soft simply continuous and g : (Y; ; E) !(Z; ; E)
is soft continuous . Then g f is soft simply continuous.
1. Since g is soft continuous , then for every (A; E) 2
M
1
s S O(Z) since f is soft simply irresolute , then f
f
1
M
thus g 1 (A; E) 2
M
(g 1 (A; E)) 2 s S O(X) but
(g 1 (A; E)) = (g f ) 1 (A; E) 2 s S O(X): Thus f is soft simply continuous
2. since g f is soft open map then for every (A; E) 2 then gof ( (A; E))2
since g is soft simply continuous then
g 1 (gof ((A; E))) =f ((A; E) 2
then f is soft open mape.
3. Since g is soft continuous , then for every (A; E) 2
since f
f
1
is soft simply continuous , then f
M
1
thus g 1 (A; E) 2
M
(g 1 (A; E)) 2 s S O(X) but
(g 1 (A; E)) = (g f ) 1 (A; E) 2 s S O(X): Thus f is soft simply continuous.
Proposition 3.5 If f : (X; ; E) !(Y; ; E) is soft semi continuous then f is soft
simply continuous.
Proof. Since f is soft semi continuous, then for every (F; E) 2 then f 1 (F; E) 2
S So(X) since every soft semi open set is soft simply open thus f is soft simply
continuous.
Example 3.3 Let X = fa; b; cg, set = f;; fag; fb; cg; Xg and = f;; fag; Xg.
Let be the identity soft function. Clearly f is soft simply-continuous but not
strongly soft simply-continuous. Set V = fa; b; g. Note that V is soft semi-open in
but V is not soft simply-open in .
Theorem 3.1 If f : (X; ; E) !(Y; ; E)is soft irresolute then f is strongly soft
simply-continuous .
Proof.
.Let 1 ! 2 and (F; E) 2 SSo(Y ) , then f 1 (F; E) 2 S So(X) since
every soft semi open set is soft simply open set thus f is strongly soft continuous .
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