Some Notes on Compact Sets in Soft Metric Spaces Sadi Bayramov, Cigdem Gunduz (Aras), Murat I. Yazar Department of Mathematics, Kafkas University, Kars, 36100-Turkey Department of Mathematics, Kocaeli University, Kocaeli, 41380-Turkey Department of Mathematics, Kafkas University, Kars, 36100-Turkey
[email protected],
[email protected],
[email protected]
Abstract The first aim of this study is to define soft sequential compact metric spaces and to investigate some important theorems on soft sequential compact metric space. Second is to introduce ~
net and totally bounded soft metric space and study properties of this space. Third is to define Lebesque number and soft uniformly continuous mapping and investigate some theorems in detail.
Keywords: soft set, soft sequential compact, totally bounded, soft uniformly continuous mapping.
1. Introduction Many practical problems in economics, engineering, environment, social science, medical science etc. cannot be dealt with by classical methods, because classical methods have inherent difficulties. The reason for these difficulties may be due to the inadequacy of the theories of parameterization tools. Molodtsov [12] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. Maji et al. [10] gave some operations on soft sets. Maji et al. [9] introduced some results on an application of fuzzy-soft-sets in decision making problem. M. Shabir and M. Naz [13] presented soft topological spaces and they investigated some properties of soft topological spaces. Later, many researches about soft topological spaces were studied in [7,8,11,13,14,15,16]. In these studies, the concept of soft point is expressed by different approaches. In the study we use the concept of soft point which was given in [4,14]. Recently in [4,5] S. Das and S. K. Samanta introduced a different notion of soft metric space by using a different concept of soft point and investigated some basic properties of these spaces. In the present paper, we first give, as preliminaries, some well-known results in soft set theory. We define soft sequential compact metric spaces and investigate some important ~
theorems on soft sequential compact metric space. We also introduce net and totally bounded soft metric space and study properties of this space. Sequential compactness can be expressed by totally boundedness. Finally, we define Lebesque number and soft uniformly continuous mapping and investigate some theorems in detail.
1
2. Preliminaries Throughout this paper, X refers to an initial universe, E is the set of all parameters for X . Definition 2.1. ([12]) A pair F, A is called a soft set over U , where F is a mapping given by F : A PU . In other words, the soft set is a parameterized family of subsets of the set U . For A, F may be considered as the set of elements of the soft set F, A , or as the set of approximate elements of the soft set. According to this manner, a soft set F, A is given as consisting of collection of approximations: F , A F : A. Definition 2.2. ([10]) For two soft sets F, A and G, B over U , F, A is called a soft subset of G, B if i A B , and ii A, F and G are identical approximations. This relationship is denoted by F , AG, B . Similarly, F, A is said to be a soft superset ~
of G, B , if G, B is a soft subset of F, A . This relationship is denoted by F , AG, B . ~
Definition 2.3. ([10]) Two soft sets F, A and G, B over 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 2.4. ([10]) The intersection of two soft sets F, A and G, B over U is the soft set H , C , where C A B and C , H F G . This is denoted by
F , AG, B H , C . ~
Definition 2.5. ([10]) If F, A and G, B are two soft sets, then F, A AND G, B is F , A G, B . F , A G, B is defined as H , A B , where denoted H , F G , , A B . Definition 2.6. ([10])The union of two soft sets F, A and G, B over U is the soft set, where C A B and C , F , if A B H G , if B A . F G , A B This relationship is denoted by F , AG, B H , C . ~
Definition 2.7. ([11]) The complement of a soft set F, A is denoted by F , A and is c
defined
by
F , A
c
( F c , A),
where
Fc : A P X
F ( ) X F ( ), A . c
2
is
a
mapping
given
by
Definition 2.8. ([3]) Let be the set of real numbers and B( ) be the collection of all nonempty bounded subsets of and E taken as a set of parameters. Then a mapping is called a soft real set. It is denoted by ( F , E ) . If specifically ( F , E ) is a F : E B( ) singleton soft set, then identifying ( F , E ) with the corresponding soft element, it will be called a soft real number and denoted r , s , t etc. 0 ,1 are the soft real numbers where 0(e) 0,1(e) 1 for all e E , respectively.
Definition 2.9. ([3]) For two soft real numbers (i) r s if r e s e , for all e E ; (ii)
r s if r e s e , for all e E ;
(iii)
r s if r e s e , for all e E ;
(iv)
r s if r e s e , for all e E .
Definition 2. 10. ([4,14]) A soft set ( P, E ) over X is said to be a soft point if there is exactly one e E , such that P(e) x for some x X and P(e) , e E / e. It will be denoted by xe . Definition 2.11. ([4,14]) Two soft point xe , ye are said to be equal if e e and P(e) P(e) i.e., x y . Thus xe ye x y or e e . Proposition 2.12. ([14]) The union of any collection of soft points can be considered as a soft set and every soft set can be expressed as union of all soft points belonging to it; (F , E) xe . xe ( F , E )
Definition 2.13.([13]) Let be the collection of soft sets over X , then is said to be a soft topology on X if (1) , X 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 X. Definition 2.14. ([8]) Let X , , E be a soft topological space over X . Then soft interior of
F , E , denoted by F , E
, is defined as the union of all soft open sets contained in F , E .
Definition 2.15. ([8]) Let X , , E be a soft topological space over X . Then soft closure of
F, E , F, E .
denoted by F , E , is defined as the intersection of all soft closed super sets of
Definition 2.16. ([7]) Let
X , , E
Y , , E
and
be two soft topological spaces,
f : X , , E Y , , E be a mapping. For each soft neighborhood H , E of
3
f x
e
, E , if
there exists a soft neighborhood F , E of ( xe , E ) such that f
F , E H , E , then
f is
said to be soft continuous mapping at ( xe , E ) . If f is soft continuous mapping for all ( xe , E ) , then f is called soft continuous mapping. Let X be the absolute soft set i.e., F e X , e E , where F , E X and SP( X ) be the ( E )* denote the set of all non-negative soft real
collection of all soft points of X and numbers.
Definition 2.17. ([4]) A mapping d : SP( X ) SP( X ) ( E )* , is said to be a soft metric on the soft set X if d satisfies the following conditions: (M1) d ( xe1 , ye2 ) 0 for all xe1 , ye2 X , (M2) d ( xe1 , ye2 ) 0 if and only if xe1 ye2 , (M3) d ( xe1 , ye2 ) d ( ye2 , xe1 ) for all xe1 , ye2 X , (M4) For all xe1 , ye2 , ze3 X , d ( xe1 , ze3 ) d ( xe1 , ye2 ) d ( ye2 , ze3 ) . The soft set X with a soft metric d on X is called a soft metric space and denoted by ( X , d , E) . Definition 2.18. [5] Let ( F , E )() S ( X ) , then the collection of all soft elements of ( F , E ) will be denoted by SE ( F , E ) . For a collection B of soft elements of X , the soft set generated by B is denoted by SS ( B) . Definition 2.19. [5] A mapping d : SE ( X ) SE ( X ) ( E)* , is said to be a soft metric on the soft set X if d satisfies the following conditions: (M1) d ( x, y) 0 for all x, y X , (M2) d ( x, y) 0 if and only if x y, (M3) d ( x, y) d ( y, x) for all x, y X , (M4) For all x, y, z X , d ( x, z ) d ( x, y) d ( y, z ) . The soft set X with a soft metric d on X is called a soft metric space and denoted by ( X , d , E) . Note 2.20. The metric defined in the definition 2.18 is different from the metric given in the definition 2.19 interms of the soft point concept. In this study we use the metric that is given in the definition 2.18.
Definition 2.21. ([4]) Let X , d , E be a soft metric space and be a non-negative soft real
number. B( xe , ) ye X : d ( xe , ye ) SP( X ) is called the soft open ball with center
xe and radius and B[ xe , ] xe X ; d ( xe , ye ) SP( X ) is called the soft closed ball with center xe and radius .
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Definition 2.22. ([4]) Let x ,n n be a sequence of soft points in a soft metric space X , d , E .
The sequence x ,n n is said to be convergent in X , d , E if there is a soft point y X such that d x ,n , y 0 as n .
This means for every 0 , chosen arbitrarily, a natural number N N ( ) ,such that 0 d x ,n , y , whenever n N .
Theorem 2.23. ([4]) Limit of a sequence in a soft metric space, if exist is unique. Definition 2.24. ([4]) (Cauchy Sequence). A sequence considered as a Cauchy sequence in
x
,n n
of soft points in
X , d , E is
X if corresponding to every 0, m N such that
d x ,i , x , j , i, j m ,i.e., d x ,i , x , j 0 as i, j .
Definition 2.25. ([4]) (Complete Metric Space). A soft metric space
X , d, E
is called
complete if every Cauchy Sequence in X converges to some point of X . The soft metric space X , d , E is called incomplete if it is not complete.
Definition 2.26. ([16]) Let X , , E be a soft topological space and W X (1) A family C iJ
F : i J of open soft sets is called an open cover of X, if it satisfies A i
( FA )i E , for each e E . A finite subfamily of a soft open cover
F : i J of A i
W is called a finite subcover of
F : i J . A i
(2) W is called soft compact if every soft open cover of W has a finite subcover.
3. Compact sets on soft metric spaces In this section, we study some important properties of soft metric spaces. ~ ~ ~ ~ Definition 3.1. Let X , d , E be a soft metric space. X , d , E is called soft sequential ~
compact metric space if every soft sequence has a soft subsequence that converges in X .
In the following, we give relations between sequential compactness of parameter metric spaces and soft sequential compactness of soft metric spaces. ~ ~ Proposition 3.2. If X , d , E is a soft sequential compact, then compact, for each e E .
5
X , de
is sequential
~ ~ Proof. Let X , d , E be a soft sequential compact and x n
be any sequence in X , d e , for
n
is obtained in X , d , E . Then there exists a soft subsequence as x . Hence subsequence x converges in X , d . each e E . By using the sequence x n n , soft sequence xen
~
~
n
nk e n
nk
e
The converse of the Proposition 3.2 above may not be true in general. This is shown by the following example. Example 3.3. Let E , X 0,1 and consider the soft metric d xe , ye e e x y , ~
where is natural number set. It is clear that X , d e is sequential compact, for each e E .
1 However, soft sequence n does not have a convergent soft subsequence in 2 n n
~ ~ X ,d, E .
~ ~ ~ ~ Proposition 3.4. Let X , d , E be a soft metric space. X , d , E is soft sequential compact if and only if every infinite soft set F, E has a soft cluster point.
Proof. Let F, E be infinite soft set and xenn be a soft sequence in F, E . Then xenn
converges to a soft point z
has a convergent soft subsequence as xennk . Assume that xennk
k
k
e0
.
~ ~ Since B ze0 , xennk B ze0 , F , E , ze0 is a soft cluster point in F, E . k n : Let xen n be an arbitrary soft sequence. If the sequence is finite, then fixed convergent
F , E , then we say that x
subsequence of soft sequence can be found . Let xenn
n
n en n
is a
soft set. According to condition, this soft set has a soft cluster point as ze0 . Since ~ B ze0 , k F , E ,
~ ~ ~ xennk B ze0 , k F , E , for each k 0 . Thus the soft subsequence k converges to ze0 .
we choose
x nk e nk
~ ~ Definition 3.5. Let X , d , E be a soft metric space, S be a soft set of soft points. If the ~ ~ ~ ~ ~ condition X B xe , is satisfied, then S is said to be a soft net in X , d , E . x e S ~ ~ ~ Definition 3.6. Let X , d , E be a soft metric space. If there exists a finite net of ~ ~ ~ ~ ~ ~ X , d , E , for each 0 , X , d , E is said to be totally bounded.
6
~ ~ Lemma 3.7. Let X , d , E be a totally bounded soft metric space, A be an infinite soft set.
Then there is an infinite soft set B A such that d B , for each 0 . ~
~
~
~
~
~ ~ Proof. Let 0 be arbitrary. Since X , d , E is totally bounded, there exists a finite net 3 n n ~ ~ ~ ~ as H xe11 , xe22 ,..., xenn1 . Then since X B xei i , , A B xei i , A is obtained. For i 1 i 1 n ~ ~ 1 i n , at least one of the soft sets A B xei i , A must have infinite elements. If we i 1 ~
~
denote this set by B , it is clear that d B . ~
~
~ ~ ~ ~ Theorem 3.8. Let X , d , E be a soft metric space. X , d , E is totally bounded if and only if every soft sequence has a Cauchy soft subsequence in X . ~ ~ Proof. Let X , d , E be totally bounded and xenn n be any soft sequence. If the soft n sequence A xen n is finite, the proof is completed. Assume that A xenn n is infinite. From
the Lemma 3.7, there exists an infinite soft set B1 A such that d B1 1 . We choose number n1 such that xenn1 B1 . If we apply Lemma 3.7 to B1 , we obtain an infinite B2 B1 ~
~
1
~
such that d B1 ~
1 . Here we take the number n2 n1 such that xenn2 B2 . Thus we get soft 2 2 .
subsequence xennk
k
is a Cauchy soft subsequence. For arbitrary 0 , ~
Now, let us show that the sequence xennk
~
k
~
~
we choose number k0 such that
1 ~ . Then since xennkk , xennmm Bk 0 , for each k , m k0 , k0 ~
d xennk , xennm
k
m
~
1 ~ k0
is satisfied. This means that xennkk is Cauchy soft subsequence. ~ ~ ~ ~ Assume that X , d , E is not totally bounded soft metric space. That means, X , d , E ~
~
~
~
has not finite 0 net, for some 0 0 . Let xe11 X be an arbitrary soft point. Then there can ~
~
~
be found a soft point xe22 X such that d xe1 , xe22 0 . Since the soft set xe1 , xe2 1
~
~
0 net, there is a soft point xe3 X such that 3
~
~
~
~
d xe1 , xe33 0 , d xe22 , xe33 o . 1
7
1
2
is not
~
~
Thus we constitute a soft sequence xekk such that d xei i , xejj 0 , for all i, j . It is clear that
x has not a Cauchy soft subsequence which k ek
contradicts the fact that every soft sequence has a Cauchy soft subsequence given in assumption. ~ ~ ~ ~ Theorem 3.9. Let X , d , E be a soft metric space. X , d , E is soft sequentially compact ~ ~ metric space if and only if X , d , E is soft complete and soft totally bounded. ~ ~ Proof. Let X , d , E be a soft sequentially compact metric space. Then every soft
~
sequence xenn has a soft subsequence that converges in X . Since the soft subsequence is a
~ ~ Cauchy sequence, then by Theorem 3.8, X , d , E is totally bounded. If xenn is a Cauchy ~ ~ soft sequence in X , d , E and xenn has a convergent soft subsequence, then it is also convergent. ~ ~ Let X , d , E be a soft complete and totally bounded metric space and xenn be an ~ ~ arbitrary soft sequence. Since X , d , E is totally bounded, xenn has a Cauchy soft ~ ~ subsequence. Since X , d , E is soft complete, the Cauchy soft subsequence converges. Then ~ ~ X , d , E is soft sequentially compact metric space.
~ ~ Definition 3.10. Let X , d , E be a soft metric space and a family U be a soft open cover of ~ ~ ~ ~ the space X , d , E . A number 0 is called Lebesque number of U , if there exists ~ ~ F , E U such that B xe , F , E , for all soft point xe X . ~ ~ Proposition 3.11. If X , d , E is a soft sequentially compact metric space then every soft ~
open cover from X has a Lebesque number. ~
Proof. Assume that soft open cover U has not a Lebesque number. Then for any n , there ~ 1 n n exists xen , for each F , E U , where B xen , F , E is not satisfied. Thus we obtain a n
8
~ ~ soft sequence xenn satisfying above condition. Since X , d , E is soft sequentially compact n metric space, the soft sequence xen has a soft subsequence xennk converging to xe . Let
k
~ 2 xe F , E U . Since F, E is soft open set, there is soft open ball B xe , such that m ~ 2 B xe , F , E . Also since the soft subsequence xennk converges to xe , there is a number k m ~ 2 k0 , such that xennkk B xe , , whenever k k0 . We take the number k k0 as nk m . Then m ~ ~ 1 2 B xennk , B xe , F , E U k nk m is obtained and this contradicts with the choice of the soft point xennk .
k
~ ~ Theorem 3.12. Let X , d , E be a soft metric space. Then the following statements are equivalent: ~ ~ a) X , d , E is soft compact. ~ ~ b) X , d , E is sequentially compact. ~ ~ ~ ~ Proof. a b Let X , d , E be a soft compact metric space. Consider X , d , E is not a soft ~
sequentially compact metric space. Then there is an infinite soft set A which does not have a ~ ~ ~ cluster point in X , d , E . Thus, there is a soft number rxe such that B xe , rxe A xe , for
~ ~ A is a soft open cover in X , d , E and this soft open xe A ~ ~ cover has not a finite soft subcover. But this implies X , d , E is not soft compact, which is a contradiction. ~ ~ b a Let X , d , E be a soft sequentially compact and U be any soft open cover in ~ ~ ~ ~ ~ ~ X , d , E . Then by Proposition 3.11, U has a Lebesque number 0 . Since X , d , E is ~
all xe A . A family B xe , rxe
~c
~
~
~ ~ totally bounded, X , d , E has finite net as xe11 , xe22 ,..., xenn . For each k 1,2,..., n 3
9
~ ~ k 2 ~ d B xek , 3 3 ~
is
satisfied.
Then
we
obtain
~ n k X B xek , Fk , E , 3 k 1 k 1 n
~
a
soft
~ k B xek , Fk , E U . 3
set
Since
~ ~ X , d , E is a soft compact metric space.
~ ~ ~ ~ Definition 3.13. Let X , d1 , E1 and Y , d 2 , E2 be two soft metric spaces. The mapping ~ ~ ~ ~ f , : X , d1 , E1 Y , d 2 , E2 is called soft uniformly continuous mapping if given any ~
~
~
~
~
~
~
0 , there exists a 0 ( depending only on ) such that for any soft point xe , ye X when d1 xe , ye satisfied then d 2 f x e , f y e . ~
~
~
~
~ ~ ~ ~ Proposition 3.14. If f , : X , d1 , E1 Y , d 2 , E2 is soft uniformly continuous mapping, then f : X , d1e Y , d2 e is uniformly continuous mapping, for each e E . Proof. The proof is straightforward.
The converse of the Proposition 3.14. above may not be true in general.
f , : X , d1 , E1 Y , d 2 , E2 ~
Theorem 3.15. If
~
~
~
is soft continuous mapping and
~ ~ X , d1 , E1 is soft sequentially compact metric space, then f , is soft uniformly continuous mapping. ~ ~ Proof. Since X , d1 , E1 is soft sequentially compact metric space, it is also soft compact
metric space. For any 0 , since f , is soft continuous, for any soft point xe there exists ~
~
a soft number xe 0 ~
d1 xe , ye 2 xe ~
~
such that for every soft point ye that satisfies the condition
d 2 f x e , f y e ~
~
we
have
~
2
.
Then
the
family
~ ~ ~ ~ ~ is a soft open cover in X , d1 , E1 . Since X , d1 , E1 is soft compact, U B xe , xe ~ x e X
~ ~ this soft open cover has a finite soft subcover as B xe11 , xe11 ,..., B xenn , xenn . We take
10
~ ~ ~ min xe1 ,..., xen .
1
n
Now consider only two soft points ye , ze X such that d1 ye , ze . Assume that ~
~
~
~ ~ ~ ye B xeii , xeii , 1 i n . Then d1 ye , xei i xei i and
d1 ze , xei i d1 ze , ye d1 ye , xei i xei i 2 xei i ~
~
~
~
~
is satisfied. Since f , is soft continuous at soft point xei i ,
~
2
d 2 f y e , f x ~
i
ei
Thus
~
2 . i
ei
d f x
d 2 f y e , f z e d 2 f y e , f xi ~
~
and d 2 f z e , f x ~
~
~
ei
2
As a result , f , is a soft uniformly continuous mapping.
i
ei
, f z e . ~
5. Conclusion The soft set theory was proposed by Molodtsov offers a general mathematical tool for dealing with uncertain and vague objects. Many researchers have contributed towards the topologization of soft set theory. This study contributes some important theorems on soft ~
sequential compact metric spaces. Later we give the concepts of net and totally bounded soft metric space. We continue to investigate some important properties of totally bounded soft metric spaces. Finally we introduce the concepts of Lebesque number and soft uniformly continuous mapping and investigate some theorems in detail.
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