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J. Math. Comput. Sci. 6 (2016), No. 5, 757-766 ISSN: 1927-5307
COMPLETENESS AND COMPACTNESS OF THE CARTESIAN PRODUCT OF TWO SOFT METRIC SPACES
JINGJING BAI∗, MEIMEI SONG
College of Science, Tianjin University of Technology, Tianjin 300384, China
Copyright c2016 Jingjing Bai and Meimei Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. In this paper, we prove that the Cartesian product of two soft metric spaces is a soft metric space then we consider the completeness and compactness of the Cartesian product space.
Keywords: Soft metric space; Cartesian product; Cauchy sequence; Complete soft metric space; soft sequential compact metric space.
2010 AMS Subject Classification:46A19, 46A30.
1. Introduction
Because of various uncertainties arising in our real world, methods of classical mathematics may not be successfully applied to solve them. Thus, new mathematical theories such as prob-ability theory, interval mathematics and fuzzy set theory have been introduced by researchers. But each of these theories has its certain inherent difficulties as mentioned by Molodtsov [1]. In 1999, Molodtsov [1] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties and studied some properties of this theory. From then on, many researchers started to study the theory of soft sets and a lot of activities have been shown in soft sets theory.
∗Corresponding author Received January 4, 2016
In 2010, in order to introduce soft set relations, Babitha and Sunil gave a definition of the Carte-sian product of two soft sets. In 2013, Das and Samanta introduced the notion of soft metric space by using a different concept of soft point and investigated some basic properties of soft metric space. In this paper, our purpose is to study the completeness and compactness of the Cartesian product space of two soft metric spaces corresponding to a common finite parameter set.
2. Preliminaries
In this section, we will recall some basic notions and examples about soft set theory.
Definition 2.1[1] LetU be an initial universe set,E be a set of parameters andP(U) denotes the power set ofU. A pair(F,E)is called a soft set (overU), whereF is a mapping given by
F:E →P(U).
Example 1[1] Soft set(F,E)describes the attractiveness of the houses which Mr. X is going to buy.U is the set of houses under consideration whereU={h1,h2,h3,h4,h5,h6},E is the set of
parameters. Each parameter is a word or a sentence. E={beauti f ul,wooden,in the green surround
-ings,in good repair}={e1,e2,e3,e4}. The mapping F :E →P(U) be defined by F(e1) = {h1,h4,h5}, F(e2) ={h1,h2,h3}, F(e3) ={h4,h5,h6}, F(e4) ={h2,h3,h4,h5,h6}. Then the soft set(F,E) ={beauti f ul houses={h1,h4,h5},wooden houses={h1,h2,h3},in the green sur
-roundings houses={h4,h5,h6},in good repair houses={h2,h3,h4,h5,h6}}.
Definition 2.2[1] A soft set (F,E)over U is said to be a null soft set denoted by Φif for all
e∈E,F(e) =φ.
Definition 2.3[1] A soft set(F,E)overU is said to be an absolute soft set denoted byUe if for
alle∈E,F(e) =U.
Definition 2.4[10] For two soft sets(F,A)and(G,B)over a common universeU, we say that
(F,A)is a soft subset of(G,B)ifA⊂Band∀e∈A, F(e)⊂G(e).
H(e) =
F(e) i f e∈A−B
G(e) i f e∈B−A
F(e)∪G(e) i f e∈A∩B
We write(F,A)∪e(G,B) = (H,C).
Definition 2.6[8] An intersection of two soft sets(F,A)and(G,B)over a common universeU, is the soft set(H,C)whereC=A∩B, and∀e∈C,H(e) =F(e)∩G(e), we write(F,A)∩e(G,B) =
(H,C).
Definition 2.7[2] LetRbe the set of real numbers,B(R)the collection of all non-empty bounded subsets ofRandAtaken as a set of parameters. Then a mapping fe:A→B(R)is called a soft
real set. If specifically ef is a single valued mapping on A taking values in the set R, then
the pair (ef,A) or simply feis called a soft real number. If ef is a single valued mapping onA
taking values in the set R+ of nonnegative real numbers, then the pair (fe,A) or simply feis
called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers (corresponding toA) byR(A)∗. We will denote a particular type of soft real numbers by ¯r, ¯s, ¯t, Such as ¯0 is a soft real number where ¯0(λ) =0, for allλ ∈A.
Example 2[2](Example of soft real number) Consider the example 1. If a mapping ef :E → B(R) is defined by ef(e) = the number of houses which satisfy the contition e. Then we get
e
f(e1) ={3}, ef(e2) = {3}, ef(e3) = {3}, fe(e4) = {5}. Then (ef,E) can be taken as a soft
real number such that (ef,E) ={beauti f ul houses=3,wooden houses =3,in the green sur -roundings houses=3,in good repair houses=5}.
Definition 2.8[3] For two soft real numbers ef, e
g, we define
(a) fe≤e ge if fe(λ) ≤ ge(λ), for allλ ∈A;
(b) fe≥e ge if fe(λ) ≥ ge(λ), for allλ ∈A;
(c) ef <e ge if ef(λ)<ge(λ), for allλ ∈A;
(d) ef >e ge if ef(λ)>ge(λ), for allλ ∈A.
Definition 2.10[3] A soft pointPx
λ is said to belongs to a soft set (F,A)if λ ∈Aand P(λ) =
{x} ⊂F(λ). We writePλx ∈e(F,A).
Definition 2.11[3] Two soft pointsPx
λ,P
y
µ are said to be equal if λ =µ andP(λ) =P(µ), that isx=y. ThusPx
λ 6=P
y
µ ⇔x6=yorλ 6=µ.
Let X be an initial universal set and A be the non-empty set of parameters. Let Xe be the
absolute soft set i.e.F(λ) =X,∀λ ∈A, where(F,A) =Xe. LetSP(Xe)denotes the collection of
all soft points ofXeandR(A)∗denotes the set of all non-negative soft real numbers. We get the
following definition:
Definition 2.12[3] A mappingd : SP(Xe)×SP(Xe)→R(A)∗, is said to be a so f t metricon the
soft setXeifd satisfies the following conditions:
(M1)d(Pλx,Pµy)≥e e0,for allPλx, P y
µ∈eXe;
(M2)d(Px
λ,P
y
µ) = e0 ⇔ Px
λ = P
y
µ;
(M3)d(Px
λ,P
y
µ) = d(P
y
µ,Pλx), for allPλx, P
y
µ∈eXe;
(M4)For allPλx, Pµy, Pγz∈eXe, d(Pλx,Pγz)≤e d(Pλx, P y
µ) + d(P
y
µ,P
z
γ).
The soft set Xe with a soft metric d is called a so f t metric spaceand denoted by (Xe,d,A)or
(Xe,d).
Definition 2.13[3] Let {Pxn
λn} be a sequence of soft points in a soft metric space(Xe,d). The
sequence {Pxn
λn} is said to be convergent in (Xe,d) if there is a soft point P x
λ ∈e Xe such that
d (Pxn
λn , P x
λ) → ¯0 as n →∞. This means for everyeε ≥e ¯0, chosen arbitrarily, there exists a
natural numberN=N(eε)such that ¯0≤e d(P xn
λn , P x
λ)≤e eε, whenevern>N. Definition 2.14[3] A sequence{Pxn
λn}of soft points in(Xe,d)is considered as a Cauchy sequence
in Xe if corresponding to everyeε ≥e ¯0, ∃ m∈N such thatd (P xi
λi, P xj
λj) ≤e eε, ∀ i, j ≥ m,i.e. d(Pxi
λi, P xj
λj) → ¯0 asi, j → ∞.
Definition 2.15[3] A soft metric space(Xe,d)is called complete if every Cauchy sequence inXe
converges to some soft point ofXe.
Definition 2.16 A soft metric space (Xe,d,A)is called soft sequential compact metric space if
3. Main results
In this section, we discuss the completeness and compactness of the cartesian product of two soft metric spaces.
Definition 3.1[4] Let(F,A) and(G,B) be two soft sets overU, then the cartesian product of
(F,A)and(G,B)is defined as(F,A)×(G,B) = (H,A×B)whereH: A×B→P(U×U)and
H(a,b) =F(a)×G(b),(a,b)∈A×B, i.e.H(a,b) ={(hi,hj): hi∈F(a), hj∈F(b)}.
Example 3[4] Consider the soft set(F,A)which describes the ”cost of the houses” and the soft set(G,B)which describes the ”attractiveness of the houses”.
Suppose thatU={h1,h2,h3,h4,h5,h6},A={very costly; costly; cheap},
B={beauti f ul,wooden,in the green surroundings,in good repair}
LetF(very costly) ={h2,h4},F(costly) ={h1},F(cheap) ={h3,h5,h6},
G(beauti f ul) ={h1,h4,h5},G(wooden) ={h1,h2,h3},
G(in the green surroundings) ={h4,h5,h6},G(in good repair) ={h2,h3,h4,h5,h6}. Now(F,A)×(G,B) = (H,A×B)where a typical element will look like
H(very costly, beauti f ul) ={h2,h4} × {h1,h4,h5}={(h2,h1),(h2,h4),(h2,h5),(h4,h1),
(h4,h4),(h4,h5)}
Definition 3.2Let(Xe, d1, A)and(Ye, d2, A)be two soft metric spaces,Ais a finite parameter
set. The cartesian product of (Xe, d1, A) and (Ye, d2, A) is the product space(Xe×Ye, d, A)
whereXe×Yeis the cartesian product ofXeandYe,dis a mapping fromSP(Xe×Ye) × SP(Xe×Ye)
into R(A)∗ given by: d ((Px1
λ1, P y1
µ1), (P x2
λ2, P y2
µ2)) = max{d1(P x1
λ1, P x2
λ2),d2(P y1
µ1, P y2
µ2)}, where
(Px1
λ1, P y1
µ1), (P x2
λ2, P y2
µ2)∈SP(Xe×Ye).
Lemma 3.1Let(Xe, d1, A)and(Ye, d2, A)be two soft metric spaces, then the cartesian product
(M2)d((Px1
λ1, P y1
µ1), (P x2
λ2, P y2
µ2)) =¯0⇔ max{d1(P x1
λ1, P x2
λ2),d2(P y1
µ1, P y2
µ2)}=¯0 ⇔d1(Px1
λ1, P x2
λ2) =d2(P y1
µ1, P y2
µ2) =¯0 ⇔Px1
λ1 =P x2
λ2 and P y1
µ1 =P y2
µ2
⇔(Px1
λ1, P y1
µ1) = (P x2
λ2, P y2
µ2)
(M3)is obviously established by the symmetry ofd1andd2;
(M4)For all(Pxi
λi, P yi
µi)∈e SP(Xe×Ye) (i=1,2,3) d((Px1
λ1, P y1
µ1), (P x3
λ3, P y3
µ3))
=max{d1(Px1
λ1, P x3
λ3), d2(P y1
µ1, P y3
µ3)}
e
≤max{d1(Px1
λ1, P x2
λ2) +d1(P x2
λ2, P x3
λ3), d2(P y1
µ1, P y2
µ2) +d2(P y2
µ2, P y3
µ3)}
e
≤max{d1(Px1
λ1, P x2
λ2), d2(P y1
µ1, P y2
µ2)}+max{d1(P x2
λ2, P x3
λ3), d2(P y2
µ2, P y3
µ3)}
=d((Px1
λ1, P y1
µ1), (P x2
λ2, P y2
µ2)) +d((P x2
λ2, P y2
µ2), (P x3
λ3, P y3
µ3))
Then(Xe×Ye, d, A)is a soft metric space.
Lemma 3.2 {(Pxn
λn, P yn
µn)} is a Cauchy sequence in (Xe×Ye, d, A), if and only if {P xn
λn} is a
Cauchy sequence in(Xe, d1, A)and{Pµynn}is a Cauchy sequence in(Ye, d2, A).
Proof. “⇒” {(Pxn
λn, P yn
µn)}is a Cauchy sequence in(Xe×Ye, d, A),
then for anyeε ≥e ¯0, ∃m∈Nsuch thatd((P xi
λi, P yi
µi), (P xj
λj, P yj
µj))≤e eε, ∀i, j ≥ m,
That ismax{d1(Pxi
λi, P xj
λj), d2(P yi
µi, P yj
µj)≤e eε, ∀i, j ≥ m.
So we haved1(Pλxii, P xj
λj)≤e eε andd2(P yi
µi, P yj
µi)≤e eε asi, j ≥ m,
then{Pxn
λn}is a Cauchy sequence inXeand{P yn
µn}is a Cauchy sequence inYe.
“⇐” Let{Pxn
λn}is a Cauchy sequence inXeand{P yn
µn}is a Cauchy sequence inYe.
Then for anyeε ≥e ¯0, there existsm1∈N, such thatd1(Pλxii, P xj
λj)≤e eε whereasi, j≥m1,
in the similar way, we can find am2∈N, such thatd2(Pyi
µi, P yj
µi)≤e eε asi, j ≥ m.
Letm=max{m1,m2}, then wheni, j≥m, we have
d((Pxi
λi, P yi
µi), (P xj
λj, P yj
µj)) =max{d1(P xi
λi, P xj
λj),d2(P yi
µi, P yj
µi)}≤e eε,
Theorem 3.1 (Xe×Ye, d, A) defined as above is a complete soft metric space if and only if
(Xe, d1, A)and(Ye, d2, A)are two complete soft metric spaces. Proof. “⇒” Let {Pxn
λn} is a Cauchy sequence in (Xe, d1, A), {P yn
µn} is a Cauchy sequence in (Ye, d2, A).
Then by Lemma 3.2, we have{(Pxn
λn,P yn
µn)}is a Cauchy sequence in(Xe×Ye, d, A).
Since(Xe×Ye, d, A)is complete, so there exists a soft point(Px
λ,P
y
µ)inXe×Ye, such that
for anyeε≥e ¯0, ∃N≥0,d((P xn
λn,P yn
µn),(P x
λ,P
y
µ))≤e eε whereasn≥N.
That ismax{d1(Pλxnn,Pλx),d2(Pµynn,P y
µ)}≤e eε asn≥N,
then we haved1(Pxn
λn,P x
λ)≤e eε andd2(P yn
µn,P y
µ)≤e eεwhenn≥N, andP x
λ ∈Xe, P
y
µ∈Ye,
so(Xe, d1, A)and(Ye, d2, A)are complete.
“⇐” Let{(Pxn
λn,P yn
µn)}is a Cauchy sequence in(Xe×Ye, d, A),
Then by Lemma 3.2, we have{Pxn
λn}is a Cauchy sequence in(Xe, d1, A), {Pyn
µn}is a Cauchy sequence in(Ye, d2, A).
Since(Xe, d1, A)and(Ye, d2, A)are complete, so there existPλx∈XeandP y
µ ∈Ye,
such that for anyeε ≥e ¯0, ∃N1>0, such thatd1(Pλxnn,Pλx)≤e eε asn ≥N1,
∃N2>0, such thatd2(Pyn
µn,P y
µ)≤e eε asn ≥N2.
LetN=max{N1,N2}, then whenn≥N, we have
d((Pxn
λn,P yn
µn),(P x
λ,P
y
µ)) =max{d1(P
xn
λn,P x
λ),d2(P
yn
µn,P y
µ)}≤e eε and(P x
λ,P
y
µ)∈Xe×Ye.
So(Xe×Ye, d, A)is complete.
Theorem 3.2(Xe×Ye, d, A)defined as above is a soft sequential compact metric space if and
only if(Xe, d1, A)and(Ye, d2, A)are two soft sequential compact metric spaces. Proof.“⇒” Let{Pxn
λn}is a soft points sequence in(Xe, d1, A), andP yn
µn is a soft points sequence
in(Ye, d2, A).
Then by the definition of the Cartesian product, we have{(Pxn
λn,P yn
µn)}is a soft points sequence
in(Xe×Ye, d, A).
Since (Xe×Ye, d, A) is a soft sequential compact metric space, so there is a subsequence
{(Pxnk
λnk,P
ynk
µnk)}converges to(Pλx,P
y
µ)∈eXe×Ye, that is for anyeε ≥e ¯0, ∃N>0, such that d((Pxnk
λnk,P
ynk
µnk),(Pλx,P
y
µ)) =max{d1(P
xnk
λnk,P
x
λ),d2(P
ynk
µnk,P
y
Sod1(P xnk
λnk,P
x
λ)≤e eε andd2(P ynk
µnk,P
y
µ)}≤e eε asn>N.
Then we can get a subsequence{Pxnk
λnk}of{P
xn
λn}which converged inXeand a subsequence{P ynk
µnk}
of{Pyn
µn}which converged inYe.
“⇐” For any soft points sequence{(Pxn
λn,P yn
µn)}in(Xe×Ye, d, A), {Pxn
λn}is a soft points sequence in(Xe, d1, A)and{P yn
µn}is a soft points sequence in(Ye, d2, A).
Because of(Xe, d1, A)is a soft sequential compact metric space,
we can get a subsequence{Pxnk
λnk}of{P
xn
λn}which converged toP x
λ ∈e Xe,
for the indexnk,{Pµynk
nk}is a subsequence of{P yn
µn}and also is a soft points sequence inYe,
since(Ye, d2, A)is a soft sequential compact metric space, so we can find a subsequence{P ynkm
µnkm}
of{Pµynk
nk}and a soft pointP y
µ inYe, such that{P ynkm
µnkm}converges toP
y
µ. Since{Pxnk
λnk}is a convergent sequence and it converges toP
x
λ, so{P
xnkm
λnkm}converges toP
x
λ. Then we have a subsequence{Pxnkm
λnkm,P
ynkm
µnkm}of{(P
xn
λn,P yn
µn)}which converged to(P x
λ,P
y
µ)
e
∈Xe×Ye.
Definition 3.3 d 0, d are two soft metrics on Xe×Ye, if for any (P x1
λ1,P y1
µ1), (P x2
λ2,P y2
µ2) ∈e Xe×Ye
d0((Px1
λ1,P y1
µ1),(P x2
λ2,P y2
µ2))≤e d((P x1
λ1,P y1
µ1),(P x2
λ2,P y2
µ2)), that is for anyη∈A, we haved 0
((Px1
λ1,P y1
µ1),
(Px2
λ2,P y2
µ2))(η)≤ d((P x1
λ1,P y1
µ1), (P x2
λ2,P y2
µ2))(η), then we sayd 0
is stronger thand.
Example 4 (Xe, d1, A) and (Ye, d2, A) are two soft metric spaces, (Xe×Ye, d, A) defines as
definition 3.2,(Px
λ1,P y
µ1), (P x
λ2,P y
µ2)∈Xe×Ye, ¯0<e t¯<e ¯1.
Letd 0((Px1
λ1,P y1
µ1), (P x2
λ2,P y2
µ2)) =t d¯ 1(P x1
λ1,P x2
λ2) + (1−t)d2(P y1
µ1,P y2
µ2),
we can deduced0 is a soft metric onXe×Ye, and for anyη∈A,we have d 0((Px1
λ1,P y1
µ1), (P x2
λ2,P y2
µ2))(η) = [t d¯ 1(P x1
λ1,P x2
λ2) +1−t d2(P y1
µ1,P y2
µ2)](η)
e
≤max{d1(Px1
λ1,P x2
λ2),d2(P y1
µ1,P y2
µ2)}(η)
=d((Px1
λ1,P y1
µ1), (P x2
λ2,P y2
µ2))(η)
thend 0 is stronger thand.
Properties Let (Xe, d, A) is a soft metric space, d
0
is another soft metric on Xe, if for any
sequence of soft points{Pxn
λn}and a soft pointP x
λ,d
0
is stronger thand, then we have: (i) if{Pxn
λn}is a Cauchy sequence in(Xe, d, A), then it is a Cauchy sequence in(Xe, d 0
(ii) ifd(Pxn
λn,P x
λ)→¯0, thend
0 (Pxn
λn,P x
λ)→¯0. Then proof is straightforward.
Theorem 3.3(Xe×Ye, d, A) is a soft sequential compact space, then(Xe×Ye, d
0
, A)is also a soft sequential compact metric space, whereasd 0is stronger thand.
Proof. For any sequence of soft points {(Pxn
λn, P yn
µn)} in Xe×Ye, since (Xe×Ye, d, A) is a soft
sequential compact space, there exists a subsequence {(Pxnk
λnk, P
ynk
µnk)} converges to (Pλx, P
y
µ)∈
e
X×Ye, because ofd
0
is stronger thand, by Properties(ii), we haved0((Pxnk
λnk, P
ynk
µnk),(Pλx, P
y
µ))→ ¯0, then we complete the proof.
Conflict of Interests
The authors declare that there is no conflict of interests.
Acknowledgements
The authors are grateful to the reviewers for their valuable comments which helped in rewriting the paper in its present form. The authors are also grateful to the editor of this journal for their kind help.
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