Volume 2008, Article ID 424320,18pages doi:10.1155/2008/424320
Research Article
Some Properties of
r, s
-
T
0
and
r, s
-
T
1
Spaces
S. E. Abbas1 and Biljana Krsteska2
1Department of Mathematics, Faculty of Science, Sohag University, Sohag, P.O. Box 82524, Egypt 2Faculty of Mathematics and Natural Science, Ss. Cyril and Methodius University, Gazi Baba b.b.,
P.O. Box 162, 1000 Skopje, Macedonia
Correspondence should be addressed to S. E. Abbas,sabbas73@yahoo.com
Received 5 May 2008; Revised 23 July 2008; Accepted 1 September 2008
Recommended by Petru Jebelean
We define r, s-quasi-T0, r, s-sub-T0, r, s-T0 and r, s-T1 spaces in an intuitionistic fuzzy
topological space and investigate some properties of these spaces and the relationships between them. Moreover, we study properties of subspaces and their products.
Copyrightq2008 S. E. Abbas and B. Krsteska. 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.
1. Introduction and preliminaries
Kubiak 1 and ˇSostak 2 introduced thefundamental concept of a fuzzytopological structure, as an extension of bothcrisp topology and fuzzy topology3, in the sensethat not only the objectsare fuzzified, but also the axiomatics. In4,5, ˇSostak gave some rules and showed how such an extension can be realized. Chattopadhyay et al.6have redefined the same concept under the name gradation of openness. A general approach to the study of topological-type structures on fuzzy power sets was developed in1,7–10.
As a generalization of fuzzy sets, the notion of intuitionisticfuzzy sets was introduced by Atanassov11. By using intuitionistic fuzzy sets, C¸ oker 12, and C¸ oker and Dimirci
13 defined the topology of intuitionisticfuzzy sets. Recently, Mondal and Samanta 14
introduced the notion of intuitionistic gradation of openness of fuzzy sets, where to each fuzzy subset there is a definite grade of openness and there is a grade of nonopenness. Thus, the concept of intuitionistic gradation of openness is a generalization of the concept of gradation of openness and the topology of intuitionistic fuzzy sets.
In this paper, we definer, s-quasi-T0,r, s-sub-T0,r, s-T0, andr, s-T1spaces in an
that
xty
t, ifyx,
0, ify /x. 1.1
The set of all fuzzy points inX is denoted byPtX. A fuzzy point xt ∈ λ if and only if t < λx. A fuzzy setλis quasi-coincident withμ, denoted byλqμ, if there existsx∈Xsuch thatλx μx>1. Ifλis not quasi-coincident withμ, we denote it by λqμ.
Definition 1.1see14. An intuitionistic gradation of opennessIGO, for shortonXis an ordered pairτ, τ∗of functions fromIXtoIsuch that
IGO1τλ τ∗λ≤1, for allλ∈IX,
IGO2τ0 τ1 1,τ∗0 τ∗1 0,
IGO3τλ1∧λ2≥τλ1∧τλ2andτ∗λ1∧λ2≤τ∗λ1∨τ∗λ2, for eachλ1, λ2∈IX, IGO4τi∈Δλi≥i∈Δτλiandτ∗i∈Δλi≤i∈Δτ∗λi, for eachλi∈IX, i∈Δ.
The tripletX, τ, τ∗is called an intuitionistic fuzzy topological spaceIFTS, for short.
τ and τ∗ may be interpreted as gradation of openness and gradation of nonopenness, respectively.
An IFTSX, τ, τ∗is called stratified if
Sτα 1 andτ∗α 0 for eachα∈I.
Let U,U∗ and τ, τ∗be IGOs on X. We say U,U∗ is finer than τ, τ∗ τ, τ∗ is coarser thanU,U∗ifτλ≤ Uλandτ∗λ≥ U∗λfor allλ∈IX.
Theorem 1.2see3,15. LetX, τ, τ∗be an IFTS. For eachr ∈I0, s∈I1, λ∈IX, an operator C:IX×I0×I1→IXis defined as follows:
Cλ, r, s μ|μ≥λ, τ1−μ≥r, τ∗1−μ≤s . 1.2
Then it satisfies the following properties:
1C0, r, s 0,C1, r, s 1,for allr∈I0,s∈I1; 2Cλ, r, s≥λ;
3Cλ1, r, s≤ Cλ2, r, s, ifλ1≤λ2;
4Cλ∨μ, r, s Cλ, r, s∨ Cμ, r, s,for allr∈I0,s∈I1; 5Cλ, r, s≤ Cλ, r, s, ifr≤r,s≥s, wherer, r ∈I0,s, s ∈I1; 6CCλ, r, s, r, s Cλ, r, s.
Definition 1.3see14. A functionf :X, τ, τ∗→Y,U,U∗is said to be as follows:
1IF continuous ifτf−1μ≥ Uμandτ∗f−1μ≤ U∗μ, for eachμ∈IY;
2IF open ifτμ≤ Ufμandτ∗μ≥ U∗fμ, for eachμ∈IX;
Definition 1.4see16. Let 0/∈ΘXbe a subset ofIX. A pairβ, β∗of functionsβ, β∗:ΘX→I is called an IF topological base onXif it satisfies the following conditions:
B1βλ β∗λ≤1,∀λ∈ΘX,
B2β1 1 andβ∗1 0,
B3βλ1∧λ2≥βλ1∧βλ2andβ∗λ1∧λ2≤β∗λ1∨β∗λ2, for eachλ1, λ2∈ΘX.
An IF topological baseβ, β∗always generates an IGO,τβ, τβ∗∗onXin the following
sense.
Theorem 1.5see16. Letβ, β∗be an IF topological base forX. Define the functionsτβ, τβ∗∗ : IX→Ias follows: for eachμ∈IX,
τβμ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
i∈J
βμi, if μ
i∈J
μi, μi∈ΘX,
1, if μ0,
0, otherwise,
1.3
whereis taken over all families{μi∈ΘX|μi∈Jμi},
τ∗ β∗μ
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
i∈J β∗μ
i, if μ
i∈J
μi, μi∈ΘX,
0, if μ0,
1, otherwise,
1.4
whereis taken over all families{μi∈ΘX|μi∈Jμi}. Then
1 X, τβ, τβ∗∗is an IFTS;
2A mapf :Y, τ, τ∗→X, τβ, τβ∗∗is IF continuous if and only ifβλ ≤ τf−1λand β∗λ≥τ∗f−1λ, for allλ∈Θ
X.
Lemma 1.6see17. LetX be a product of the family{Xi | i ∈ Γ}of sets, and for eachi ∈ Γ, πi:X→Xia projection map. For eachλ∈IX,i, j∈Γ, andλi∈IXi, the following properties hold:
1πiπi−1λi∧λ λi∧πiλ;
2ifxi∈Xiλixi αifori∈Fwith each finite index subsetFofΓ− {j}and putαi∈Fαi,
then
ax∈Xi∈Fπi−1λix α;
bπii∈Fπi−1λi α.
Definition 1.7. LetX, τ, τ∗be an IFTS,μ∈IX,xt∈PtX,r∈I0, ands∈I1. It holds that
Qxt, r, sμ∈IX |xtqμ, τμ≥r, τ∗μ≤s . 1.5
2. Some properties of product intuitionistic fuzzy topological spaces
Theorem 2.1. Let{Xi, τi, τi∗}i∈Γbe a family of IFTSs, letXbe a set and for eachi∈Γ,fi:X→Xi a map. Let
ΘX
0 /μ n
j1
f−1
kj
νkj
|τkj
νkj
>0,∀kj∈K
, 2.1
for every finite setK{k1, . . . , kn} ⊂Γ. Define the functionsβ, β∗:ΘX→IonXby
βμ
n
j1
τkj
νkj
|μn
j1
f−1
kj
νkj
,
β∗μ
n
j1
τ∗ kj
νkj
|μn
j1
f−1
kj
νkj
,
2.2
whereandare taken over all finite subsetsK{k1, k2, . . . , kn} ⊂Γ. Then,
1 β, β∗is an IF topological base onX;
2the IGO,τβ, τβ∗∗generated byβ, β∗is the coarsest IGO onX for which eachi∈Γ,fiis
IF continuous;
3a mapf:Y, τ1, τ1∗→X, τβ, τβ∗∗is IF continuous if and only if for eachi∈Γ,fi◦fis IF
continuous.
Proof. B1It is trivial.
B2Sinceλfi−1λfor eachλ∈ {0,1},β1 β0 1 andβ∗1 β∗0 0.
B3For all finite subsetsK{k1, . . . , kp}andJ {j1, . . . , jq}ofΓsuch that
λp
i1
f−1
ki
λki
, μq
i1
f−1
ji
μji
, 2.3
we have
λ∧μ
p
i1
f−1
ki
λki
∧
q
i1
f−1
ji
μji
. 2.4
Furthermore, we have for eachk∈K∩J,
f−1
k
λk∧fk−1μkfk−1λk∧μk. 2.5
Putλ∧μmi∈K∪Jf−1
miμiρmi,where
ρmi
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
λmi ifmi∈K−K∩J,
μmi ifmi∈J−K∩J,
λmi∧μmi ifmi∈K∩J.
We have
βλ∧μ≥
j∈K∪J τjρj
≥
p
i1
τki
λki
∧
q
i1
τji
μji
,
β∗λ∧μ≤ j∈K∪J
τ∗ j ρj ≤ p
i1
τ∗ ki
λki
∨
q
i1
τ∗ ji
μji
.
2.7
Then,βλ∧μ≥βλ∧βμandβ∗λ∧μ≤β∗λ∨β∗μ.
2For eachλi∈IXi, one family{fi−1λi},andi∈Γ, we have
τβfi−1
λi≥βfi−1
λi≥τiλi, τ∗
β∗
f−1
i
λi≤β∗fi−1
λi≤τi∗
λi.
2.8
Thus, for eachi ∈ Γ,fi : X, τβ, τβ∗∗ → Xi, τi, τi∗is IF continuous. Let fi : X, τ◦, τ◦∗ →
Xi, τi, τi∗ be IF continuous, that is, for each i ∈ Γ and λi ∈ IXi,τ◦fi−1λi ≥ τiλiand τ◦∗f−1
i λi≤τi∗λi. For all finite subsetsK{k1, . . . , kp}ofΓsuch thatλ
p
i1fk−i1λki, we
have
τ◦λ≥p i1
τ◦f−1
ki
λki
≥p
i1
τki
λki
,
τ◦∗λ≤p i1
τ◦∗f−1
ki
λki
≤p
i1
τ∗ ki
λki
.
2.9
It impliesτ◦λ≥βλandτ◦∗λ≤β∗λfor eachλ∈IX. ByTheorem 1.52,τ◦ ≥τβ andτ◦∗ ≤τβ∗∗.
3 ⇒Letf :Y, τ1, τ1∗→X, τβ, τβ∗∗be an IF continuous. For eachi∈Γandλi ∈IXi,
we have
τ1
fi◦f−1λiτ1
f−1f−1
i
λi≥τβfi−1
λi≥τiλi, τ∗
1
fi◦f−1λiτ1∗
f−1f−1
i
λi≤τβ∗∗
f−1
i
λi≤τi∗
λi.
2.10
Hence,fi◦f :Y, τ1, τ1∗→Xi, τi, τi∗is IF continuous.
⇐For all finite subsetsK{k1, . . . , kp}ofΓsuch thatλpi1fk−i1λki, sincefki◦f: Y, τ1, τ1∗→Xki, τki, τk∗iis IF continuous,
τ1
f−1f−1
ki
λki
≥τki
λki
, A
τ∗
1
f−1f−1
ki
λki
≤τ∗ ki
λki
Hence, we have
τ1
f−1λτ 1
f−1
p
i1
f−1
ki
λki
τ1
p
i1
f−1f−1
ki
λki
≥p
i1
τ1
f−1f−1
ki
λki
≥p
i1
τki
λki
, byA,
τ∗
1
f−1λτ∗
1
f−1
p
i1
f−1
ki
λki
τ∗
1
p
i1
f−1f−1
ki
λki
≤p
i1
τ∗
1
f−1f−1
ki
λki
≤p
i1
τ∗ ki
λki
, byB.
2.11
It impliesτ1f−1λ≥ βλandτ∗
1f−1λ≤ β∗λfor allλ ∈IX. ByTheorem 1.52,
f:Y, τ1, τ1∗→X, τβ, τβ∗∗is IF continuous.
Definition 2.2. LetX, τ, τ∗be an IFTS andA ⊂ X. The triple A, τ|A, τ∗|Ais said to be a subspace ofX, τ, τ∗ifτ|A, τ∗|Ais the coarsest IGO onAfor which the inclusion mapiis IF continuous.
Definition 2.3. LetX be the product i∈ΓXi of the family{Xi, τi, τi∗ | i ∈ Γ}of IFTSs. The coarsest IGO, τ, τ∗ τi,τi∗onX for which each the projections πi : X→Xi is IF continuous, is called the product IGO of{τi, τi∗ |i∈Γ}andX, τ, τ∗is called the product IFTS.
Lemma 2.4. Let Y,U,U∗ be an IFTS and β, β∗ an IF topological base on X. If f :
X, β, β∗→Y,U,U∗is a function such thatβλ≤ Ufλandβ∗λ≥ U∗fλfor allλ∈ΘX, thenf :X, τβ, τβ∗∗→Y,U,U∗is IF open.
Proof. Suppose there existsμ∈IXsuch that
τβμ>Ufμ or τβ∗∗μ<U∗
fμ, 2.12
then there exists a family{λi∈ΘX|μi∈Γλi}such that
τβμ≥
i∈Γ
βλi>Ufμ or τβ∗∗μ≤
i∈Γ
β∗λ
On the other hand, sinceβλ≤ Ufλandβ∗λ≥ U∗fλ∀λ∈ΘX, then we have
i∈Γ
βλi≤
i∈Γ
Ufλi≤ U i∈Γ
fλiU
f
i∈Γ
λiUfμ,
i∈Γ
β∗λ i≥
i∈Γ
U∗fλ
i≥ U∗
i∈Γ
fλiU∗
f
i∈Γ
λiU∗fμ.
2.14
It is a contradiction. Hencefis IF open.
Theorem 2.5. LetX, τβ, τβ∗∗be a product space of a family{Xi, τi, τi∗|i∈Γ}of IFTS’s. Then the following statements are equivalent:
1a projection mapπj:X, τβ, τβ∗∗→Xj, τj, τj∗is IF open;
2for every μ i∈Γπi−1λisuch thatx∈Xiλix αi for eachαi ∈ I and i ∈ Γ◦ such
that a finite index subset Γ◦ of Γ− {j} and τiλi > 0, theni∈Γ◦τiλi ≤ τjα and
i∈Γ◦τi∗λi≥τj∗α,whereα
i∈Γ◦αi.
Proof. 1⇒2: For everyμi∈Γπi−1λisuch thatx∈Xiλix αifor eachαi ∈Iandi∈Γ◦
such that a finite index subsetΓ◦ofΓ− {j}. ByLemma 1.62b, we have, forαi∈Γ◦αi,
πjμ πj
i∈Γ◦ π−1
i
λiα. 2.15
Sinceμ∈ΘX, byTheorem 2.1, we have
i∈Γ◦
τiλi≤βμ≤τβμ,
i∈Γ◦ τ∗
i
λi≥β∗μ≥τβ∗∗μ. 2.16
Furthermore, sinceπjis IF open, we have
τβμ≤τjπjμτjα, τβ∗∗μ≥τj∗
πjμτj∗α. 2.17
Hence,i∈Γ◦τiλi≤τjαandi∈Γ◦τi∗λi≥τj∗α.
2⇒1: FromLemma 2.4, we only show thatβμ≤τjπjλandβ∗μ≥τj∗πjλ for all λ ∈ ΘX. Suppose that there exists ν ∈ ΘX such that βν > τjπjν or β∗ν < τ∗
jπjν. Then there exists a finite index subsetΓ◦ofΓ− {j}withνπj−1λj∧i∈Γ◦π
−1
i λi
if necessary, we can takeλj1such that
βν≥τjλj∧
i∈Γ◦
τiλi> τjπjν,
β∗ν≤τ∗ j
λj∨
i∈Γ◦ τ∗
i
λi< τj∗
πjν.
On the other hand, byLemma 1.62, we have
πjν πj
π−1
j
λj∧
i∈Γ◦ π−1
i
λi
λj∧πj i∈Γ◦
π−1
i
λi
λj∧α,
2.19
wherex∈Xiλix αi andα i∈Γ◦αi. Since
i∈Γ◦τiλi≤ τjαand
i∈Γ◦τi∗λi ≥τj∗α, we
have
τjπjντjλj∧α
≥τjλj∧τjα
≥τjλj∧
i∈Γ◦
τiλi,
τ∗ j
πjντj∗
λj∧α
≤τ∗ j
λj∧τj∗
α
≤τ∗ j
λj∨
i∈Γ◦ τ∗
i
λi.
2.20
It is a contradiction.
Theorem 2.6. Let X, τ, τ∗ be a product space of a family {X
i, τi, τi∗ | i ∈ Γ} of IFTSs and
Xj, τj, τj∗be stratified. Then, the following properties hold:
1 X, τ, τ∗is stratified;
2a projection mapπj:X→Xjis IF open.
Proof. 1It is clear from the following: for allα∈I,
τα≥βα
i∈Γ◦
τiλi|α
i∈Γ◦ π−1
i
λi≥τjα1,
τ∗α≤β∗α
i∈Γ◦ τ∗
i
λi|α
i∈Γ◦ π−1
i
λi≤τj∗
α0.
2.21
2 Since τjα 1 and τj∗α 0 for all α ∈ I, it satisfies the condition of Theorem 2.52.
Theorem 2.7. LetX, τ, τ∗be a product space of a family{X
i, τi, τi∗ | i ∈ Γ}of IFTSs and let
Xj, τj, τj∗be stratified. Then for everyXjXj×{yi|i /j}inXparallel toXj,πj|Xj :Xj→Xj
Proof. LetXj Xj ×{yi | i /j}. Since i : Xj→Xj and πj : Xj→Xj are IF continuous, πj◦iπj|Xjis IF continuous. Moreover,πj|Xjis bijective.
Now we only show thatπj|Xjis IF open. Suppose there existsμ∈IXj such that
τ|Xjμ> τj
πj|Xjμ
or τ∗|Xjμ< τ
∗ j
πj|Xjμ
. 2.22
Then there existsν∈IXwithμi−1νsuch that
τ|Xjμ≥τν> τj
πj|Xjμ
or τ∗|Xjμ≤τ
∗ν< τ∗ j
πj|Xjμ
. 2.23
From the definition ofτ, τ∗, there exists a family{νk∈ΘX|νk∈Kνk}such that
τν≥
k∈K
βνk> τjπj|Xjμ or τ∗ν≤
k∈K β∗ν
k< τj∗
πj|Xjμ. C
On the other hand, since each νk ∈ ΘX, there exists a finite index Fk of Γ − {j} with νk πj−1λkj ∧
i∈Fkπ−
1
i λi. Since πi−1λix yi for i /j, then for each x ∈ Xj,
i∈Fkπ− 1
i λix i∈Fkλiyi. Put αk
i∈Fkλiyi. Let μk i− 1ν
k for each k ∈ K. Then,
πj|Xjμkxj μkx|x∈Xj, πj|Xjx xj
i−1ν
kx|x∈Xj, πjx xjμki−1νk
νkx|x∈Xj, πjx xj
π−1
j
λkj
x∧
i∈Fk
π−1
i
λix|x∈Xj, πjx xj
λkj
πjx∧ i∈Fk
λiπix|x∈Xj, πjx xj
λkj
xj∧ i∈Fk
λi
yi
λkj
xj∧α k
λkj∧αk
xj.
Hence,πj|Xjμk λkj∧αk. Thus,
τjπj|Xjμkτjλkj∧αk
≥τjλkj
∧τjαk
τjλkj
≥τjλkj
∧
i∈Fk
λi , τ∗ j
πj|Xj
μk τj∗
λkj∧αk
≤τ∗ j
λkj
∨τ∗ j αk τ∗ j
λkj
≤τ∗ j
λkj
∨
i∈Fk
λi
.
2.25
From the definition ofβ, β∗, it implies
τjπj|Xj
μk≥βνk, τj∗
πj|Xj
μk≤β∗νk. 2.26
Thus,
τjπj|Xjμ
≥
k∈K τjπj|Xj
μk≥
k∈K βνk,
τ∗ j
πj|Xjμ≤
k∈K τ∗
j
πj|Xjμk≤
k∈K β∗ν
k.
2.27
It is a contradiction forC.
In an IFTS{Xi, τi, τi∗|i∈Γ},Xj Xj×{yi |i /j}need not be homeomorphic to Xfrom the following example.
Example 2.8. LetX {x1, x2, x3},Y {y1, y2}, andZ {z1, z2}be sets andW X×Y×Z
a product set. Letπ1 : W→X,π2 : W→Y, andπ3 : W→Zbe the projection maps. Define
τ1, τ1∗:IX→Iby
τ1λ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1 if λ0,1, 1
2 if λλ1, 0 otherwise,
τ∗
1λ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0 ifλ0,1, 1
2 ifλλ1, 1 otherwise,
whereλ1x1 0.5,λ1x2 0.2,and λ1x3 0.3. Also,X
j {x, y2, z2 : x ∈ X},define τ|Xj, τ∗|Xj :IXj→Iby
τ|Xjμ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1 ifμ0,1, 1
2 ifμμ1, 2
3 ifμ0.1, 1
4 if μ0.7, 0 otherwise,
τ∗| Xjμ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0 ifμ0,1, 1
2 ifμμ1, 1
3 ifμ0.1, 3
4 ifμ0.7, 1 otherwise,
2.29
whereμ1x1, y2, z2 0.5,μ1x2, y2, z2 0.2,and μ1x3, y2, z2 0.3. Then the projection
mapπj|Xj :Xj→Xis bijective IF continuous, butπj|Xj is not IF open, because
2 3 τ|Xj
0.1/≤τ1
π1|Xj
0.10. 2.30
Hence,XjandXare not homeomorphic.
Theorem 2.9. LetX, τ, τ∗be a product space of a family{X
i, τi, τi∗|i∈Γ}of IFTSs. Then, the following properties hold:
1Cτ,τ∗i∈Γλi, r, s≤i∈ΓCτi,τ∗
iλi, r, s,∀λi∈IXi, r∈I0, s∈I1; 2ifCτi,τ∗
iλi, r, s λi,∀λi∈IXi, r∈I0, s∈I1, thenCτ,τ∗
i∈Γλi, r, s i∈Γλi.
Proof. 1SupposeCτ,τ∗i∈Γλi, r, s/≤i∈ΓCτi,τ∗
iλi, r, s. Then there existx∈Xandt∈0,1
such that
Cτ,τ∗ i∈Γ
λi, r, s
x≥t > i∈Γ
Cτi,τi∗
λi, r, sx. D
Since i∈ΓCτi,τ∗
iλi, r, s < t, there exists j ∈ Γ such that
i∈ΓCτi,τi∗λi, r, s ≤
π−1
j Cτi,τi∗λi, r, s < t. Putπjx xj. It impliesCτj,τj∗λj, r, sxj < t. From the definition
ofCτj,τj∗, there existsμj∈IXj withλj ≤μjandτj1−μj≥r,τj∗1−μj≤ssuch that
Cτj,τj∗
λj, r, sxj≤μjxj< t. 2.31
On the other hand, we have
λj≤μj⇒πj−1λj≤πj−1μj
⇒
i∈Γ
λi
i∈Γ
π−1
j
λi≤πj−1
μj
⇒ Cτ,τ∗ i∈Γ
λi, r, s
≤π−1
j
μj,
becauseτ1−πj−1μj τπj−11−μj≥τj1−μj≥randτ∗1−πj−1μj τ∗πj−11−μj≤ τ∗
j1−μj≤s. Hence,
Cτ,τ∗ i∈Γ
λi, r, s
x≤π−1
j
μjx μjxj< t. 2.33
It is a contradiction forD. Hence,
Cτ,τ∗
i∈Γ
λi, r, s
≤
i∈Γ
Cτi,τi∗
λi, r, s. 2.34
2It is clear from the following:
i∈Γ
λi≤ Cτ,τ∗ i∈Γ
λi, r, s
≤
i∈Γ
Cτi,τi∗
λi, r, s
i∈Γ
λi. 2.35
3. Some properties ofr, s-T0andr, s-T1spaces
Definition 3.1. An IFTSX, τ, τ∗is said to be as follows.
1 r, s-quasi-T0space if for eachxt, xm∈PtXandt < m, there existsλ∈ Qxm, r, s such thatxtqλ.
2 r, s-sub-T0 space if for eachx /y ∈ X, there existst ∈ I0 such that there exists
λ∈ Qxt, r, ssuch thatytqλ, or there existsμ∈ Qyt, r, ssuch thatxtqμ.
3 r, s-T0space if for eachxt, ym ∈PtX, there existsλ∈ Qxt, r, ssuch thatymqλ, or there existsμ∈ Qym, r, ssuch thatxtqμ.
4 r, s-T1space if for eachxt, ym ∈PtXsuch thatxt/≤ym, there existsλ∈ Qxt, r, s such thatymqλ.
Theorem 3.2. LetX, τ, τ∗be an IFTS. Then the following statements are equivalent:
1 X, τ, τ∗isr, s-T0space;
2for eachxt, ym∈PtX,Qxt, r, s/Qym, r, s;
3for eachxt, ym∈PtX, thenxt/∈ Cτ,τ∗ym, r, sorym/∈ Cτ,τ∗xt, r, s.
Proof. 1⇒2: It is trivial.
2⇒3: Letλ∈ Qxt, r, sandλ /∈ Qym, r, s. Sinceλ /∈ Qym, r, s, we have
ym≤1−λ, τλ≥r, τ∗λ≤s. 3.1
3⇒1: Letxt, ym ∈ PtXand xt∈ C/ τ,τ∗ym, r, s. Thent > Cτ,τ∗ym, r, sximplies xtq1− Cτ,τ∗ym, r, s. SinceCτ,τ∗ym, r, s {μ|μ≥ym, τ1−μ≥r, τ∗1−μ≤s}. Since τ1−μ≥τ1−μandτ∗1−μ≤τ∗1−μ, we haveτ1− Cτ,τ∗ym, r, s≥rand τ∗1− Cτ,τ∗ym, r, s≤s. Hence,1− Cτ,τ∗ym, r, s∈ Qxt, r, s. Sinceym∈ Cτ,τ∗ym, r, sand
ymq1− Cτ,τ∗ym, r, s. Thus,X, τ, τ∗isr, s-T0space.
We can prove the following corollaries as a similar method asTheorem 3.2.
Corollary 3.3. LetX, τ, τ∗be an IFTS. Then the following statements are equivalent:
1 X, τ, τ∗isr, s-quasi-T0space;
2for eachxt, xm∈PtX,Qxt, r, s/Qxm, r, s;
3for eachxt, xm∈PtX, thenxt/∈ Cτ,τ∗xm, r, sorxm/∈ Cτ,τ∗xt, r, s.
Corollary 3.4. LetX, τ, τ∗be an IFTS. Then the following statements are equivalent:
1 X, τ, τ∗isr, s-sub-T0space;
2for eachx /y∈X, there existst∈I0such thatQxt, r, s/Qyt, r, s;
3for eachx /y∈X, there existst∈I0such thatxt/∈ Cτ,τ∗yt, r, soryt/∈ Cτ,τ∗xt, r, s.
Example 3.5. LetX{x, y}be a set. We define an IGOτ, τ∗onXas follows:
τλ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1 if λ1 or 0, 1
2 if λx0.7, 0 otherwise,
τ∗λ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
0 ifλ1 or 0, 1
2 ifλx0.7, 1 otherwise.
3.2
For each x /y ∈ X, there exists 0.4 ∈ I0 such thatx0.7 ∈ Qx0.4,1/2,1/2 and y0.4qx0.7.
Hence, X, τ, τ∗ is 1/2,1/2-sub-T0 space. On the other hand, since Qy0.5,1/2,1/2 Qy0.6,1/2,1/2 {1}, byCorollary 3.32,X, τ, τ∗is not1/2,1/2-quasi-T0space.
Theorem 3.6. LetX, τ, τ∗be an IFTS. Then the following statements are equivalent:
1 X, τ, τ∗isr, s-T1space;
2for eachxt∈PtX,xtCτ,τ∗xt, r, s;
3for eachλ∈IX,λ{μ|λ≤μ, τμ≥r, τ∗μ≤s}.
Proof. 1⇒2: We only show thatCτ,τ∗xt, r, s ≤ xt. Let ym ∈ Cτ,τ∗xt, r, s. Suppose that ym/≤xt. SinceX, τ, τ∗isr, s-T1space, there existsλ∈ Qym, r, ssuch thatxtqλ. It implies xt≤1−λwithτλ≥r andτ∗λ≤s. Hence,Cτ,τ∗xt, r, s≤1−λ. Sinceym∈ Cτ,τ∗xt, r, s≤
1−λ, we haveλ /∈ Qym, r, s. It is a contradiction. Hence,ym ≤ xt. Sinceym ∈ Cτ,τ∗xt, r, s
impliesym≤xt, thenCτ,τ∗xt, r, s≤xt.
2⇒3: Letρ{μ|λ≤μ, τμ≥r, τ∗μ≤s}. We only show thatρ≤λ. Suppose there existx∈Xandt∈0,1such that
Then,λ≤1−xt. SincextCτ,τ∗xt, r, s,
τ1−xtτ1− Cτ,τ∗xt, r, s≥r, τ∗1− Cτ,τ∗xt, r, s≤s. 3.4
Hence,ρ≤1−xt. It is a contradiction.
3⇒1: For eachxt, ym ∈ PtX such that xt/≤ym, 1−xt/≥1 −ym. From 3, since 1−ym{μ|1−ym≤μ, τμ≥r, τ∗μ≤s}, there existsμ1−ym∈IXsuch that
τ1−ym≥r, τ∗1−ym≤s. 3.5
Moreover, since 1−xt/≥1−ym, we have xtq1−ym. Thus 1−ym ∈ Qxt, r, s such that
ymq1−ym. Hence,X, τ, τ∗isr, s-T1space.
Theorem 3.7. LetX, τ, τ∗be a stratified IFTS. ThenX, τ, τ∗isr, s-quasi-T
0 space for allr ∈
I0, s∈I1.
Proof. Letxt, xm∈PtXsuch thatt < m. Then there existsα∈I0such that
t≤1−α < m. 3.6
SinceX, τ, τ∗is stratified IFTS, we haveτα 1 andτ∗α 0. Hence,α∈ Qxm, r, ssuch thatxtqα.
Theorem 3.8. 1Everyr, s-T0space is bothr, s-quasi-T0andr, s-sub-T0. 2Everyr, s-T1space isr, s-T0.
Proof. 1For eachxt, xm∈PtXsuch thatt < m. Byr, s-T0space, there existsλ∈ Qxt, r, s such thatxmqλ. Sincet < m, we havextqλ. So,Xisr, s-quasi-T0.
2For eachxt, ym∈PtX, ifxt≤/ym, byr, s-T1space, there existsλ∈ Qxt, r, ssuch thatymqλ. Also, ifym/≤xt, byr, s-T1space, there existsμ∈ Qym, r, ssuch thatxtqμ. Hence, Xisr, s-T0.
The converse ofTheorem 3.8is not true from the following examples.
Example 3.9. LetX{x, y}be a set. We define an IGOτ, τ∗onXas follows:
τλ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1 ifλαforα∈I, 1
3 ifλμpq, 0 otherwise,
τ∗λ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
0 ifλαforα∈I, 2
3 ifλμpq, 1 otherwise,
3.7
where for each 0< p≤0.4,μpqx pandμpqy q, 0≤q < p. SinceX, τ, τ∗is a stratified IFTS, byTheorem 3.7,X, τ, τ∗isr, s-quasi-T0space for eachr∈I0ands∈I1.
Ifr >1/3,s <2/3, andt∈I0, then for eachxt, yt∈PtX, we have
ByTheorem 3.22andCorollary 3.42,X, τ, τ∗is neitherr, s-sub-T0norr, s-T0 forr >
1/3 ands <2/3.
If 0< r ≤1/3, 2/3 ≤s < 1,andx /y ∈X, there exists 0.7 ∈I0such that there exists
μ2/50 ∈ Qx0.7, r, swith y0.7qμ2/50. Hence, X, τ, τ∗isr, s-sub-T0 for 0 < r ≤ 1/3 and
2/3≤s <1.
Forx0.3, y0.3 ∈PtX, 0< r≤1/3, and 2/3≤s <1, we haveQx0.3, r, s Qy0.3, r, s {α|0.7< α}. Hence, it is notr, s-T0for 0< r ≤ 1/3 and 2/3 ≤ s <1. For 0 < r ≤1/3 and
2/3≤s <1,X, τ, τ∗is bothr, s-quasi-T0andr, s-sub-T0, but notr, s-T0.
Example 3.10. LetX{x, y}be a set. We define an IGOτ, τ∗onXas follows:
τλ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1 ifλαforα∈I, 1
2 ifλμpq, 0 otherwise,
τ∗λ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
0 ifλαforα∈I, 1
2 ifλμpq, 1 otherwise,
3.9
where for each 0< p <1,μpqx pandμpqy q, 0≤q < p. Letzt, zm ∈PtXwitht /m forzxory. We have
Q
zt,12,12
/
Q
zm,12,12
. 3.10
Forxt, ym ∈PtX, forp >1−t, we haveμp0∈ Qxt,1/2,1/2withymqμp0. Hence,X, τ, τ∗
is1/2,1/2-T0space. On the other hand, lety0.5/≤x0.5. For eachμpq∈ Qy0.5,1/2,1/2, since
q0.5 > 1 andp > q, we havex0.5qμpq, that is,Qy0.5,1/2,1/2 ⊂ Qx0.5,1/2,1/2. Thus X, τ, τ∗is notr, s-T
1space.
Theorem 3.11. Every subspace ofr, s-quasi-T0(resp.,r, s-sub-T0,r, s-T0, andr, s-T1) space
isr, s-quasi-T0(resp.,r, s-sub-T0,r, s-T0, andr, s-T1) space.
Proof. LetX, τ, τ∗ber, s-T1space. Letat, bm∈PtAsuch thatat/≤bm. Then,at, bm∈PtX
such thatat/≤bm. SinceX, τ, τ∗isr, s-T1, there existsλ ∈ Qat, r, ssuch thatbmqλ. Since τAi−1λ ≥τλ ≥randτA∗i−1λ≤ τ∗λ≤ s, we havei−1λ∈ Qτ|A,τ∗|Aat, r, ssuch that
bmqi−1λ. The others are similarly proved.
We can prove the following theorem as a similar method asTheorem 3.11.
Theorem 3.12. Every IF homeomorphic space ofr, s-quasi-T0 (resp.,r, s-sub-T0,r, s-T0, and r, s-T1) space isr, s-quasi-T0(resp.,r, s-sub-T0,r, s-T0, andr, s-T1) space.
Theorem 3.13. Let{Xi, τi, τi∗|i∈Γ}be a family ofr, s-quasiT0(resp.,r, s-sub-T0,r, s-T0,
andr, s-T1) space. Letτ, τ∗be the product IGO onX
i∈ΓXi. ThenX, τ, τ∗isr, s-quasi-T0
(resp.,r, s-sub-T0,r, s-T0, andr, s-T1) space.
Proof. Letxt, ym∈PtXsuch thatxt/≤ym. Then there existsi∈Γsuch thatπixt/≤πiym. SinceXi, τi, τi∗isr, s-T1space, there existsλ∈IXi such that
λ∈ Qτi,τi∗
Sinceπixt πixtqλif and only ifxtqπi−1λ, we have
π−1
i λ∈ Q
xt, r, s, ymqπi−1λ. 3.12
Therefore,X, τ, τ∗isr, s-T1space. The others are similarly proved.
Theorem 3.14. Let {Xi, τi, τi∗ | i ∈ Γ}be a family of IFTSs. Letτ, τ∗be the product IGO on Xi∈ΓXi. IfX, τ, τ∗isr, s-sub-T0space, thenXi, τi, τi∗isr− , s -sub-T0space for each
>0 and for eachi∈Γ.
Proof. Letxj, yj ∈Xjsuch thatxj /yj. Then there existsxi ∈Xifor alli∈Γ− {j}such that
x /y∈Xand
πix
xi if i∈Γ− {j},
xj if ij, πiy
xi ifi∈Γ− {j},
yj ifij.
3.13
SinceX, τ, τ∗isr, s-sub-T0space, there existst∈0,1such that
ρ∈ Qxt, r, s, ytqρ. 3.14
Letβ, β∗be a base forτ, τ∗. Sinceτρ≥randτ∗ρ≤s, byTheorem 1.5, for >0, there exists a family{ρk|ρk∈Δρk}such that
τρ≥
k∈Δ
βρk> r− , τ∗ρ≤
k∈Δ
β∗ρ
k< s . 3.15
Sincextqρ k∈Δρk, there exists k ∈ Γsuch thatxtqρk,βρk > r− andβ∗ρk < s . Then, there exists a family{λi|ρki∈Fπi−1λi}whichFis a finite subset ofΓsuch that
βρk≥
i∈F
τiλi> r− , β∗ρk≤
i∈F τ∗
i
λi< s . E
Without loss of generality, we may assumej∈Fbecause we can takeF1F∪ {j}such that
λj 1,τj1 1,andτj∗1 0, if necessary. Sincextqρkandytqρk,
t > i∈F−{j}
1−λiπix∨1−λjxj, F
t≤
i∈F−{j}
1−λiπix∨1−λjyj. G
Ifi∈F−{j}1−λiπix≥t, it is a contradiction forFandG. Thus
i∈F−{j}
It implies
t >1−λjxj, t≤1−λjyj. 3.17
Furthermore, byE, we haveτjλj> r− andτj∗λj< s . Hence,
λj∈ Qτj,τj∗
xj
t, r− , s
, yj
tqλj. 3.18
Thus,Xj, τj, τj∗isr− , s -sub-T0space.
In the above theorem, ifX, τ, τ∗isr, s-T1 resp.,r, s-quasi-T0 andr, s-T0, it is
not true from the following example.
Example 3.15. LetX{x}andY {y}be sets. Define IGOτ1, τ1∗onXas follows:
τ1λ
1 ifλαforα∈I, 0 otherwise, τ
∗
1λ
0 ifλαforα∈I,
1 otherwise, 3.19
and IGOτ2, τ2∗onYas follows:
τ2λ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1 ifλ0 or 1, 1
2 ifλy0.2, 0 otherwise,
τ∗
2λ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
0 ifλ0 or 1, 1
2 ifλy0.2, 1 otherwise.
3.20
LetX×Y {x, y}be a productset andτ1τ2, τ1∗τ2∗the product IGO onX×Y. Since x, y0.2π−1
1 0.2 π2−1y0.2, byTheorem 1.5, we have
τ1
τ2
0.2τ1
0.2∨τ2
y0.2
1, τ1∗τ2∗0.2τ2
0.2∧τ2∗y0.2
0. 3.21
We can obtain the product IGO,τ1
τ2, τ1∗
τ∗
2as follows:
τ1
τ2λ
1 ifλαforα∈I, 0 otherwise,
τ∗
1
τ∗
2λ
0 ifλαforα∈I, 1 otherwise.
3.22
Then,X×Y, τ1τ2, τ1∗τ2∗arer, s-T1,r, s-T0, andr, s-quasi-T0for allr ∈I0, s∈I1.
ButY, τ2, τ2∗is notr, s-quasi-T0for allr∈I0, s∈I1. Hence, it is neitherr, s-T0norr, s-T1
for allr∈I0, s∈I1.
Theorem 3.16. Let{Xi, τi, τi∗|i∈Γ}be a family of IFTSs andτ, τ∗be the product IGO onX
Proof. Let X, τ, τ∗ and Xj Xj ×{yi | i / j}of X parallel toXj. Since Xj, τ|Xj, τ
∗| Xj
is a subspace of X, τ, τ∗, by Theorem 3.11, Xj, τ|Xj, τ∗|Xj is r, s-quasi-T0 resp., r, s
-sub-T0, r, s-T0, and r, s-T1. Since Xj, τj, τj∗ is stratified, by Theorem 2.7, πj|Xj :
Xj, τ|Xj, τ∗|Xj→Xj, τj, τj∗ is IF homeomorphism. FromTheorem 3.12,Xj, τj, τj∗isr, s -quasi-T0resp.,r, s-sub-T0,r, s-T0, andr, s-T1.
Acknowledgment
The authors are very grateful to the referees for their valuable suggestions in rewriting the paper in the present form.
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