Some Properties of (r,s)-T0 and (r,s)-T1 Spaces

Abstract

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.

Full text

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Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 424320, 18 pages
doi:10.1155/2008/424320
Research Article
Some Properties of ?r,s?-T0and ?r,s?-T1Spaces
S. E. Abbas1and 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.
Copyright q 2008 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 topology ?3?, in the sensethat not
only the objectsare fuzzified, but also the axiomatics. In ?4, 5?,ˇSostak gave some rules and
showed how such an extension can be realized. Chattopadhyay et al. ?6? have 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 in ?1, 7–10?.
As a generalization of fuzzy sets, the notion of intuitionisticfuzzy sets was introduced
by Atanassov ?11?. 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 define ?r,s?-quasi-T0, ?r,s?-sub-T0, ?r,s?-T0, and ?r,s?-T1spaces 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.
Throughout this paper, let X be a nonempty set, I ? ?0,1?, and I0 ? ?0,1? and I1 ?
?0,1?. For α ∈ I, α?x? ? α for all x ∈ X. A fuzzy point xtfor t ∈ I0is an element of IXsuch

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2International Journal of Mathematics and Mathematical Sciences
that
xt?y? ?
?
t,
0,
if y ? x,
if y / ? x.
?1.1?
The set of all fuzzy points in X is denoted by Pt?X?. A fuzzy point xt ∈ λ if and only if
t < λ?x?. A fuzzy set λ is quasi-coincident with μ, denoted by λqμ, if there exists x ∈ X such
that λ?x? ? μ?x? > 1. If λ is not quasi-coincident with μ, we denote it by λqμ.
Definition 1.1 ?see ?14??. An intuitionistic gradation of openness ?IGO, for short? on X is an
ordered pair ?τ,τ∗? of functions from IXto I such that
?IGO1? τ?λ? ? τ∗?λ? ≤ 1, for all λ ∈ IX,
?IGO2? τ?0? ? τ?1? ? 1, τ∗?0? ? τ∗?1? ? 0,
?IGO3? τ?λ1∧ λ2? ≥ τ?λ1? ∧ τ?λ2? and τ∗?λ1∧ λ2? ≤ τ∗?λ1? ∨ τ∗?λ2?, for each λ1,λ2∈ IX,
?IGO4? τ??
The triplet ?X,τ,τ∗? is called an intuitionistic fuzzy topological space ?IFTS, for short?.
τ and τ∗may be interpreted as gradation of openness and gradation of nonopenness,
respectively.
An IFTS ?X,τ,τ∗? is called stratified if
?S? τ?α? ? 1 and τ∗?α? ? 0 for each α ∈ I.
Let ?U,U∗? and ?τ,τ∗? be IGOs on X. We say ?U,U∗? is finer than ?τ,τ∗? ??τ,τ∗? is
coarser than ?U,U∗?? if τ?λ? ≤ U?λ? and τ∗?λ? ≥ U∗?λ? for all λ ∈ IX.
Theorem 1.2 ?see ?3, 15??. Let ?X,τ,τ∗? be an IFTS. For each r ∈ I0, s ∈ I1, λ ∈ IX, an operator
C : IX× I0× I1→IXis defined as follows:
??μ | μ ≥ λ,τ?1 − μ?≥ r,τ∗?1 − μ?≤ s?.
Then it satisfies the following properties:
i∈Δλi? ≥?
i∈Δτ?λi? and τ∗??
i∈Δλi? ≤?
i∈Δτ∗?λi?, for each λi∈ IX, i ∈ Δ.
C?λ,r,s? ?
?1.2?
?1? C?0,r,s? ? 0, C?1,r,s? ? 1, for all r ∈ I0, s ∈ I1;
?2? C?λ,r,s? ≥ λ;
?3? C?λ1,r,s? ≤ C?λ2,r,s?, if λ1≤ λ2;
?4? C?λ ∨ μ,r,s? ? C?λ,r,s? ∨ C?μ,r,s?, for all r ∈ I0, s ∈ I1;
?5? C?λ,r,s? ≤ C?λ,r?,s??, if r ≤ r?, s ≥ s?, where r,r?∈ I0, s,s?∈ I1;
?6? C?C?λ,r,s?,r,s? ? C?λ,r,s?.
Definition 1.3 ?see ?14??. A function f : ?X,τ,τ∗?→?Y,U,U∗? is said to be as follows:
?1? IF continuous if τ?f−1?μ?? ≥ U?μ? and τ∗?f−1?μ?? ≤ U∗?μ?, for each μ ∈ IY;
?2? IF open if τ?μ? ≤ U?f?μ?? and τ∗?μ? ≥ U∗?f?μ??, for each μ ∈ IX;
?3? IFhomeomorphism ifandonlyiff isbijectiveandbothf andf−1areIFcontinuous.

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S. E. Abbas and B. Krsteska3
Definition 1.4 ?see ?16??. Let 0/ ∈ΘXbe a subset of IX. A pair ?β,β∗? of functions β,β∗: ΘX→I
is called an IF topological base on X if it satisfies the following conditions:
?B1? β?λ? ? β∗?λ? ≤ 1, ∀λ ∈ ΘX,
?B2? β?1? ? 1 and β∗?1? ? 0,
?B3? β?λ1∧ λ2? ≥ β?λ1? ∧ β?λ2? and β∗?λ1∧ λ2? ≤ β∗?λ1? ∨ β∗?λ2?, for each λ1,λ2∈ ΘX.
An IF topological base ?β,β∗? always generates an IGO, ?τβ,τ∗
sense.
β∗? on X in the following
Theorem 1.5 ?see ?16??. Let ?β,β∗? be an IF topological base for X. Define the functions τβ,τ∗
IX→I as follows: for each μ ∈ IX,
⎧
⎪
where?is taken over all families {μi∈ ΘX| μ ??
⎧
⎪
where?is taken over all families {μi∈ ΘX| μ ??
?1? ?X,τβ,τ∗
β∗ :
τβ?μ? ?
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
???
1,
0,
i∈J
β?μi
??
,
if μ ?
?
i∈J
μi, μi∈ ΘX,
if μ ? 0,
otherwise,
?1.3?
i∈Jμi},
??
τ∗
β∗?μ? ?
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
???
0,
1,
i∈J
β∗?μi
,
if μ ?
?
i∈J
μi, μi∈ ΘX,
if μ ? 0,
otherwise,
?1.4?
i∈Jμi}. Then
β∗? is an IFTS;
?2? A map f : ?Y,τ,τ∗?→?X,τβ,τ∗
β∗?λ? ≥ τ∗?f−1?λ??, for all λ ∈ ΘX.
Lemma 1.6 ?see ?17??. Let X be a product of the family {Xi | i ∈ Γ} of sets, and for each i ∈ Γ,
πi: X →Xia projection map. For each λ ∈ IX, i,j ∈ Γ, and λi∈ IXi, the following properties hold:
?1? πi?π−1
?2? if?
?a??
β∗? is IF continuous if and only if β?λ? ≤ τ?f−1?λ?? and
i?λi? ∧ λ? ? λi∧ πi?λ?;
xi∈Xiλi?xi? ? αifor i ∈ F with each finite index subset F of Γ−{j} and put α ??
x∈X??
i∈Fαi,
then
i∈Fπ−1
i∈Fπ−1
i?λi???x? ? α;
i?λi?? ? α.
?b? πi??
Definition 1.7. Let ?X,τ,τ∗? be an IFTS, μ ∈ IX, xt∈ Pt?X?, r ∈ I0, and s ∈ I1. It holds that
Q?xt,r,s???μ ∈ IX| xtqμ, τ?μ? ≥ r, τ∗?μ? ≤ s?.
A fuzzy set μ ∈ Q?xt,r,s? is called ?r,s?-Q open neighborhood of xt.
?1.5?

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4International Journal of Mathematics and Mathematical Sciences
2. Some properties of product intuitionistic fuzzy topological spaces
Theorem 2.1. Let {?Xi,τi,τ∗
a map. Let
i?}i∈Γbe a family of IFTSs, let X be a set and for each i ∈ Γ, fi: X →Xi
ΘX?
?
0/ ? μ ?
n ?
j?1
f−1
kj
?νkj
?| τkj
?νkj
?> 0, ∀kj∈ K
?
,
?2.1?
for every finite set K ? {k1,...,kn} ⊂ Γ. Define the functions β,β∗: ΘX→I on X by
??n ?
??n ?
where?and?are taken over all finite subsets K ? {k1,k2,...,kn} ⊂ Γ. Then,
?1? ?β,β∗? is an IF topological base on X;
?2? the IGO, ?τβ,τ∗
IF continuous;
?3? a map f : ?Y,τ1,τ∗
continuous.
β?μ? ?
j?1
τkj
?νkj
?νkj
?| μ ?
?| μ ?
n ?
n ?
j?1
f−1
kj
?νkj
?νkj
??
??
,
β∗?μ? ?
j?1
τ∗
kj
j?1
f−1
kj
,
?2.2?
β∗? generated by ?β,β∗? is the coarsest IGO on X for which each i ∈ Γ, fiis
1?→?X,τβ,τ∗
β∗? is IF continuous if and only if for each i ∈ Γ, fi◦ f is IF
Proof. ?B1? It is trivial.
?B2? Since λ ? f−1
?B3? For all finite subsets K ? {k1,...,kp} and J ? {j1,...,jq} of Γ such that
i?λ? for each λ ∈ {0,1}, β?1? ? β?0? ? 1 and β∗?1? ? β∗?0? ? 0.
λ ?
p?
i?1
f−1
ki
?λki
?,μ ?
q?
i?1
f−1
ji
?μji
?,
?2.3?
we have
λ ∧ μ ?
?p?
i?1
f−1
ki
?λki
??
∧
?q?
i?1
f−1
ji
?μji
??
.
?2.4?
Furthermore, we have for each k ∈ K ∩ J,
f−1
k
?λk
?∧ f−1
k
?μk
?? f−1
k
?λk∧ μk
?.
?2.5?
Put λ ∧ μ ??
mi∈K∪Jf−1
mi?μi??ρmi?, where
ρmi?
⎧
⎪
⎪
⎪
⎪
⎩
⎨
λmi
μmi
λmi∧ μmi
if mi∈ K − ?K ∩ J?,
if mi∈ J − ?K ∩ J?,
if mi∈ ?K ∩ J?.
?2.6?

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S. E. Abbas and B. Krsteska5
We have
β?λ ∧ μ? ≥
?
?p?
?
?p?
j∈K∪J
τj
?ρj
?λki
τ∗
j
?
≥
i?1
τki
??
?
??
∧
?q?
i?1
τji
?μji
??
,
β∗?λ ∧ μ? ≤
j∈K∪J
?ρj
?λki
≤
i?1
τ∗
ki
∨
?q?
i?1
τ∗
ji
?μji
??
.
?2.7?
Then, β?λ ∧ μ? ≥ β?λ? ∧ β?μ? and β∗?λ ∧ μ? ≤ β∗?λ? ∨ β∗?μ?.
?2? For each λi∈ IXi, one family {f−1
?f−1
τ∗
i
i?λi?}, and i ∈ Γ, we have
??≥ β?f−1
??≤ β∗?f−1
β∗? → ?Xi,τi,τ∗
τβ
β∗?f−1
i
?λi
?λi
i
?λi
?λi
i? is IF continuous. Let fi : ?X,τ◦,τ◦∗? →
??≥ τi
??≤ τ∗
?λi
i
?,
i
?λi
?.
?2.8?
Thus, for each i ∈ Γ, fi : ?X,τβ,τ∗
?Xi,τi,τ∗
τ◦∗?f−1
have
i? be IF continuous, that is, for each i ∈ Γ and λi ∈ IXi, τ◦?f−1
i?λi?? ≤ τ∗
i?λi?? ≥ τi?λi? and
i?1f−1
i?λi?. For all finite subsets K ? {k1,...,kp} of Γ such that λ ??p
ki?λki?, we
τ◦?λ? ≥
p?
p?
i?1
τ◦?f−1
τ◦∗?f−1
ki
?λki
?λki
??≥
??≤
p?
p?
i?1
τki
?λki
τ∗
ki
?,
?.τ◦∗?λ? ≤
i?1
ki
i?1
?λki
?2.9?
It implies τ◦?λ? ≥ β?λ? and τ◦∗?λ? ≤ β∗?λ? for each λ ∈ IX. By Theorem 1.5?2?, τ◦≥ τβ
and τ◦∗≤ τ∗
?3? ?⇒? Let f : ?Y,τ1,τ∗
we have
β∗.
1?→?X,τβ,τ∗
β∗? be an IF continuous. For each i ∈ Γ and λi∈ IXi,
τ1
??fi◦ f?−1?λi
??? τ1
??f−1?f−1
i
?λi
???≥ τβ
?f−1
i
?λi
???≥ τi
?λi
i
?,
τ∗
1
??fi◦ f?−1?λi
1?→?Xi,τi,τ∗
?⇐? For all finite subsets K ? {k1,...,kp} of Γ such that λ ??p
??? τ∗
1
??f−1?f−1
i? is IF continuous.
i
?λi
???≤ τ∗
β∗?f−1
i
?λi
???≤ τ∗
?λi
?.
?2.10?
Hence, fi◦ f : ?Y,τ1,τ∗
i?1f−1
ki?λki?, since fki◦ f :
?Y,τ1,τ∗
1?→?Xki,τki,τ∗
ki? is IF continuous,
τ1
τ∗
?f−1?f−1
ki
?λki
???≥ τki
?λki
?,
?A?
1
?f−1?f−1
ki
?λki
???≤ τ∗
ki
?λki
?.
?B?