http://scma.maragheh.ac.ir
ON RARELY GENERALIZED REGULAR FUZZY CONTINUOUS FUNCTIONS IN FUZZY TOPOLOGICAL
SPACES
APPACHI VADIVEL1∗
AND ELANGOVAN ELAVARASAN2
Abstract. In this paper, we introduce the concept of rarely gen-eralized regular fuzzy continuous functions in the sense of A.P. Sostak’s and Ramadan is introduced. Some interesting properties and characterizations of them are investigated. Also, some applica-tions to fuzzy compact spaces are established.
1. Introduction and preliminaries
The concept of fuzzy set was introduced by Zadeh [18] in his classical paper. Fuzzy sets have applications in many fields such as information [12] and control [13]. Chang [2] introduced the notion of a fuzzy topology. Later Lowen [9] redefined what is now known as stratified fuzzy topology.
ˇ
Sostak [14] introduced the notion of fuzzy topology as an extension of Chang an Lowen’s fuzzy topology. Later on he has developed the theory of fuzzy topological spaces in [15] and [16]. Popa [10] introduced the notion of rarely continuity as a generalization of weak continuity [7] which has been further investigated by Long and Herrington [8] and Jafari [4] and [5]. Recently Vadivel et al. [17] introduced the concept of generalized regular fuzzy closed sets in fuzzy topological spaces in the sense of ˇSostak’s.
In this paper, we introduce the concept of rarely generalized regu-lar fuzzy continuous functions in the sense of A. P. Sostak’s [14] and Ramadan [11] is introduced. Some intresting properties and charac-terizations of them are investigated. Also, some applications to fuzzy compact spaces are established.
2010Mathematics Subject Classification. 54A40, 03E72.
Key words and phrases. Rarely generalized regular fuzzy continuous, Grf-compact space, Rarely grf-almost compact space, Rarely grf-T2-spaces.
Received: 09 April 2016, Accepted: 25 May 2016. ∗Corresponding author.
Throughout this paper, let X be a nonempty set, I = [0, 1] and
I0 = (0, 1]. Forλ∈IX, λ(x) =λfor all x∈X. For x∈X and t∈I0,
a fuzzy pointxtis defined by
xt(y) =
{
t ify=x
0 if y̸=x.
Let P t(X) be the family of all fuzzy points inX. A fuzzy point xt∈λ
iff t < λ(x). All other notations and definitions are standard, for all in the fuzzy set theory.
Definition 1.1 ([14]). A functionτ :IX →I is called a fuzzy topology
on X if it satisfies the following conditions: (O1) τ(0) =τ(1) = 1,
(O2) τ(∨
i∈Γµi)≥ ∧
i∈Γτ(µi), for any{µi}i∈Γ⊂IX,
(O3) τ(µ1∧µ2)≥τ(µ1)∧τ(µ2), for anyµ1, µ2 ∈IX.
The pair (X, τ) is called a fuzzy topological space (for short, fts ). A fuzzy set λ is called an r-fuzzy open (r-fo, for short) if τ(λ)≥ r. A fuzzy setλis called an r-fuzzy closed (r-fc, for short) set iff 1−λis an
r-fo set.
Theorem 1.2 ([3]). Let (X, τ) be a fts. Then for each λ ∈ IX and r ∈I0, we define an operator Cτ :IX×I0→IX as follows: Cτ(λ, r) = ∧
{µ ∈ IX : λ ≤ µ, τ(1−µ) ≥ r}. For λ, µ ∈ IX and r, s ∈ I0, the operator Cτ satisfies the following statements:
(C1) Cτ(0, r) = 0,
(C2) λ≤Cτ(λ, r),
(C3) Cτ(λ, r)∨Cτ(µ, r) =Cτ(λ∨µ, r),
(C4) Cτ(λ, r)≤Cτ(λ, s) ifr ≤s,
(C5) Cτ(Cτ(λ, r), r) =Cτ(λ, r).
Theorem 1.3 ([3]). Let (X, τ) be a fts. Then for each λ ∈ IX and
r ∈I0, we define an operator Iτ :IX ×I0 →IX as follows: Iτ(λ, r) =
∨
{µ∈IX :µ≤λ, τ(µ)≥r}. Forλ, µ∈IX andr, s∈I0, the operator Iτ satisfies the following statements:
(I1) Iτ(1, r) = 1,
(I2) Iτ(λ, r)≤λ,
(I3) Iτ(λ, r)∧Iτ(µ, r) =Iτ(λ∧µ, r),
(I4) Iτ(λ, r)≤Iτ(λ, s) ifs≤r,
(I5) Iτ(Iτ(λ, r), r) =Iτ(λ, r),
(I6) Iτ(1−λ, r) = 1−Cτ(λ, r) and Cτ(1−λ, r) = 1−Iτ(λ, r).
(1) a fuzzy set λ is called r-fuzzy regular open (for short, r-fro) if
λ=Iτ(Cτ(λ, r), r).
(2) a fuzzy setλis called r-fuzzy regular closed (for short,r-frc) if
λ=Cτ(Iτ(λ, r), r).
Definition 1.5([17]). Letf : (X, τ)→(Y, σ) be a function andr∈I0.
Then f is called:
(1) fuzzy regular continuous (for short, fr-continuous) if f−1
(λ) is ar-fro set inIX for each λ∈IY withσ(λ)≥r.
(2) fuzzy regular open (for short, fr-open) if f(λ) is a r-fro set in
IY for eachλ∈IX withτ(λ)≥r.
(3) fuzzy regular closed (for short, fr-closed) if f(λ) is ar-frc set in
IY for eachλ∈IX withτ(1−λ)≥r.
Definition 1.6 ([17]). Let (X, τ) be a fts. Letλ, µ∈IX and r∈I0.
(1) The r-fuzzy regular closure of λ, denoted by RCτ(λ, r), and is
defined byRCτ(λ, r) =∧{µ∈IX|µ≥λ, µ isr-frc}.
(2) Ther-fuzzy regular interiror of λ, denoted by RIτ(λ, r), and is
defined byRIτ(λ, r) =∨{µ∈IX|µ≤λ, µis r-fro }.
Definition 1.7 ([17]). Let (X, τ) be a fts. Letλ, µ∈IX andr ∈I0,
(1) Fuzzy setλis calledr-generalized regular fuzzy closed (for short,
r-grfc) set if RCτ(λ, r)≤µ, whenever λ≤µ and τ(µ)≥r.
(2) Fuzzy setλis calledr-generalized regular fuzzy open (for short,
r-grfo) set if 1−λis r-grfc.
Definition 1.8([17]). Let (X, τ) and (Y, η) be a fts’s. Letf : (X, τ)→ (Y, η) be a function.
(1) f is called generalized regular fuzzy continuous (for short, grf-continuous) iff f−1
(µ) is r-grfc for each µ ∈ IY, r ∈ I
0 with η(1−µ)≥r.
(2) f is called generalized regular fuzzy open (for short, grf-open) ifff(λ) isr-grfo for each λ∈IX, r∈I0 withτ(λ)≥r.
(3) f is called generalized regular fuzzy closed (for short, grf-closed) ifff(λ) isr-grfc for each λ∈IX, r∈I0 withτ(1−λ)≥r.
Definition 1.9 ([1]). Let (X, τ) be a fts and r ∈I0. For λ∈ IX, λis
called ar-fuzzy rare set if Iτ(λ, r) = 0.
Definition 1.10 ([1]). Let (X, τ) and (Y, η) be two fts’s. Let f : (X, τ)→(Y, η) be a function. Then f is called
(1) weakly continuous if for each µ∈IY, whereσ(µ)≥r, r∈I
0,
f−1
(2) rarely continuous if for each µ ∈ IY, where σ(µ) ≥ r, r ∈ I0,
there exists ar-fuzzy rare setλ∈IY withµ+C
σ(λ, r)≥1 and ρ∈IX, whereτ(ρ)≥r such that f(ρ)≤µ∨λ.
Proposition 1.11 ([1]). Let (X, τ) and (Y, σ) be any two fts’s,r ∈I0 andf : (X, τ)→(Y, σ) be fuzzy open and one-to-one, thenf preserves
r-fuzzy rare sets.
2. Rarely generalized regular fuzzy continuous functions
Definition 2.1. Let (X, τ) and (Y, σ) be two fts’s, and f : (X, τ)→ (Y, η) be a function. Then f is called
(1) rarely fuzzy regular continuous (for short, rarely fr-continuous) if for eachµ∈IY, whereσ(µ)≥r, r∈I
0, there exists ar-fuzzy
rare set λ∈ IY with µ+Cσ(λ, r) ≥ 1 and a r-fro set ρ ∈ IX
such thatf(ρ)≤µ∨λ.
(2) rarely generalized regular fuzzy continuous (for short, rarely grf-continuous) if for each µ ∈ IY, where σ(µ) ≥ r, r ∈ I0, there
exists a r-fuzzy rare set λ ∈ IY with µ+C
σ(λ, r) ≥ 1 and a r-grfo set ρ∈IX, such thatf(ρ)≤µ∨λ.
Remark 2.2. (1) Every weakly continuous (resp. fr-continuous) func-tion is rarely continuous [1] (resp. grf-continuous [17]) but con-verse is not true.
(2) Every rarely fr-continuous (resp. grf-continuous) function is rarely grf-continuous but converse is not true.
(3) Every rarely fr-continuous function is rarely continuous but con-verse is not true.
From the above definitions and remarks it is not difficult to conclude that the following diagram of implications is true.
weakly continuous rarely continuous rarely fr-continuous
fr-continuous grf-continuous rarely grf-continuous
Example 2.3. Let X = {a, b} = Y. Define λ1 ∈ IX, λ2 ∈ IY as
follows: λ1(a) = 0.8, λ1(b) = 0.5, λ2(a) = 0.8, λ2(b) = 0.6. Define the
fuzzy topologies τ, σ:IX →I as follows:
τ(λ) =
1 ifλ= 0 or 1, 1
10 ifλ=λ1,
0 otherwise,
σ(λ) =
1 ifλ= 0 or 1, 1
10 ifλ=λ2,
Let r= 1/10. Let f : (X, τ)→(Y, σ) be defined by f(a) =a, f(b) =a
and γ ∈ IY be a 1/10-fuzzy rare set defined byγ(a) = 0.6, γ(b) = 0.8
and a r-grfo set λ ∈ IX is defined by λ(a) = 0.2, λ(b) = 0.3, f(λ) = (0.8, 0.8)≤λ2∨γ = (0.8, 0.8). Then f is rarely grf-continuous but not
rarely fr-continuous, because λ∈IX is not ar-fro set.
Example 2.4. In Example 2.3 f is rarely continuous but not rarely fr-continuous.
Example 2.5. LetX ={a, b}=Y. Defineλ1 ∈IX, λ2∈IY as follows: λ1(a) = 0.7, λ1(b) = 0.9. Define the fuzzy topologies τ, σ :IX → I as
follows:
τ(λ) =
1 ifλ= 0 or 1, 1
10 ifλ=λ1,
0 otherwise,
σ(λ) =
1 ifλ= 0 or 1, 1
10 ifλ=λ1,
0 otherwise.
Let r = 1/10. Letf : (X, τ)→(Y, σ) be defined by f(a) =a, f(b) =b
and γ ∈ IY be a 1/10-fuzzy rare set defined byγ(a) = 0.7, γ(b) = 0.7 and a r-grfo set λ ∈ IX is defined by λ(a) = 0.2, λ(b) = 0.1, f(λ) =
(0.7, 0.9)≤λ2∨γ = (0.7, 0.9). Then f is rarely grf-continuous but not
grf-continuous, because f−1
(1−λ1) is not a r-grfo set in IX for each
(1−λ1)∈IY andσ(1−λ1)≥r.
Definition 2.6. Let (X, τ) and (Y, σ) be two fts’s, and f : (X, τ)→ (Y, η) be a function. Thenf is called weakly generalized regular fuzzy continuous (for short, weakly grf-continuous) if for each r-grfo set µ ∈
IY,r∈I
0,f−1(µ)≤Iτ(f−1(Cσ(µ, r)), r).
Definition 2.7 ([17]). A fts (X, τ) is said to be a fuzzy F RT1/2-space
if everyr-grfo set λ∈IX, r∈I0 is ar-fro set.
Proposition 2.8. Let (X, τ) and (Y, σ) be any two fuzzy topological spaces. If f : (X, τ) → (Y, σ) is both grf-open and grf-irresolute and
(X, τ) is a F RT1/2 space, then it is weakly grf-continuous.
Proof. Letλ∈IX be such thatτ(λ)≥r. Sincef is grf-open,f(λ)∈IY
is ar-grfo. Also, sincef is grf-irresolute,f−1
(f(λ))∈IX is ar-grfo set.
Since (X, τ) is F RT1/2 space, every r-grfo set is a r-fro set and also
everyr-fro set is ar-fo set, now, τ(f−1
(f(λ)))≥r. Consider
f−1
(f(λ))≤f−1
(Cσ(f(λ), r))
from which
Iτ(f−1(f(λ)), r)≤Iτ(f−1(Cσ(f(λ), r)), r).
Since
τ( f−1
(f(λ)))
≥r, f−1
(f(λ))≤Iτ
( f−1
(Cσ(f(λ), r)), r
thusf is weakly grf-continuous. □
Definition 2.9. Let (X, τ) be a fts. A grf-open cover of (X, τ) is the collection {λi∈IX, λi is r-grfo, i∈J}such that ∨i∈Jλi = 1.
Definition 2.10. A fts (X, τ) is said to be a grf-compact space if every grf-open cover of (X, τ) has a finite sub cover.
Definition 2.11. A fts (X, τ) is said to be rarely grf-almost compact if for every grf-open cover {λi ∈ IX, λi is r-grfo, i ∈J} of (X, τ), there
exists a finite subset J0 of J such that ∨i∈Jλi∨ρi = 1 where ρi ∈IX
are r-fuzzy rare sets.
Proposition 2.12. Let (X, τ) and (Y, σ) be any two fts’s, r∈I0 and f : (X, τ) →(Y, σ) be rarely grf-continuous. If (X, τ) is grf-compact then (Y, σ) is rarely grf-almost compact.
Proof. Let {λi ∈ IY, i ∈ J} be the grf-open cover of (Y, σ). Then
1 =∨i∈Jλi. Since f is rarely grf-continuous, there exists a r-fuzzy rare
set ρi ∈ IY such that λi+Cσ(ρi, r) ≥1 and a r-grfo set µi ∈IX such
that f(µi) ≤ λi ∨ρi. Since (X, τ) is grf-compact, every grf-open cover
of (X, τ) has a finite sub cover. Thus 1 ≤ ∨i∈J0µi. Hence 1 = f(1) = f(∨i∈J0µi) = ∨i∈J0f(µi) ≤ ∨i∈J0λi∨ρi. Therefore (Y, σ) is rarely
grf-almost compact. □
Proposition 2.13. Let (X, τ) and (Y, σ) be any two fts’s, r∈I0 and f : (X, τ)→(Y, σ) be rarely continuous. If(X, τ) is grf-compact then
(Y, σ) is rarely grf-almost compact.
Proof. Since every rarely continuous function is rarely grf-continuous, then proof follows immediatly from the Proposition 2.12. □
Proposition 2.14. Let (X, τ), (Y, σ) and (Z, η) be any fts’s and
r ∈ I0. If f : (X, τ) → (Y, σ) be rarely grf-continuous, grf-open and g: (Y, σ)→(Z, η)be fuzzy open and one-to-one, then g◦f : (X, τ)→ (Z, η) is rarely grf-continuous.
Proof. Let λ∈IX with τ(λ) ≥r. Sincef is grf-open, f(λ) ∈IY with
σ(f(λ))≥r. Sincef is rarely grf-continuous and there exists a r-fuzzy rare set ρ ∈IY with f(λ) +Cσ(ρ, r) ≥1 and a r-grfo set µ∈IX such
thatf(µ)≤f(λ)∨ρ. By the proposition 1.11,g(ρ)∈IZ is also ar-fuzzy rare set. Since ρ ∈IY is such that ρ < γ for all γ ∈IY withσ(γ)≥r,
and g is injective, it follows that (g◦f)(λ) +Cη(g(ρ), r) ≥ 1. Then
(g◦f)(µ) =g(f(µ))≤g(f(λ)∨ρ)≤g(f(λ))∨g(ρ)≤(g◦f)(λ)∨g(ρ).
Hence the result. □
Proposition 2.15. Let (X, τ), (Y, σ) and(Z, η) be any fts’s andr ∈
be a function such that g◦f : (X, τ)→(Z, η) is rarely grf-continuous, then g is rarely grf-continuous.
Proof. Letλ∈IX and µ∈IY be such that f(λ) =µ. Let (g◦f)(λ) =
γ ∈ IZ with η(γ) ≥ r. Since (g◦f) is grf-continuous, there exists a
rare set ρ ∈ IZ with γ +Cη(ρ, r) ≥ 1 and a r-grfo set δ ∈ IX such
that (g◦f)(δ) ≤γ ∨ρ. Since f is grf-open, f(δ) ∈ IY is a r-grfo set.
Thus there exists a r-fuzzy rare set ρ ∈ IZ with γ+C
η(ρ, r) ≥ 1 and
a r-grfo set f(δ) ∈ IY such that g(f(δ)) ≤ γ ∨ρ. Hence g is rarely
grf-continuous. □
Proposition 2.16. Let (X, τ) and (Y, σ) be any two fuzzy topological spaces and r ∈I0. If f : (X, τ)→ (Y, σ) be rarely grf-continuous and
(X, τ) be a fuzzy F RT1/2-space, then f is rarely fr-continuous.
Proof. The proof is trivial. □
Definition 2.17. A fts (X, τ) is said to be a rarely grf-T2-space if for
each pair λ, µ ∈ IX with λ ̸= µ there exist r-grfo sets ρ1, ρ2 ∈ IX
with ρ1 ̸=ρ1 and a r-fuzzy rare setγ ∈IX with ρ1+Cτ(γ, r)≥1 and ρ2+Cτ(γ, r)≥1 such thatλ≤ρ1∨γ and µ≤ρ2∨γ.
Proposition 2.18. Let (X, τ) and (Y, σ) be any two fuzzy topological spaces and r ∈I0. If f : (X, τ)→(Y, σ) be grf-open and injective and
(X, τ) be a rarely grf-T2 space, then(Y, σ) is also a rarely grf-T2 space.
Proof. Let λ, µ ∈ IX with λ ̸= µ. Since f is injective, f(λ) ̸= f(µ). Since (X, τ) is a rarely grf-T2-space, there exist r-grfo setsρ1, ρ2∈IX
with ρ1 ̸= ρ1 and a r-fuzzy rare set γ ∈ IX with ρ1 +Cτ(γ, r) ≥ 1
and ρ2 +Cτ(γ, r) ≥ 1 such that λ ≤ ρ1 ∨γ and µ ≤ ρ2 ∨γ. Since f is grf-open, f(ρ1), f(ρ2) ∈ IY are r-grfo sets with f(ρ1) ̸= f(ρ2).
Since f is grf-open and one-to-one, f(γ) is also a r-fuzzy rare set with
f(ρ1) +Cσ(γ, r)≥1 andf(ρ2) +Cσ(γ, r)≥1 such thatf(λ)≤f(ρ1∨γ)
and f(µ)≤f(ρ1∨γ). Thus (Y, σ) is a grf-rarelyT2-space. □
3. Conclusion
ˇ
Sostak’s fuzzy topology has been recently of major interest among fuzzy topologies. In this paper, we have introduced rarely general-ized regular fuzzy continuous functions in fuzzy topological spaces of
ˇ
Sostak’s. We have also introduced grf-compact spaces, rarely grf-almost compact spaces and rarely-grf-T2-spaces, some intresting properties and
characterizations of them are investigated.
References
1. B. Amudhambigai, M.K. Uma, and E. Roja,On rarelyeg-continuous functions in smooth fuzzy topological spaces, The Journal of Fuzzy Mathematics., 20 (2) (2012) 433-442.
2. C.L. Chang,Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968) 182-190. 3. K.C. Chattopadhyay and S.K. Samanta,Fuzzy topology, Fuzzy Sets and Systems,
54 (1993) 207-212.
4. S. Jafari,A note on rarely continuous functions, Univ. Bacau. Stud. Cerc. St. Ser. Mat., 5 29-34.
5. S. Jafari,On some properties of rarely continuous functions, Univ. Bacau. Stud. Cerc. St. Ser. Mat., 7 65-73.
6. S.J. Lee and E.P. Lee,Fuzzyr-regular open sets and fuzzy almostr-continuous maps, Bull. Korean Math. Soc., 39(3) (2002) 441-453.
7. N. Levine, Decomposition of continuity in topological spaces, Amer. Math. Monthly., 60 44-46.
8. P.E. Long and L.L. Herrington,Properties of rarely continuous functions, Glas-nik Math., 17 (37) 147-153.
9. R. Lowen,Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976) 621-633.
10. V. Popa,Sur certain decompositionde la continuite dans les espaces topologiques, Glasnik Mat. Setr III., 14 (34) 359-362.
11. A.A. Ramadan, S.E. Abbas, and Y.C. Kim,Fuzzy irresolute mappings in smooth fuzzy topological spaces, The Journal of Fuzzy Mathematics., 9 (2001) 865-877. 12. P. Smets,The degree of belief in a fuzzy event, Inform. Sci., 25 (1981) 1-19. 13. M. Sugeno, An introductory survey of fuzzy control, Inform. Sci., 36 (1985)
59-83.
14. A.P. ˇSostak,On a fuzzy topological structure, Rend. Circ. Matem. Palermo Ser II., 11 (1986) 89-103.
15. A.P. ˇSostak, On the neighborhood structure of a fuzzy topological spaces, Zh. Rodova Univ. Nis. Ser. Math., 4 (1990) 7-14.
16. A.P. ˇSostak, Basic structures of fuzzy topology, J. Math. Sci., 78 (6) (1996) 662-701.
17. A. Vadivel and E. Elavarasan,Applications ofr-generalized regular fuzzy closed sets, Annals of Fuzzy Mathematics and Informatics (Accepted).
18. L.A. Zadeh,Fuzzy sets, Information and control., 8 (1965) 338-353.
1
Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu-608 002, India.
E-mail address: avmaths@gmail.com 2
Research scholar, Department of Mathematics, Annamalai Univer-sity, Annamalainagar, Tamil Nadu-608 002, India.
