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Completely continuous functions in intuitionistic fuzzy topological spaces

01 Dec 2003-Czechoslovak Mathematical Journal (Institute of Mathematics, Academy of Sciences of the Czech Republic)-Vol. 53, Iss: 4, pp 793-803
TL;DR: This paper introduces and study the concept of fuzzy completely continuous functions between intuitionistic fuzzy topological spaces.
Abstract: In this paper, after giving the basic results related to the product of functions and the graph of functions in intuitionistic fuzzy topological spaces, we introduce and study the concept of fuzzy completely continuous functions between intuitionistic fuzzy topological spaces.

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Czechoslovak Mathematical Journal
I. M. Hanafy
Completely continuous functions in intuitionistic fuzzy topological spaces
Czechoslovak Mathematical Journal, Vol. 53 (2003), No. 4, 793–803
Persistent URL: http://dml.cz/dmlcz/127841
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Czechoslovak Mathematical Journal, 53 (128) (2003), 793–803
COMPLETELY CONTINUOUS FUNCTIONS
IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

, El-Arish
(Received November 6, 2000)
Abstract. In this paper, after giving the basic results related to the product of functions
and the graph of functions in intuitionistic fuzzy topological spaces, we introduce and study
the concept of fuzzy completely continuous functions between intuitionistic fuzzy topological
spaces.
Keywords: intuitionistic fuzzy set, intuitionistic fuzzy topological space, fuzzy completely
continuous function, fuzzy near compactness, fuzzy product space
MSC 2000 : 54A40, 54D20
1. Introduction
In [1], [2], Atanassov introduced the fundamental concept of an intuitionistic fuzzy
set. Coker in [4], [5] introduced the notion of an intuitionistic fuzzy topological
space, fuzzy continuity, fuzzy near compactness and some other related concepts.
Completely continuous functions and results related to the product in fuzzy topo-
logical spaces were introduced in [7] and [3], respectively. In this paper, some of
results related to the product of functions and the graph of functions are obtained
in intuitionistic fuzzy topological spaces. Mainly we introduce and study completely
continuous functions between intuitionistic fuzzy topological spaces. Some coun-
terexamples are given and also, Theorem 3.14 in [4] is strengthened.
793

2. Preliminaries
Throughout this section, we shall present the fundamental definitions and results
of intuitionistic fuzzy sets as given by Atanassov [2] and Coker [5].
Definition 1 ([2]). Let X be a nonempty fixed set. An intuitionistic fuzzy set
(IFS, for short) A is an object having the form A = {hx, µ
A
(x), γ
A
(x)i: x X}
where the functions µ
A
: X I and γ
A
: X I denote respectively the degree of
membership (namely µ
A
(x)) and the degree of nonmembership (namely γ
A
(x)) of
each element x X to the set A, and 0 6 µ
A
(x) + γ
A
(x) 6 1 for each x X.
Obviously, every fuzzy set A on a nonempty set X is an IFS having the form
A = {hx, µ
A
(x), 1 γ
A
(x)i: x X}.
Definition 2 ([2]). Let X be a nonempty set and let the IFS’s A and B be in
the form A = {hx, µ
A
(x), γ
A
(x)i: x X}, B = {hx, µ
B
(x), γ
B
(x)i: x X} and let
{A
j
: j J} be an arbitrary family of IFS’s in X. Then
(i) A 6 B iff x X [µ
A
(x) 6 µ
B
(x) and γ
A
(x) > γ
B
(x)];
(ii) A = {hx, γ
A
(x), µ
A
(x)i: x X};
(iii)
T
A
j
= {hx,
V
µ
A
j
(x),
W
γ
A
j
(x)i: x X};
(iv)
S
A
j
= {hx,
W
µ
A
j
(x),
V
γ
A
j
(x)i: x X};
(v) 1
˜
= {hx, 1, 0i : x X} and 0
˜
= {hx, 0, 1i : x X};
(vi)
A = A, 0
˜
= 1
˜
and 1
˜
= 0
˜
.
Definition 3 ([5]). Let X and Y be two nonempty sets and f : X Y a
function.
(i) If B = {hy, µ
B
(y), γ
B
(y)i: y Y } is an IFS in Y , then the preimage of B
under f is denoted and defined by f
1
(B) = {hx, f
1
(µ
B
)(x), f
1
γ
B
)(x)i:
x X}.
(ii) If A = {hx, λ
A
(x), υ
A
(x)i: x X} is an IFS in X, then the image of A under f
is denoted and defined by f(A) = {hy, f (λ
A
)(y), f
(υ
A
)(y)i: y Y } where
f
(υ
A
) = 1 f(1 υ
A
).
In (i), (ii), since µ
B
, γ
B
, λ
A
, υ
A
are fuzzy sets, we explain that
f
1
(µ
B
)(x) = µ
B
(f(x)),
and
f(λ
A
)(y) =
(
sup λ
A
(x) if f
1
(y) 6= 0,
0 otherwise.
Definition 4. An intuitionistic fuzzy topology (IFT, for short) on a nonempty
set X is a family Ψ of IFS’s in X satisfying the following axioms:
794

(i) 0
˜
, 1
˜
Ψ;
(ii) A
1
A
2
Ψ for any A
1
, A
2
Ψ;
(iii)
S
A
j
Ψ for any {A
j
: j J} Ψ.
In this case the pair (X, Ψ) is called an intuitionistic fuzzy topological space (IFTS,
for short) and each IFS in Ψ is known as an intuitionistic fuzzy open set (IFOS, for
short) in X.
Definition 5. The complement A of IFOS A in IFTS (X, Ψ) is called an intu-
itionistic fuzzy closed set (IFCS, for short).
Definition 6. Let (X, Ψ) be an IFTS and A = hx, µ
A
(x), γ
A
(x)i an IFS in X.
Then the fuzzy interior and the fuzzy closure of A are defined by
cl(A) =
T
{K : K is an IFCS in X and A 6 K} and
int(A) =
S
{G: G is an IFOS in X and G 6 K}.
Definition 7. An IFS A of all IFTS X is called
(i) an intuitionistic fuzzy regular open set (IFROS, for short) of X if int cl(A) = A.
(ii) an intuitionistic fuzzy regular closed set (IFRCS, for short) of X if cl int(A) = A.
Obviously, an IFS A in an IFTS X is IFROS iff
A is IFRCS.
Definition 8. Let (X, Ψ) and (Y, Φ) be two IFTS’s and f : X Y a function.
Then
(i) f is fuzzy continuous iff the preimage of each IFS in Φ is an IFS in Ψ ([5]);
(ii) f is fuzzy strongly continuous iff for each IFS A in X, f (cl(A)) 6 f(A) ([4]);
(iii) f is fuzzy almost open iff the image of each IFROS in X is an IFOS in Y ([4]).
Definition 9. An IFTS (X, Ψ) is called fuzzy nearly compact iff every fuzzy
open cover of X has a finite subcollection such that the interior of closures of IFS’s
in this subcollection covers X.
3. Basic results
Definition 10. A subfamily β of IFTS (X, Ψ) is called a base for Ψ if each IFS
of Ψ is a union of some members of β.
Definition 11. Let X, Y be nonempty sets and A = hx, µ
A
(x), γ
A
(x)i, B =
hy, µ
B
(y), γ
B
(y)i IFS’s of X and Y , respectively. Then A × B is an IFS of X × Y
defined by
(A × B)(x, y) = h(x, y), min(µ
A
(x), µ
B
(y)), max(γ
A
(x), γ
B
(y))i.
795

Notice that
1
˜
(A × B)(x, y) = h(x, y), max(γ
A
(x), γ
B
(y)), min(µ
A
(x), µ
B
(y))i.
Lemma 12. If A is an IFS of X and B is an IFS of Y , then
(i) (A × 1
˜
) (1
˜
× B) = A × B;
(ii) (A × 1
˜
) (1
˜
× B) = 1
˜
A × B;
(iii) 1
˜
A × B = (A × 1
˜
) (1
˜
× B).

. Let A = hx, µ
A
(x), γ
A
(x)i, B = hy, µ
B
(y), γ
B
(y)i.
(i) Since A×1
˜
= hx, min(µ
A
, 1
˜
), max(γ
A
, 0
˜
)i = hx, µ
A
(x), γ
A
(x)i = A and similarly
1
˜
× B = hy, min(1
˜
, µ
B
), max(0
˜
, γ
B
)i = B, we have
(A × 1
˜
) (1
˜
× B) = A(x) B(y)
= h(x, y), µ
A
(x) µ
B
(y), γ
A
(x) γ
B
(y)i = A × B.
(ii) Similarly to (i).
(iii) Obvious by putting A, B instead of A, B in (ii).
Definition 13. Let (X, Ψ) and (Y, Φ) be IFTS’s. The intuitionistic fuzzy
product space (IFPTS, for short) of (X, Ψ) and (Y, Φ) is the cartesian product
X × Y of IFS’s X and Y together with the IFT ξ of X × Y which is generated
by the family {P
1
1
(A
i
), P
1
2
(B
j
): A
i
Ψ, B
j
Φ and P
1
, P
2
are projections
of X × Y onto X and Y, respectively} (i.e. the family {P
1
1
(A
i
), P
1
2
(B
j
): A
i
Ψ, B
j
Φ} is a subbase for IFT ξ of X × Y ).
Remark 1. In the above definition, since P
1
1
(A
i
) = A
i
×1
˜
and P
1
2
(B
j
) = 1
˜
×B
j
and A
i
× 1
˜
1
˜
× B
j
= A
i
× B
j
, the family β = {A
i
× B
j
: A
i
Ψ, B
j
Φ} forms a
base for IFPTS ξ of X × Y .
Definition 14. Let f
1
: X
1
Y
1
and f
2
: X
2
Y
2
. The product f
1
× f
2
:
X
1
× X
2
Y
1
× Y
2
is defined by
(f
1
× f
2
)(x
1
, x
2
) = (f
1
(x
1
), f
2
(x
2
)) (x
1
, x
2
) X
1
× X
2
.
Definition 15. Let f : X Y be a function. The graph g : X X × Y of f
is defined by
g(x) = (x, f(x)) x X.
796

Citations
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Journal ArticleDOI
TL;DR: The originality and value of IFT are here demonstrated, which show that the results obtained by various authors on intuitionistic fuzzy topological spaces (IFTSs), are not redundant with others for the ordinary fuzzy sense.
Abstract: Purpose – In 2000, Wang and He published an important result on the theory of intuitionistic fuzzy sets (IFSs) Indeed, they showed that every IFS may be regarded as an L‐fuzzy set for some appropriate lattice L This paper aims to show that, nevertheless, the results obtained by various authors on intuitionistic fuzzy topological spaces (IFTSs), are not redundant with others for the ordinary fuzzy senseDesign/methodology/approach – The most important definitions and results on intuitionistic fuzzy topology (IFT) one compared with the result obtained by Wang and HeFindings – That these results are not redundantResearch limitations/implications – Clearly, this paper is devoted to IFTSsPractical implications – The main applications are in the mathematical fieldOriginality/value – The originality and value of IFT are here demonstrated

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TL;DR: In this article, a relation between the intuitionistic fuzzy topology (IFT) on an IFS and the neutrosophic topology is studied, and the possible relation between IFT and NSs is shown.
Abstract: Purpose – Recently, Smarandache generalized the Atanassov's intuitionistic fuzzy sets (IFSs) and other kinds of sets to neutrosophic sets (NSs). Also, this author defined the notion of neutrosophic topology on the non‐standard interval. One can expect some relation between the intuitionistic fuzzy topology (IFT) on an IFS and the neutrosophic topology. This paper aims to show that this is false.Design/methodology/approach – The possible relation between the IFT and the neutrosophic topology is studied.Findings – Relations on neutrosophic topology and IFT are found.Research limitations/implications – Clearly, this paper is confined to IFSs and NSs.Practical implications – The main applications are in the mathematical field.Originality/value – The paper shows original results on fuzzy sets and topology.

76 citations

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Abstract: On 2005, F. Smarandache generalized the Atanassov's intuitionistic fuzzy setsto neutrosophic sets. Also, this author and some co-workers introduced the notion of interval neutrosophic set, which is an instance of neutrosophic set and studied various properties. The notion of neutro- sophic topology on the non-standard interval is also due to Smarandache. We study in this paper relations between interval neutrosophic sets and topology.

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TL;DR: The basic concepts of the so-called “intuitionistic fuzzy topological spaces” are constructed, the definitions of fuzzy continuity, fuzzy compactness, fuzzy connectedness and fuzzy Hausdorff space are introduced, and several preservation properties and some characterizations concerning fuzzy compactity and fuzzyconnectedness are obtained.

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"Completely continuous functions in ..." refers background or methods in this paper

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  • ...Then (i) f is fuzzy continuous iff the preimage of each IFS in Φ is an IFS in Ψ ([5]); (ii) f is fuzzy strongly continuous iff for each IFS A in X , f(cl(A)) 6 f(A) ([4]); (iii) f is fuzzy almost open iff the image of each IFROS in X is an IFOS in Y ([4])....

    [...]

  • ...Preliminaries Throughout this section, we shall present the fundamental definitions and results of intuitionistic fuzzy sets as given by Atanassov [2] and Coker [5]....

    [...]

  • ...Throughout this section, we shall present the fundamental definitions and results of intuitionistic fuzzy sets as given by Atanassov [2] and Coker [5]....

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K.K Azad1

430 citations


"Completely continuous functions in ..." refers background in this paper

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  • ...In ordinary topology, it is well known that the closure of the product is the product of the closures, while this property is not true in fuzzy setting (see [3])....

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TL;DR: The introduction and study of the concepts of certain classes of functions between fuzzy topological spaces, characterized and investigated mainly in the light of the notions of q-neighbourhoods, quasi-coincidence, fuzzy δ-closure and ds-NEighbourhips.

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"Completely continuous functions in ..." refers background in this paper

  • ...Completely continuous functions and results related to the product in fuzzy topological spaces were introduced in [7] and [3], respectively....

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Frequently Asked Questions (1)
Q1. What are the contributions in this paper?

In this paper, after giving the basic results related to the product of functions and the graph of functions in intuitionistic fuzzy topological spaces, the authors introduce and study the concept of fuzzy completely continuous functions between intuitionistic fuzzy topological spaces.