On Covering Properties in Intuitionistic Fuzzy Topological Spaces:
A survey
FRANCISCO GALLEGO LUPIÁÑEZ
Department of Mathematics
University Complutense
Ciudad Universitaria, Madrid 28040
SPAIN
Abstract: - We show here some results on covering properties in intuitionistic fuzzy topological spaces. In
1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy
set. D. Çoker constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Çoker and
other mathematicians studied compactness, connectedness, continuity, separation, convergence and
paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He showed that every
intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the
results obtained by above authors are not redundant with other for ordinary fuzzy sense.
Key-Words: - Mathematics; Topology; Fuzzy topology; Fuzzy sets; Atanassov's intuitionistic fuzzy sets;
Covering properties
Received: April 15, 2024. Revised: September 6, 2024. Accepted: October 7, 2024. Published: November 6, 2024.
1 Introduction
In 1983, K.T. Atanassov proposed a
generalization of the notion of fuzzy set: the
concept of "intuitionistic fuzzy set" (IFS) [1].
Some basic results on intuitionistic fuzzy sets
were published in [2, 3], and the book [4]
provides a comprehensive coverage of virtually
all results until 1999 in the area of the theory
and applications of intuitionistic fuzzy set. D.
Çoker and M. Demirci [5] defined and studied
the basic concept of intuitionistic fuzzy point,
later D. Çoker [6, 7] constructed the
fundamental theory on "intuitionistic fuzzy
topological spaces" (IFTSs), and D. Çoker and
other mathematicians [8-38] studied
compactness, connectedness, continuity,
separation, convergence and paracompactness
in intuitionistic fuzzy topological spaces.
Finally, G.- J. Wang and Y.Y. He [39] showed
that every intuitionistic fuzzy set may be
regarded as an L-fuzzy set for some appropriate
lattice L. Nevertheless, the results obtained by
above authors are not redundant with other for
ordinary fuzzy sense.
On the other hand, currently, various authors
continue working about intuitionistic fuzzy
topological spaces. Indeed, K.T. Atanassov and
co-workers studied relation with various logic
properties [40-45], and many authors studied
some properties in intuitionistic fuzzy
topological spaces [46-56].
In this paper, we do a survey about covering
properties in intuitionistic fuzzy topological
spaces.
Papers authored by us about this subject, were
exposed previously in [57].
2 Definitions and Results
Firstly, we list previous definitions:
Definition 1. Let
X
a nonempty set. An
intuitionistic fuzzy set (IFS) in
X
is an object
from the form
, ( ), ( )
AA xXA x x x

∣
where the functions
: [0,1]
AX
and
: [0,1]
A
vX
define of degree of membership
and the degree of non- membership of an
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
69
element
xX
, respectively, and for each
xX
,
0 ( ) ( ) 1
AA
xx

Definition 2. [3] Let
X
be a nonempty set,
and
A
be an IFS in
X
. We determine for it the
four numbers
K=max{ ( )}
A
xX x
L=min{ ( )}
A
xX x
k=min{ ( )}
A
xX x
l=max{ ( )}
A
xX x
And the IFS
( ) , , |C A x K L x X
and
( ) , , |I A x k l x X
called closure and
interior of
A
Theorem 1. [23] Let
X
be a non-empty set,
S
the family of all IFS in
X
, and the mapping
: S S
()A C A
If we denote
F
|)F I S (F FCF
then
FF
{ | }
F
is an IFT on
X
(in Çoker's
sense).
Remark 1. [23] The closure operator of
Atanassov defines a Çoker's intuitionistic fuzzy
topology, but if we take an IFTS the closures of
IFS of it in Atanassov's sense are not
necessarily IFCSs in the IFTS.
Theorem 2. [23] Let
X
be a non-empty set,
S
the family of all IFSs in
X
, and
: S S
,
()
AA
be a mapping which
verify:
(1)
()
AA
(2)
( ( )) ( )
AA
(3)
(0 ) 0
(4)
( ) ( ) ( )
A B A B
Then, if
{
F
F
| ( ) }
IFS F F
and
FF
{ | }
F
we have that
is an IFT on
X
in
Çoker's sense, and, for every IFS
A
in
X
,
()
A
is the closure of
A
in
( , )
X
.
Proposition 1. [26] Let
( , )
X
,
( , )Y
be IFTSs,
:f X Y
be a map and
p
either an IFP or
VIFP in
X
. Then
f
is fuzzy continuous at
p
if
and only if every intuitionistic fuzzy net
s
in
X
that converges to
p
in
( , )
X
has the property
that
fs
converges to
()fp
in
( , )Y
.
Note. It is clear that, the result by Wang and
He [40] does not make this proposition
redundant. The reason for this is fundamentally,
that Çoker's definition of
-neighborhoods for
IFPs and VIFPs are not equivalent to the
corresponding concept of neighborhood of a
fuzzy point, based on the containment of fuzzy
points in fuzzy sets.
Definition 3. [24]
Let
, ( ), ( )
AA xXA x x x

∣
and
, ( ), ( )
BB xXB x x x

∣
be two IFSs.
We say that
A
quasi-coincides with
B
, denoted
by
AqB
, if
A
quasi-coincides with
B
and
'
A
quasi-coincides with
'
B
. (See also [20]
and [46-47])
Theorem 3. [24] Let
( , )
X
be an IFTS, let
p
be an IFP of
X
and let
()
Qp
U
be the family of
all the Q-nighborhoods of
p
in
( , )
X
, then:
(1)
()
Q
Np
U
implies that
pqN
.
(2)
12
, ( )
Q
N N p
U
imply that
12 ()
Q
N N p
U
.
(3) if
()
Q
Np
U
and
NM
, then
()
Q
Mp
U
.
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
70
(4) if
()
Q
Np
U
, then exists
()
Q
Mp
U
,
MN
, such that, for every IFP
e
which
quasi-coincides with
M
, we have that
()
Q
Me
U
.
Definition 4. [22] An IFTS
( , )
X
will be
called regular if for each IFP
p
and each IFCS
C
such that,
0pC
, there exist IFOSs
M
and
N
such that
pM
,
CN
, and
0MN
.
Definition 5. [22] An IFTS
( , )
X
will be
called normal if for each IFCSs
1
C
and
2
C
such
that
12
0CC
there exist IFOSs
1
M
and
2
M
such that
ii
CM
(i=1,2) and
12
0MM
.
Proposition 2. [22] Let
( , )
X
be a
2
T
IFTS. If
( , )
X
is normal, then also, it is a
regular IFTS.
Compactness in intuitionistic fuzzy
topological spaces is defined and studied in [6,
11-13, 19, 29, 31-33, 35].
Definition 6. [6] Let
( , )
X
be an IFTS. It is
called fuzzy compact if every fuzzy open cover
of
( , )
X
has a finite subcover.
Çoker [6] obtained some properties on
compactness in intuitionistic fuzzy topological
spaces analogous to them for compactness in
topological spaces. Çoker and Eş [12-13],
Hanafy [19], Ramadan [29], Thakur [38] and
other authors [28], [31], [33-35], [37] defined
and studied some variations of this concept.
Definition 7. [27] Let
( , )
X
be an IFTS and
,,
jj
GG Jx

U
∣j
and
,,
ii
AA Ix

V
∣i
be two families of
IFOSs in
X
. We will say that
V
refines
U
(or
V
is a refinement of
U
), if for each
Ii
there
exists some
Jj
such that
, , , ,
i i j j
A A G G
xx
.
Definition 8. [27] Let
( , )
X
be an IFTS and
,,
jj
GG Jx

U
∣j
be a family of IFSs in
X
. We will say that
U
is locally finite in an
IFS
A
of
X
if, for each intuitionistic fuzzy
point
pA
, there exists an
-neighbourhood
N
of
p
such that
, , 0
jj
GG
Nx


for all
Jj
in the complement of a finite subset of
J
.
Definition 9. [27] If
( , )
X
is an IFTS and
A
is an IFS of
X
, we will say that
A
is
paracompact if for each fuzzy open cover
,,
jj
GG Jx

U
∣j
of
A
and for each
(0,1]r
, there exists a refinement of
U
which
is locally finite in
A
and a fuzzy open cover of
Ar
. We will say that
X
is paracompact if
1
is a paracompact IFS.
Note. There is no problem with these
concepts and the result of Wang and He [39],
because here we used
-neighbourhoods.
Proposition 3. [27] If
0
( , )X
is a fuzzy
topological space in Lowen’s sense and
0
, ,1
AA
x
∣A
the associate IFT
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
71
on
X
. If
1
c
is a
-paracompact fuzzy set
of
0
( , )X
, then
( , )
X
is a paracompact IFTS.
3 Conclusion
Atanassov´s intuitionistic fuzzy set is a still
emerging concept with very interesting
applications to various areas as decision
making, information theory, intelligent systems,
pattern recognition, medical diagnosis,…
Anyone can quickly check this looking for the
subject “intuitionistic fuzzy set” in some
database. And also occurs for intuitionistic
fuzzy topological spaces. Future research in this
field could show yet other surprising new
applications in various scientific or
technological areas (for example, some papers
on intuitionistic fuzzy topological spaces
authored by us, are been cited for other authors
in papers on differential equations, logic,
algebra, pattern recognition, decision
making,...)
References:
[1] K.T. Atanassov, Intuitionistic fuzzy sets, in VII
ITKR's Session, Sofia (June 1983).
[2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy
Sets Syst 1986, 20, 87-96
[3] K.T. Atanassov, On four intuitionistic fuzzy
topological operators, Mathware Soft Comput. 2001,
8, 65-70.
[4] K.T. Atanassov, On Intuitionistic Fuzzy Sets
Theory, Springer-Verlag: Heidelberg, 2012
[5] D. Çoker, and M. Demirci. On intuitionistic
fuzzy points, Notes IFS, 1995 1, 2, 79-84
[6] D. Çoker, An introduction to intuitionistic fuzzy
topological spaces, Fuzzy Sets Syst.1997, 88, 81-89.
[7] D. Çoker, An introduction to fuzzy subspaces in
intuitionistic fuzzy topological spaces, J. Fuzzy
Math. 1996, 4, 749-764.
[8] S Bayhan, and D. Çoker, On
1
T
and
2
T
separation axioms in intuitionistic fuzzy
topological spaces, J. Fuzzy Math. 2003, 11, 581-
592.
[9] D. Çoker, and M. Demirci, An introduction to
intuitionistic fuzzy topological spaces in Šostak’s
sense, Busefal, 1996, 67, 61-66.
[10] D. Çoker, and M. Demirci, On fuzzy inclusion
in the intuitionistic sense. Fuzzy Math. }. 1996, 4,
701-714.
[11] A.H. Eş, and D. Çoker, On several types of
degrees of fuzzy compactness in fuzzy topological
spaces in Sostak's sense. J.Fuzzy Math.}.1995, 3,
481-491.
[12] D. Çoker, and A.H. Eş, On fuzzy compactness
in intuitionistic fuzzy topological spaces, J. Fuzzy
Math.1995, 3, 899-909.
[13] A.H. Eş, and D. Çoker, D. More on fuzzy
compactness in intuitionistic fuzzy topological
spaces, Notes IFS, 1996, 2, 1, 4-10.
[14] H. Gürçay, D. Çoker, and A.H. Eş, On fuzzy
continuity in intuitionistic fuzzy topological spaces
J. Fuzzy Math. 1997, 5, 365-378.
[15] S. Özçaġ, and D. Çoker, On connectedness in
intuitionistic fuzzy special topological spaces,
Internat. J. Math.& Math. Sci. 1998, 21, 33-40.
[16] N. Turanh, and D. Çoker, Fuzzy connectedness
in intuitionistic fuzzy topological spaces, Fuzzy Sets
Syst. 2000, 116, 369-375.
[17] Seok Jong Lee and Eun Pyo Lee, The category
of intuitionistic fuzzy topological spaces, Bull.
Korean Math. Soc. 2000, 37, 63-76.
[18] I.M. Hanafy, Completely continuous functions
in intuitionistic fuzzy topological spaces, Czech.
Math. J. 2003, 53 (128) 793-803.
[19] I.M. Hanafy. Intuitionistic fuzzy γ-
compactness. J. Fuzzy Math. 2003, 11, 313-323.
[20] K. Hur , J.H. Kim, and J.H. Ryou,
Intuitionistic fuzzy topological spaces. J. Korea Soc.
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
72
Math. Educ. Ser. B Pure Appl. Math. 2004, 11, 243–
265.
[21] F.G. Lupiáñez, Hausdorffness in intuitionistic
fuzzy topological spaces, J. Fuzzy Math. 2004, 12
521-52
[22] F.G. Lupiáñez, Separation in intuitionistic
fuzzy topological spaces, Int. J. Pure Appl. Math.
2004, 17, 24-34.
[23] F.G. Lupiáñez, Intuitionistic fuzzy topological
operators and topology, Int. J. Pure Appl. Math.,
2004, 17, 35-40.
[24] F.G. Lupiáñez, Quasi-coincidence for
intuitionistic fuzzy points, Internat. J. Math. &
Math. Sci, 2005, no10,1539-1542.
[25] F.G. Lupiáñez, On intuitionistic fuzzy
topological spaces, Kybernetes, 2006, 35, 743-747.
[26] F.G. Lupiáñez, Nets and filters in intuitionistic
fuzzy topological spaces, Information Sciences,
2006, 176, 2396-2404.
[27] F.G. Lupiáñez, Covering properties in
intuitionistic fuzzy topological spaces, Kybernetes,
2007, 36, 749-753.
[28] S.E. Abbas, On intuitionistic fuzzy
compactness., Inf. Sciences. 2005, 173, 75-91
[29] AA. Ramadan, S.E. Abbas, and A.A. Abd El-
Latif, Compactness in intuitionistic fuzzy
topological spaces, Intern. J. Math. Math. Sci. 2005,
19-32.
[30] R. Saadati, and Jin Han Park, On the
intuitionistic fuzzy topological spaces. Chaos
Solitons Fractals 2006, 27, 331-344.
[31] M.N. Mukherjee, and S. Das, Intuitionistic
fuzzy almost compactness in intuitionistic fuzzy
topological spaces, J. Fuzzy Math. 2008, 16, 583-
598.
[32] M.N. Mukherjee, and S. Das, A note on α-
compactness in intuitionistic fuzzy topological
spaces. J. Fuzzy Math. 2009, 17, 877-883
[33] K.K. Azad, S. and S. Mittal, On Hausdorffness
and compactness in intuitionistic fuzzy topological
spaces. Mat. Vesnik. 2011, 63, 145-155.
[34] Z.M. Zhang, Generalized intuitionistic fuzzy
rough sets based on intuitionistic fuzzy coverings,
Information Sciences 2012, 198 , 186-206
[35] N. Gowrisankar, N. Rajesh, and V.
Vijayabharathi, On intuitionistic fuzzy γ-
compactness. J. Fuzzy Math. 2013, 21, 279-287.
[36] A.A.Q. Al-Qubati, Covering dimension of
intuitionistic fuzzy topological spaces. Ann. Fuzzy
Math. Inform., 2014, 7, 485–493.
[37] P. Saranya, M.K. Uma, and E. Roja,. View on
intuitionistic rough paracompactness and
intuitionistic rough nearly paracompactness. Ann.
Fuzzy Math. Inform. 2015, 9, 639-648.
[38] M. Thakur, and S.S. Thakur, C-compactness in
intuitionistic fuzzy topology, J. Xi’an
Univ.Architecture & Technology 2021 13, no 3,
148-156.
[39] G.-J. Wang, and Y.-Y. He, Intuitionistic fuzzy
sets and L-fuzzy sets, Fuzzy Sets Syst, 2000, 110,
271-274.
[40] K. Atanassov, Intuitionistic fuzzy modal
topological structure, Mathematics 2022, 10, 3313.
https:// doi.org/10.3390/math10183313
[41]K.Atanassov,Intuitionistic fuzzy modal topologi
cal structures based on two
new intuitionistic fuzzy modal operators, J.
Multiple-Valued Logic Soft Computing 2023, 41,
.227-240.
[42] K. Atanassov, On intuitionistic fuzzy temporal
topological structures, Axioms 2023, 12, 182.
https://doi.org/10.3390/ axioms12020182
[43] K. Atanassov, N. Angelova, and T. Pencheva,
On two intuitionistic fuzzy modal topological
structures, Axioms 2023, 12, 408.
https://doi.org/10.3390/ axioms12050408
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
73
[44] K.T. Atanassov and R. Tsvetkov,
New intuitionistic fuzzy operations, operators
and topological structures, Iranian J. Fuzzy Systems
2023, 20 (7), 37-53.
[45] K. Atanassov, Intuitionistic fuzzy modal
multi-topological structures and intuitionistic fuzzy
multi-modal multi-topological structures,
Mathematics 2024, 12, 361. https://doi.org/10.3390/
math12030361
[46] J. Kim, P.K. Lim, J.G. Lee, and K. Hur,
Intuitionistic topological spaces. Ann. Fuzzy Math.
Inform. 2018, 15, 29-46.
[47] J.G. Lee, P.K. Kim, J. Kim, and K. Hur,
Intuitionistic continuous,closed and open mappings,
Ann. Fuzzy Math. Inform. 2018, 15, no. 2, 101-122.
[48] A.A.Q. Al-Qubati, M.E. Sayed, and H.F.Al-
Qahtani, Small and large inductive dimensions
of intuitionistic fuzzy topological spaces,
Nanoscience and Nanotechnology Letters, 2020, 12,
413-417
[49] S. S. Thakur, Ch. P. Rathorb, and M.Thakur ,
Generalized e-closed sets and generalized e-
continuity in intuitionistic fuzzy topology, J. Math.
Computer Sci. 2022, 25, 219-231.
[50] Jin Tae Kim and Seok Jong Lee,
Generalized fuzzy closed sets on intuitionistic
fuzzy topological spaces, Journal Chungcheong
Math. Soci. 2022 35 (3), 243-254.
[51] M. Mostafavi, Intuitionistic topological spaces
with L-gradations of openness and nonopenness
with respect to LT-norm T and LC-conorm C on X. ,
J. Ramanujan Soc. Math. Math. Sci. 2022, 9, no. 2,
131-152.
[52] A. A. Q. Al-Qubati, and M. El Sayed, Door
spaces in intuitionistic fuzzy topological spaces,
Intern. J. Fuzzy Logic Intelligent Systems 2022 22,
No. 3, 296-302.
[53] S. Tarsuslu, A study
on intuitionistic fuzzy topological operators, Italian
J. Pure Appl. Math. 2023, (49), 863-875.
[54] S.M. Sudha, β** generalized homeomorphisms
in intuitionistic fuzzy topological spaces, Adv. Appl.
Math. Sci. 2021, 21.831-843.
[55] T. Menahadevi, P. Maragatha Meenakshi, N.
Rajesh, and B. Brundha, Regularly open sets in
intuitionistic fuzzy topological spaces, J. Math.
Computer Science, 2023, 30, 10-18.
[56] G. Sivaraman, and V.J. Jasmy, A theoretical
approach on intuitionistic fuzzy Hausdorff
space. Proyecciones, 2023, 42, no. 2, 319-338
[57] F.G. Lupiáñez, Some recent results on
Atanassov’s intuitionistic fuzzy topological spaces.
In Computational Intelligence in Decision and
Control (Proc. 8th International FLINS Conf.);
World Scientific: Singapore, 2008, pp. 229-234.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
PROOF
DOI: 10.37394/232020.2024.4.5
Francisco Gallego Lupiáñez
E-ISSN: 2732-9941
74