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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
Maejo International
Journal of Science and Technology
ISSN 1905-7873
Available online at www.mijst.mju.ac.th
Full Paper
On
-open set and
topological spaces
-open set in intuitionistic
Basker Palaniswamy 1, * and Kokilavani Varadharajan 2
1
Department of Mathematics, Jansons Institute of Technology, Coimbatore-641 659, Tamil Nadu,
India
2
Department of Mathematics, Kongunadu Arts and Science College, Coimbatore - 641 029, Tamil
Nadu, India
* Correspondingauthor, e-mail: [email protected]; [email protected]
Received: 11 November 2014 / Accepted: 26 March 2016 / Published: 15 July 2016
Abstract: We introduce some new types of sets called intuitionistic semi-open set,
open set, intuitionistic pre-open set and
Also, we discuss some of their properties.
Keywords: intuitionistic semi-open set,
open set
-
-open set in intuitionistic topological spaces.
-open set, intuitionistic pre-open set,
-
_______________________________________________________________________________________
INTRODUCTION
After Atanassov [1] introduced the concept of ‘intuitionistic fuzzy sets’ as a generalisation
of fuzzy sets, it became a popular topic of investigation in the fuzzy set community. Many
mathematical advantages of intuitionistic fuzzy sets have been discussed. Coker [2] generalised
topological structures in fuzzy topological spaces to intuitionistic fuzzy topological spaces using
intuitionistic fuzzy sets. Later many researchers have studied topics related to intuitionistic fuzzy
topological spaces.
On the other hand, Coker [3] introduced the concept of ‘intuitionistic sets’ in 1996. This is a
discrete form of intuitionistic fuzzy set, where all the sets are entirely crisp sets. Still, it has
membership and non-membership degrees, so this concept gives us more flexible approaches to
representing vagueness in mathematical objects including those in engineering fields with classical
set logic. In 2000 Coker [4] also introduced the concept of intuitionistic topological spaces with
intuitionistic set and investigated the basic properties of continuous functions and compactness. He
and his colleague [5, 6] also examined separation axioms in intuitionistic topological spaces.
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PRELIMINARIES
Definition 1 [3]. Let be a non-empty fixed set. An intuitionistic set ( ) is an object having the
form
, where
and
are subsets of satisfying
. The set
is
called the set of members of , while
is called the set of non-members of .
Every crisp set on a non-empty set is obviously an having the form
, and
one can define several relations and operations between
as follows:
Definition 2 [3]. Let
let
be a non-empty set,
be an arbitrary family of
if and only if
and
if and only if
;
;
;
and
in , where
;
;
be
. Then
if and only if
;
;
;
and
on , and
and
;
.
The following are the basic properties of inclusion and complementation:
Corollary 1 [3]. Let
for each
,
and
be
in
. Then
for each
;
;
;
;
.
;
Now we generalise the concept of ‘intuitionistic fuzzy topological space’ to intuitionistic sets:
Definition 3 [2]. An intuitionistic topology (
satisfying the following axioms:
) on a non-empty set
is a family
of
in
,
for any
,
for any arbitrary family
.
In this case the pair
is called an intuitionistic topological space (
known as an intuitionistic open set (
) in .
Example 1 [4]. Any topological space
is obviously an
whenever we identify a subset in with its counterpart
Example 2 [4]. Let
where
) and any
in the form
as before.
and consider the family
. Then
is an
Example 3 [4]. Let
,
on .
,
. Then
is
,
and
and consider the family
where
in
,
,
is an
on .
Definition 4 [4]. The complement of an intuitionistic open set
called an intuitionistic closed set (
) in .
Now we define closure and interior operations in
and
:
in an
is
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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
Definition 5 [7]. Let
closure of are defined by
be an
and
be an
in . Then the interior and
,
.
In this paper we use
shown that
is an
and is an
instead of
and
is an
in if and only if
Example 4 [4]. Let
where
and
in
Proposition 2 [4]. Let
(a)
;
(c)
(e)
(g)
(i)
;
, then we can write
.
in
we have
be an
;
in an
(a) intuitionistic semi-open (
(b) intuitionistic -open (
in . Then the following properties hold:
;
;
;
is said to be
-open) if
;
-open) if
(c) intuitionistic pre-open (
;
-open) if
-open (
,
.
-open,
-open) sets of an
is denoted by
).
An IS
in an
(
,
Example 5. Let
and
is an
.
-OPEN SET
Definition 6. An IS
The family of all
and
and , be
(b)
;
(d)
(f)
; (h)
(j)
.
;
-OPEN SET AND
if
,
and
Proposition 1 [4]. For any IS
(
in
is said to be
-closed (
-closed,
-closed)
).
and consider the IT
. Let
, where
. Here
-open set since
Example 6. Let
and
.
and consider the IT
Let
,
where
is an
-
. Here
open set since
.
Example 7. Let
and
since
. It can also be
if and only if
and consider the family
. If
and
ON
instead of
, and is an
.
and consider the IT
Let
.
. Here
is an
where
-open set
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Theorem 1. (a) Every
-open set is
be an IOS in
Proof. (a) Let
. Since
Hence
(b)
-open set; (b) every
and
Let
be
-open set.
, then
is an
-open
-open set is
.
-open set.
set
in
.
. Therefore,
Then
is an
it
follows
that
-open set.
Remark 1. The separate converses need not be true in general, which is shown by the following
examples.
and consider the IT
Let
Example 8. Let
and
open set but not an IOS since
and consider the IT
Let
and
-open set since
and
. Here
is not contained in
Theorem 2. The finite union of
Proof. Let
is an
, where
-open set
.
Example 9. Let
but not an
. Here
, where
is an
-
.
-open set is always an
be two
and
-open set.
-open sets. Then
imply that
is an
Proposition
. Therefore,
-open set.
Let
3.
be
ITS
an
and
let
.
Then
.
Proof. If
, then
.
Conversely, let
and so
. Take
be such that
. Then
and
. Then
.
Theorem 3. Let
be an ITS. A sub-set
open set and
-open set.
Proof. Necessity: Let A be
that
of
be
-open set if and only if it is
-open set. Then we have
and
Sufficiency: Let
is an
. Hence
-open set and
. This implies
is
Let
is said to be an
be an ITS and
.
-open set and
-open set.
-open set. Then we have
. This shows that
Definition 7. A sub-set of
open set, which is equivalent.
-
be a sub-set
is an
-open set.
-closed set if and only if
. Then
is an
is an
-
-closed set if and only if
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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
Definition 8. Let
(a)
-interior of
.
be an ITS and be a sub-set . Then
is the union of all
-open sets contained in
and it is denoted by
.
(b)
-closure of
.
is the intersection of all
-closed sets containing
and it is denoted by
.
(c)
-interior of
.
is the union of all
-preopen sets contained in
and it is denoted by
.
(d)
-closure of
.
is the intersection of all
-preclosed sets containing
and it is denoted by
.
(e)
-interior of
.
is the union of all
-semiopen sets contained in
and it is denoted by
.
(f)
by
-closure of
.
is the intersection of all
-semiclosed sets containing
and it is denoted
.
Remark 2. It is clear that
is an
Theorem 4. An IS
(a)
(b)
-open set and
in an
.
.
is an
-closed set.
. Then
Proof. (a) and (b) are clear.
Observation 1. The following statements are true for every
(a)
.
(b)
.
and :
Proof. Obvious.
Observation 2. Let
be a sub-set of a space
(a)
. Then
.
(b)
.
(c)
.
(d)
.
(e)
.
(f)
.
Proof. Obvious.
Example 10. Let
and
and consider the IT
Let
, where
. Then
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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
(a)
.
(b)
.
(c)
.
(d)
.
(e)
.
(f)
.
Observation 3. Let A be a sub-set of a space
. Then
(a)
.
(b)
.
(c)
.
Example 11. Let
and
and consider the IT
Let
(a)
, where
. Then
.
,
,
,
,
(1)
(2)
From (1) and (2),
.
(b)
,
,
(3)
(4)
From (3) and (4),
.
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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
(c)
,
(5)
,
(6)
,
(7)
,
(8)
,
(9)
From (5), (6), (7), (8) and (9),
.
Definition 9. An IS
in an
is said to be
(a) intuitionistic semi-preopen (
-open or
-open) if there exists
such that
. The family of all
-open sets of an
will be denoted by
.
(b) intuitionistic semi-preclosed (
-closed or
-closed) if there exists an intuitionistic
preclosed set such that
. The family of all
-closed sets of an
will be denoted by
.
Let
. Then
Example 12.
Let
and
,
,
is an IT on and
in
. Then
be an
.
, and hence
.
Theorem 5. For every
Proof. Straightforward.
Theorem 6. Every
Proof. Let
be
in
-open set is an
, we have
.
-open set.
-open set in
. Then it follows that
. Hence
is an
-open set.
Remark 3. The converse of the above theorem need not be true in general. It is shown by the
following example.
Example 13. Let
but not an
and
-open set since
Theorem 7. For every
Proof. Straightforward.
and consider the IT
Let
is not contained in
in
, we have
Theorem 8. Let
be an
. Then
(a) Any union of
-open sets is an
-open set.
(b) Any intersection of
-closed sets is an
-closed set.
. Here
is an
, where
-open set
.
.
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Proof. (a) Let
be a collection of
-open sets of
such
that
and
(b) This is from (a) by taking compliments.
for
. Hence
Theorem 9. For any IS
. Then there exists
each
.
,
, then we can take
so that
and assume that there exists
, and so
Theorem 10. Let
(a)
be an
follows
that
.
in an
Proof. If
Let be an in
Then
set by Theorem 8(a).
It
if and only if
for every
such that
, which is an
.
.
-open
. Then
.
(b)
.
Proof. (a) Assume that
be such that
that
so that
(b) This follows from (a).
Definition 10. Let
(a)
-interior of
.
for every
. Obviously,
and
. From
. Hence
be an ITS and be a sub-set . Then
is the union of all
-open sets contained in
. Let
, it follows
.
and it is denoted by
.
(b)
-closure of
.
is the intersection of all
-closed sets containing
and it is denoted by
.
Observation 4. Let
be a sub-set of a space
(a)
. Then
.
(b)
.
Example 14. Let
and
and consider the IT
Let
(a)
(b)
Observation 5. Let
(a)
, where
. Then
.
.
be a sub-set of a space
. Then
.
(b)
.
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Example 15. Let
and
(a)
and consider the IT
Let
.
, where
. Then
.
(10)
,
(11)
From (10) and (11),
.
(b)
.
,
(12)
,
(13)
(14)
,
(15)
(16)
From (12), (13), (14), (15) and (16),
.
CONCLUSIONS
We have introduced the sets in intuitionistic topological spaces called intuitionistic semiopen set, intuitionistic -open set, intuitionistic pre-open set,
-interior of ,
-closure of ,
-interior of ,
-closure of ,
-interior of ,
-closure of , intuitionistic semipreopen set, intuitionistic semi-preclosed set,
-interior of and
-closure of , and studied
some of their properties.
ACKNOWLEDGEMENTS
We would like to express our sincere gratitude to the referees and editor for their valuable
suggestions and comments which improved the paper.
REFERENCES
1. K. T. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets Syst., 1986, 20, 87-96.
2. D. Coker, “An introduction to intuitionistic fuzzy topological spaces”, Fuzzy Sets Syst., 1997,
88, 81-89.
3. D. Coker, “A note on intuitionistic sets and intuitionistic points”, Tr. J. Math., 1996, 20, 343351.
4. D. Coker, “An introduction to intuitionistic topological spaces”, Bull. Stud. Exchanges Fuzz.
Appl. 2000, 81, 51-56.
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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196
5. S. Bayhan and D. Coker, “On separation axioms in intuitionistic topological spaces”, Int. J.
Math. Math. Sci., 2001, 27, 621-630.
6. S. Bayhan and D. Coker, “Pairwise separation axioms in intuitionistic topological spaces”,
Hacet. J. Math. Stat., 2005, 34S, 101-114.
7. S. J. Lee and J. M. Chu, “Categorical property of intuitionistic topological spaces”, Commun.
Korean Math. Soc., 2009, 24, 595-603.
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