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ISSN XXXX XXXX © 2016 IJESC
Research Article
Volume 6 Issue No. 8
Fuzzy Topological Spaces and Fuzzy Multifunctions
Sentamilselv i.M 1 , Mary sophia.S2
Assistant Professor1 , Research Scholar2
Depart ment of Mathemat ics
Vivekanandha College of Arts & Science for Wo men (Autonomous), Namakkal, Tamilnadu, India
Abstract:
In this paper, we introduced several fuzzy topological concepts and examine their properties. Our basic tool in that process is the
notion of the fuzzy neighbourhood of a point and also introduce the notion of fuzzy mu ltifunctions, linear fuzzy mult ifunctions and
study their properties.
Keywords: fu zzy topological space, fuzzy boundary, connected, fuzzy mu lt ifunction, fu zzy upper semicontinuous, fuzzy lower
semicontinuous, linear fuzzy mu ltifunctions, fuzzy balanced, fu zzy absorbing .
All elements of
- open sets).
INTRODUCTION
The fundamental concept of a fuzzy set was first introduced by
Zadeh. Since then work has been done by many authors, in
several directions, which has resulted in the formation of a new
mathematical field called “fuzzy mathematics”. The theory of
fuzzy
topological spaces
is a branch
of such
mathematics.Chang was the first to introduce the notion of
fuzzy topology. Several others continued the work in this area.
We remark that general point set topology can be regarded as a
special case of fuzzy topology, where all membership functions
are just characteristic functions.In this dissertation entitled,
‘Fuzzy Topological S paces and Fuzzy Multifunctions ’, we
introduced several fuzzy topological concepts and examine
their properties. Our basic tool in that process is the notion of
the fuzzy neighbourhood of a point and also introduce the
notion of fuzzy mu ltifunctions, linear fuzzy mu ltifunctions and
study their properties.
briefly
PRELIMINARIES
In this chapter, we discussed some basic defin itions which
are needed for the further definition.
Clearly then
1.
2.
are set to be open fuzzy set (or
Then an element
is said to be a closed
fuzzy set (or b riefly - closed ) if and only if
is -open.
2.3 Definiti on
A map
fro m
closure operator if for all
Kuratowski axio ms:
(i)
(ii)
into
(i.e.,
idempotent),
the
is said to be a
it satisfies the four
closure
operator
is
(iii)
(iv)
An equivalent way to define
is
is always
is the following:
-closed and
-closed.
2.4 Definiti on
A fuzzy topology on
this is the collection
2.1 Definiti on
Let
be a set. A fuzzy set in
is an element in
that is a function from into
Actually this is the
membership function of a fuzzy set of
2.2 Definiti on
A fuzzy topology to be a subset
of
such that
and let
Then
if and only if there is a
is called an interior
such that
The least upper bound of all interior fuzzy sets of
the interior of
and it is denoted by
is
-open).
is called
(clearly
2.5 Definiti on
(1)
(2)
If
(3)
If
Then the pair
Now let
fuzzy set of
then
for all
The fuzzy boundary of
then
denoted by
This is defined to be the infrimu m o f all -closed sets with
the property
for all
for wh ich we have
is called a fuzzy topol ogical space.
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Clearly,
is
-closed and
Hence, indeed
is a Hausdorff fu zzy topological space.
2.6 Definiti on
If
a map
continuous)
Hence the proof
and
are two fuzzy topological space,
is said to be fuzzy continuous ( or briefly if for all
3.3 Definiti on
Let
be an fuzzy topological space. We define
the following notions.
2.7 Definiti on
(i)
A fuzzy set in an fu zzy topological space
is
said to be a neighborhood of the points
if and only if
there is a
such that
and
We will denote a neighborhood of
by
then it is called an open neighborhood of
3.
A fuzzy set is said to be fuzzy dense ( dense) if and only if
A fuzzy set is said to be fuzzy boundary
( -boundary) if and only if
A fuzzy set
is said to be fuzzy nowhere
dense
( -nowhere dense) if and only
if
is F-boundary.
(ii)
(iii)
If
FUZZY TOPOLOGICAL SPACE
In this chapter, we discussed about some theorems on
fuzzy topological space.
3.4 Theorem
Let
and be an fuzzy topological space. Then
3.1 Definiti on
An fuzzy topological space
is said to be
Haus dorff if and only if for all
such that
is
- boundary if and only if
(ii)
is
- boundary if and only if
there exist
Proof
and
(i)
3.2 Theorem
If
space,
(i)
is
a
Hausdorff
fu zzy
-open bijection, then
is an
is also a Hausdorff
Proof
So
Given that,
is a Hausdorff fu zzy topological
space,
an fuzzy topological space. And
is an
-open bijection.
is also a
, since
.
is
-boundary.
(ii)
Now suppose that
Fro m part (i) we have that
is -boundary.
But
So
Since
is closed,
and
(3.1)
we know
Since by hypothesis,
is a Hausdorff fuzzy topological space,
(3.2)
Fro m (3.1) and (3.2), we conclude that
there exists
such that
and
But
Now,
since
is an -open map
and
both belong to
So
wh ich imp lies that
Therefore, by part (i) of this theorem,
Also
Finally,
(see
which means that
Next, let
and
Let
that
fu zzy
But
Let
proved
we can show that
topological space.
To prove that, fu zzy topological space
Hausdorff fu zzy topological space.
we
Theorem:2.6),
topological
an fuzzy topological space. And
As
and
we finally deduce that
is -boundary.
Hence the proof
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3.5 Theorem
If
Let
and
be fu zzy topological space.
Then -boundary sets are stable under
-continuous
surjections
If, in addition, is
-closed then the
family of -nowhere dense sets is also stable.
So
is
-nowhere dense, then (
.
But
Hence
Proof
Which means (theorem 2.8 (i)) that
Given that,
and
be fuzzy topological
space. Then -boundary sets are stable under -continuous
surjections
If, in addition, is -closed
To prove that, the family of
also stable.
Therefore
is -boundary.
is an - nowhere dense set.
Hence the proof
-nowhere dense sets is
3.6 Definiti on
First we will show that for
If
Fro m the definition of fu zzy interiors we have
, then
is said to be dense in
if and
only if
3.7 Definiti on
An fuzzy topological space
-compact for
=
if and only if for all
exists a neighborhood
But because
is
-continuous and
is said to be locally
of
such that
there
is -compact.
,we have that
3.8 Theorem
So
is
If
-open in
is a locally
space,
-compact fuzzy topological
is a Hausdorff fuzzy topological space and
Now d irectly fro m the definit ion of the fu zzy interior
is an
-continuous and
-open surjection then
is
also locally -co mpact.
we deduct that
Proof
Applying
in both sides and since
Given that,
for all
topological space,
(fro m the surjective of ),
space and
We deduct that
is a locally
-compact fuzzy
is a Hausdorff fu zzy topological
is an -continuous and -open surjection.
as claimed.
To prove that, is also locally -co mpact.
Now let
be an -boundary set.
Let
We know fro m theorem 3.4 (i) that
and
such that
which
means that
Since by defin ition
Let
be a neighborhood of
Since
is
-open, then
such that
is -compact.
is a neighborhood of
Also
So
But
(3.3)
, So
But since
is
-continuous,
Hence again by theorem 3.4 (i), we conclude that
is
is
- boundary.
Finally, assume that in addit ion
is
-compact and so it is also -closed.
(3.4)
-closed.
Then (
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Hence
But again by the -continuity of
2163
we have that
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(3.5)
Therefore, fro m (3.3),(3.4) and (3.5) above,
Hence
we
is disconnected,
which contradicts the hypothesis of the proposition.
conclude that
So
Hence the proof
is -co mpact.
Since
FUZZY MULTIFUNCTIONS
4.
was arbitrary, we conclude that
is indeed
locally -compact.
In this chapter, we discussed about some theorems on
fuzzy mu ltifunctions and linear fuzzy mu lt ifunctions.
Hence the proof
4.1 Definiti on
3.9 Definiti on
Let
An fuzzy topological space
such that
and
an fuzzy topological space. Let
is said to be
connected if and only if there do not exist
-open sets,
A subset
is
to be connected if it is connected as a fuzzy topological
be an ordinary topological space and
a mapping such that for every
we say that
i)
is a fuzzy multifunction ( -mult ifunction).
every neighborhood
neighborhood of
3.10 Theorem
and
be fuzzy topological space,
is -continuous and
is connected then
ii)
is also
is said to be
every
connected.
or
there exists
such that
a
for all
i.e.,
-lower semicontinuous if and only if for
such that
neighborhood
of
there exists a
such that
for all
.
Proof
Given that,
is
and
be fuzzy topological space,
-continuous and
4.2 Theorem
is connected.
is an
To prove that,
is also connected.
Suppose not. Then we can find
-upper semicontinuous mult ifunction
if and only if for all
is open
in
such that
Proof
and
Then since
to
is a fu zzy set. Then
is said to be -upper semicontinuous if and only if for
subspace of
If
be
is
-continuous, both
and
belong
Also we have that
First
Let
suppose
that
is
-upper
semicontinuous.
Then fro m definit ion 3.1(i)
We know that there exists a neighborhood
for all
So
claimed.
of
Which means that
such that
is open as
The other direction is just the definition of
semicontinuity.
-upper
Hence the proof
And
4.3 Definiti on
An fuzzy topological space
regular if and only if for
set such that
that
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is said to be quasi
a fuzzy point and
(i.e.,
and -closed
there exist
and
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such
4.4 Definiti on
So we have that for
A fuzzy multifunction
and only if when
neighborhood
(i.e.,
of
( ) wh ich imp lies that
is said to be closed if
and
is indeed fuzzy convex.
there is a
such that
Hence the proof
(i.e.,
for which if
4.7 Definiti on
4.5 Definiti on
A fuzzy mult ifunction
A fuzzy set
is said to be
(i)
fuzzy bal anced if and only if
is said to be linear if
and only if :
for
(ii)
i)
ii)
iii)
fuzzy
absorbing
if
and
only
if
for all
If
then
4.8 Definiti on
iv)
If
A fuzzy set
then
is said to be fuzzy cone if and
only if for every
If, in addition,
is
fuzzy convex, then it is called a fuzzy convex c one.
4.9 Theorem
4.6 Theorem
If
If
is a linear fu zzy mult ifunction with the
property that for
convex then
then if
is a cone and
mu ltifunction then
is a linear fuzzy
is a fuzzy cone.
is
Proof
is fu zzy convex.
Given that,
Proof
Let
and
is a cone and
We have
is a linear
fuzzy mu ltifunction.
To prove that,
Let
For any
is a fu zzy cone.
We have
]( )
But
since by hypothesis
So
Hence
is a cone in
for all
is indeed a fuzzy cone.
Hence the proof
for all
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CONCLUS ION
In this dissertation, we introduced several fuzzy
topological concepts and examine their properties and also
introduced the notion of fuzzy multifunctions and linear fuzzy
mu ltifunctions. We discussed about some properties of fuzzy
mu ltifunctions and linear fuzzy mult ifunctions. In Chapter III,
the concept of fuzzy topological spaces. In Chapter II, we have
discussed some basic definitions of fuzzy topological spaces. In
chapter I, the definit ions which are necessary for above were
given.
BIB LIOGRAPHY
1.
K. AZAD, On fu zzy semicountinuity, fu zzy almost
continuity and fuzzy weak continuity, J. Math. Anal. Appl.
82 (1981), 14-32.
2.
C. CHANG, Fu zzy topological spaces, J. Math. Anal.
Appl. 24 (1968), 191-201.
3.
A. KATSARAS AND D. LIU, Fu zzy vector spaces and
fuzzy topological vector spaces, J. Math. Anal. Appl. 58
(1977), 135-146.
4.
NIKOLAOS S. PAPAGEORGIOU, fu zzy topology and
fuzzy mu ltifunctions, J. Math. Anal. Appl.109(1985),
397-425..
5.
C. WONG, Fuzzy points and topology, J. Math. Anal.
Appl. 46 (1974), 314-328.
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