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J. Bangladesh Acad. Sci., Vol. 40, No. 2, 117-124, 2016
ON R0 SPACE IN L-TOPOLOGICAL SPACES
RAFIQUL ISLAM AND M. S. HOSSAIN1
Department of Mathematics, Pabna University of Science and Technology, Pabna,
Bangladesh
ABSTRACT
R0 space in L-topological spaces are defined and studied. The authors give eight definitions of
R0 space in L-topological spaces and discuss certain relationship among them. They showed that
all of these satisfy ‘good extension’ property. Moreover, some of their other properties are
obtained.
Key words: L-fuzzy set, L-fuzzy topology
INTRDUCTION
The concept of R0-property first defined by Shanin (1943) and there after Naimpally
(1967), Dude (1974), Hutton (1975 a, b), Dorsett (1978), Ali (1990) and Caldas (2000).
Khedr (2001) and Ekici (2005) defined and studied many characterizations of R0properties. Later, this concept was generalized to ‘fuzzy R0-propertise’ by Zhang (2008),
Keskin (2009) and Roy (2010) and many other fuzzy topologists. In this paper the
workers defined possible eight notions of R0 space in L-topological spaces and they
showed that this space possesses many nice properties which are hereditary productive
and projective.
Definition: Let X be a non-empty set and I = [0, 1]. A fuzzy set in X is a function
which assigns to each element
, a degree of membership,
(Zadeh 1965).
Definition: Let X be a non-empty set and L be a complete distributive lattice with 0
and 1. An L-fuzzy set in X is a function
which assigns to each element
, a degree of membership,
(Goguen 1967).
Definition: Let
complement of
in
Definition: If
then the authors refer to
1967).
1
be an L-fuzzy set in
(Goguen 1967).
.Then
and
is an L-fuzzy sets in
defined by
as a constant L-fuzzy sets and denoted it by
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
is called the
itself (Goguen
118
ISLAM AND HOSSAIN
In particular, these are the constant L-fuzzy sets 0 and 1.
Definition: An L-fuzzy point
function
in
where
Definition: An L-fuzzy point
if and only if
and
al. 1997).
is a special L-fuzzy sets with membership
(Liu et al.1997).
is said to belong to an L-fuzzy set
. That is
in
(Liu et
Definition: Let
,
be a non-empty set and
be the collection of all
mappings from
into , i. e. the class of all fuzzy sets in . A fuzzy topology on
is defined as a family of members of , satisfying the following conditions:
if
for each
then
if
then
. The pair
is called a fuzzy topological space (fts, in short) and the
members of are called -open (or simply open) fuzzy sets. A fuzzy set
is called a
-closed (or simply closed) fuzzy set if
(Chang 1968).
Definition: Let X be a non-empty set and L be a complete distributive lattice with 0
and 1. Suppose that
be the sub collection of all mappings from
to
.Then is called L-topology on
if it satisfies the following conditions:
(i)
(ii) If
(iii) If
of
then
for each
then
.
Then the pair
is called an L-topological space (lts, for short) and the members
are called open L-fuzzy sets. An L-fuzzy sets
is called a closed L-fuzzy set if
(Jin-xuan and Bai-lin 1998).
Definition: Let
by and defined as
The interior of
al.1997).
be an L-fuzzy set in lts
. Then the closure of
is denoted
.
written
is defined by
(Liu et
Definition: An L-fuzzy singleton in
is an L-fuzzy set in
which is zero
everywhere except at one point say , where it takes a value say
with
and
. The authors denote it by
and
iff
(Zadeh 1965).
Definition: An L-fuzzy singleton
is said to be quasi-coincident (q-coincident, in
short) with an L-fuzzy set
in , denoted by
iff
. Similarly, an
L-fuzzy set
in
is said to be q-coincident with an L-fuzzy set
in , denoted by
ON R0 SPACE IN L-TOPOLOGICAL SPACES
if and only if
for all
, where
not q-coincident with an L-fuzzy set
in
Definition: Let
is a fuzzy set in
119
for some
denote an L-fuzzy set
(Liu et al.1997).
. Therefore
iff
in
is said to be
be a function and
be fuzzy set in
which membership function is defined by
if
. Then the image
(Chang 1968).
Definition: Let
be a real-valued function on an L-topological space. If
is open for every real , then
is called lower-semi continuous function
(lsc, in short) (Liu et al.1997).
Definition: Let
from
into
and
be two L-topological space and
. Then is called
(i) Continuous iff for each open L-fuzzy set
(ii) Open iff
.
for each open L-fuzzy set
(iii) Closed iff
.
is s-closed for each
(iv) Homeomorphism iff
al.1997).
is bijective and both
be a mapping
.
where
is closed L-fuzzy set in
and
are continuous (Liu et
Definition: Let
be a nonempty set and
be a topology on . Let
be
the set of all lower semi continuous (lsc) functions from
to
(with usual
topology). Thus
for each
. It can be shown that
is
a
L-topology on . Let “P” be the property of a topological space
and LP be its
L-topological analogue. Then LP is called a “good extension” of P “if the statement
has P iff
has LP” holds good for every topological space
(Liu et
al. 1997).
Definition: Let
be a family of L-topological space. Then the space
is called the product lts of the family
where
denote the
usual product L-topologies of the families
of L-topologies on
(Li 1998).
R0 SPACES IN L-TOPOLOGY
The authors now give the following definitions of R0 L-topological spaces.
Definition: An lts
(a)
if
is called
whenever
with
then
120
ISLAM AND HOSSAIN
such that
(b)
.
if
whenever
then
with
such that
(c)
.
if for any pair of distinct L-fuzzy points
with
then
such that
(d)
whenever
.
if for all pairs of distinct L-fuzzy singletons
whenever
(e)
with then
and
such that .
if for any pair of distinct L-fuzzy points
with
(f)
then
whenever
such that
.
if for any pair of distinct L-fuzzy points
with
whenever
then
such
that
.
(g)
if
whenever
then
(h)
with
such that
if
whenever
such that
.
with
then
.
“GOOD EXTENSION”, HEREDITARY PROPERTY AND PRODUCT L-TOPOLOGY
Now
all
the
definitions
,
and
are ‘good extensions’ of
Theorem: Let
, as shown below:
be a topological space. Then
is
iff
is
, for
Proof: Let the topological space
fuzzy topological
is
with
. Let
, then it is clear that
and
,
with
. Since
, for any
is
, then there exist
. Now consider the characteristics function
with
. Thus
Conversely, let
. Choose
be
,
and
characteristic function
is
, The workers shall prove that the
. Choose
with
see that
be
.
, the workers shall prove that
, with
. Also it is clear that
, then
. The authors
is
such that
is
, but we know that the
. Since
Again since
ON R0 SPACE IN L-TOPOLOGICAL SPACES
is
121
lower semi continuous function then
,
. Hence
and from above, we get
is
.
Similarly the authors showed that
are
extension’ property.
also
hold
‘good
122
ISLAM AND HOSSAIN
Theorem: Let
be an lts,
and
is
, then
(a)
is
.
(b)
is
(c)
is
is
.
(d)
is
is
.
(e)
is
(f)
is
(g)
is
(h)
is
is
.
is
.
is
is
is
.
.
.
Proof: The authors proved only (b). Suppose
is L-topological space and
. The workers shall prove
is
. Let
,
, and
with
. Then
with
as
.
Consider
be the extension function of
on the set X, then it is clear that
. Since
is
. Then
such that
.
For
,
we
find
such
that
as
. Hence it is clear that the subspace
is
. Similarly, (a), (c), (d), (e), (f), (g), (h) can be proved.
Theorem: Given
product of L-topological space
is
where
be a family of L-topological space. Then the
is
iff each coordinate space
.
Proof: Let each coordinate space
authors show that the product space is
with
Choose
be
. Suppose
. Then the
,
, and
, but they have
, then there exist at least one
such that
and
,
.
Since
is
for each
, then
such that
. Now take
where
and
for
,
then
such that
. Hence the product L-topological space
is
.
Conversely, let the product of L-topological space
is
. The
workers shall prove each coordinate space
is also
. Choose
and
with
. Now, construct
such that
where
for
and
. Then
. Further, let
be a projection map from
into
.
Now,
the
workers
observe
that
ON R0 SPACE IN L-TOPOLOGICAL SPACES
Then
Then
that
123
, i.e for
, with
. Since the product space
is
.
such that
. Now choose any L-fuzzy point
in .
a basic open L-fuzzy set
such that
which implies
or that
and hence
and
.
Further,
, since for
and hence from
,
. Thus we have
. Now consider
then
with
. This showing that
is
.
Moreover one can verify that
is
is
is
is
is
is
is
is
is
is
is
is
is
Hence,
is
the
.
workers
see
that
properties are productive and projective.
MAPPING IN L-TOOLOGICAL SPACES
The present authors show that
and continuous maps for
Theorem: Let
and
property is preserved under one-one, onto
.
be two L-topological space and
be one-one, onto, L-continuous and L-open map, then(a)
is
is
(b)
is
(c)
is
is
.
(d)
is
is
.
(e)
is
is
is
.
.
.
124
ISLAM AND HOSSAIN
(f)
is
(g)
is
(h)
is
is
.
is
.
is
Proof: Suppose
. Let
is onto,
one.
is
,
such that
and
.
. The authors shall prove that
is
with
. Since
and
as
is one-
Now
and
Since
is
L-continuous
then
and
,
.
Since
is
, then
such that
. Now
.
Since
is L-open,
. Now, it is clear that
such that
. Hence, it is clear that the L-topological space
is
. Similarly (a), (c), (d), (e), (f), (g), (h) can be proved.
Theorem: Let
and
be two L-topological spaces and
be one-one, L-continuous and L-open map, then(a)
is
(b)
is
(c)
is
is
.
(d)
is
is
.
(e)
is
(f)
is
(g)
is
(h)
is
Proof: Suppose
. Let
is one-one map then
is
.
is
.
is
.
is
is
is
is
.
.
.
. The workers shall prove that
and
with
,
.
is
. Since
ON R0 SPACE IN L-TOPOLOGICAL SPACES
Now
125
as
is one-one
And
So, the authors have
L-open
map.
, with
Since
is
such
. This implies that
and
as
is L-continuous and
. Now it is clear that
such that
,
. Hence the L-topological space
. Similarly (a), (c), (d), (e), (f), (g), (h) can be proved.
, as
is
that
is
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(Received revised manuscript on 30 August, 2016)