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Abstract
In this paper, graded fuzzy topological spaces based on the notion of neighbourhood system of graded fuzzy neighbourhoods at ordinary points are introduced and studied. These graded fuzzy neighbourhoods at ordinary points and usual subsets played the main role in this study.
AMS Subject Classifications:
Public Interest Statement
Separation axioms depend on the concept of neighbourhoods and so, for the fuzzy case, fuzzy neighbourhoods or valued fuzzy neighbourhoods means neighbourhoods with some degree in [0, 1] . These grades to be a fuzzy neighbourhood forced the fuzzy separation axioms to be graded. In the fuzzy case, separation axioms are not sharp concepts. For example, there is no topological space, but there are
topological spaces depending on the existence of the fuzzy neighbourhood with grade
at a point or the existence of the fuzzy neighbourhood with grade
at the other distinct point. In this paper, I introduced these graded fuzzy separation axioms. The main section was for defining a fuzzy neighbourhood system of fuzzy graded neighbourhoods at ordinary points and usual subsets.
1. Introduction
Kubiak (Citation1985) and Sǒstak (Citation1985) introduced the fundamental concept of a fuzzy topological structure as an extension of both crisp topology and fuzzy topology Chang (Citation1968), in the sense that both objects and axioms are fuzzified and we may say they began the graded fuzzy topology. Bayoumi and Ibedou (Citation2001,Citation2002,Citation2002b,Citation2004) introduced and studied the separation axioms in the fuzzy case in Chang’s topology (Citation1968) using the notion of fuzzy filter defined by Gähler (Citation1995a,Citation1995b).
Now, we will try to investigate fuzzy topological spaces in sense of Sstak, not using fuzzy filters but starting from a neighbourhood system of graded fuzzy neighbourhoods at ordinary points and usual sets. From that neighbourhood system, we can build a fuzzy topology in sense of S
stak and moreover, this fuzzy topology is itself the fuzzy topology in sense of Chang associated with the fuzzy neighbourhoterest Satementod filter
(Gähler, Citation1995b) at ordinary point
defined by Gähler. Interior operator and closure operator are defined using these graded fuzzy neighbourhoods; also their associated fuzzy topologies coincide with this fuzzy topology in sense of Chang associated with the fuzzy neighbourhood filter of Gähler. Fuzzy continuous, fuzzy open and fuzzy closed mappings are defined with grades according to these graded fuzzy neighbourhoods.
Separation axioms in the fuzzy case are introduced based on these graded fuzzy neighbourhoods and thus, axioms are graded. These axioms satisfy common results and implications. These graded axioms are a good extension in sense of Lowen (Citation1978). In Fuzzy neuro systems for machine learning for large data sets (Citation2009) and DCPE Co-Training for Classification (Citation2012), there are some applications based on fuzzy sets.
2. Preliminaries
Throughout the paper, let and
.
A fuzzy topology is defined by Kubiak (Citation1985) and Sǒstak (Citation1985):
(1) |
| ||||
(2) |
| ||||
(3) |
|
For each fuzzy set , the weak
cut-off f is given by
; the strong
cut-off f is the subset of X,
.
If T is an ordinary topology on X, then the induced fuzzy topology on X is given by .
fuzzy filters. Let X be a non- empty set. A fuzzy filter on X (Eklund, Citation1992; Gähler, Citation1995a) is a mapping such that the following conditions are fulfilled:
(F1) |
| ||||
(F2) |
|
A non-empty subset of
is called a prefilter on X (Lowen, Lowen), provided that the following conditions are fulfilled:
(1) |
| ||||
(2) |
| ||||
(3) |
|
is a prefilter on X.
Proposition 2.1
(Gähler, Citation1995a) There is a one-to-one correspondence between fuzzy filters on X and the families
of prefilters on X which fulfill the following conditions:
(1) |
| ||||
(2) |
| ||||
(3) | For each |
Proposition 2.2
(Eklund, Citation1992) Let A be a set of fuzzy filters on X. Then, the following are equivalent.
(1) | The infimum | ||||
(2) | For each non-empty finite subset | ||||
(3) | For each |
Recall that and
for all
.
Fuzzy neighbourhood filters. For each fuzzy topological space and each
, the mapping
defined by (Gähler, Gahler):
is a fuzzy filter on X, called the fuzzy neighbourhood filter of the space at the point x, and for short is called a fuzzy neighbourhood filter at x. The mapping
is defined by
for all
. The fuzzy neighbourhood filters fulfil the following conditions:
(1) |
| ||||
(2) |
|
The fuzzy neighbourhood filter at an ordinary subset F of X is the fuzzy filter on X defined in Bayoumi and Ibedou (Citation2002b), by means of
,
as:
The fuzzy filter is defined by
holds for all
. Also, recall that the fuzzy filter
and the fuzzy neighbourhood filter
at a fuzzy subset
of X are defined by
(1)
(1)
respectively. holds for all
(Bayoumi & Ibedou, Citation2004).
For each fuzzy topological space the closure operator
which assigns to each fuzzy filter
on X, the fuzzy filter
is defined by
(2)
(2)
is called the closure of
.
is isotone, hull and idempotent operator, that is for all fuzzy filters
and
on X, we have (Gähler, Citation1995b):
(3)
(3)
(4)
(4)
3. Neighbourhood systems
Definition 3.1
A family of fuzzy sets
is said to be a neighbourhood system with grade
on X if it satisfies the following conditions:
(Nb1) | For all | ||||
(Nb2) |
| ||||
(Nb3) |
| ||||
(Nb4) |
| ||||
(Nb5) | If |
Lemma 3.2
These families of prefilters at
satisfy the following conditions:
(Pr1) |
| ||||
(Pr2) |
| ||||
(Pr3) | For every |
Proof
Clear.
Remark 3.3
For any subset A of X, let us define by
, that is
iff
iff
.
,
,
,
,
. For all
in
, we have
.
For any , we have
,
and
implies that
,
and
implies that
. Also, for all
, we have either
or
.
fuzzy open sets, fuzzy open sets.
Let us define an fuzzy open set as follows:
(5)
(5)
An fuzzy closed set is the complement of an
fuzzy open set.
A set is said to be fuzzy open if it is
fuzzy open for all
. In other words, if for all
and for all
, we have
and
It is called a fuzzy closed if it is the complement of a fuzzy open set. These notations are restricted to the usual open and closed sets in fuzzy topology and usual topology.
Starting from a neighbourhood system with grade
, we can define an interior operator and a closure operator as follows:
(6)
(6)
(7)
(7) For every
,
satisfying (Nb1) to (Nb4) is exactly a prefilter on X of all neighbourhoods of
with grade
. That is,
is a family of prefilters with grade
at every
constructing after adding condition (Nb5) a neighbourhood system on X with grade
. The pair
is called a neighbourhood space with a grade
.
From Lemma 2.1 and from the correspondence given in Proposition 2.1 between the fuzzy filters and the families satisfying the conditions (Pr1) to (Pr3), we can say this family is a representation of the fuzzy neighbourhood filter
as a family of prefilters. This is given by the following two conditions (Nb) and (Pr):
(Pr) |
| ||||
(Nb) |
|
Clearly, both the interior operator and closure operator satisfy the common axioms of interior operator and closure operator, respectively. A fuzzy topology on X could be generated by this interior operator given by (2.2) or this closure operator given by (2.3), using the properties of stated in (Nb1)—(Nb5). That fuzzy topology is exactly the fuzzy topology
associated with the fuzzy neighbourhood filters
given by an interior operation as in (2.2) so that
Also, we can consider(8)
(8)
and then, (2.1) for an fuzzy open set could be rewritten as
(9)
(9)
That is, from a neighbourhood system of graded neighbourhoods, we can deduce interior operation by which it is introduced a graded fuzzy topology and the converse is true.
From (1.2) and (1.4) for all and all
, we can define
by
(10)
(10)
and equivalently,(11)
(11)
For all and all
, we have
.
Definition 3.4
Let and
be fuzzy topological spaces, and
a map. Then, for some
, f is called (
) fuzzy continuous if for all (
) fuzzy open set
with respect to
, we have
is an (
) fuzzy open set with respect to
for all
.
f is called fuzzy continuous if for all fuzzy open set with respect to
, we have
is a fuzzy open set with respect to
for all
.
Definition 3.5
Let and
be fuzzy topological spaces. Then, the mapping
is called (
) fuzzy open ((
) fuzzy closed) mapping if the image f(g) of the (
) fuzzy open ((
) fuzzy closed) set g with respect to
is (
) fuzzy open ((
) fuzzy closed) set with respect to
.
The mapping is called fuzzy open (fuzzy closed) mapping if the image f(g) of the fuzzy open (fuzzy closed) set g with respect to
is fuzzy open (fuzzy closed) set with respect to
.
Now, we define the continuity locally at a point between two fuzzy topological spaces using these graded neighbourhoods.
Definition 3.6
Let and
be two fuzzy topological spaces. Then, the mapping
is called
fuzzy continuous at a point
provided that for all
,
there exists such that
for some
. f is
fuzzy continuous if it is
fuzzy continuous at every
. f is an fuzzy continuous if it is
fuzzy continuous for all
.
This is an equivalent definition with Definition 2.2 for the fuzzy continuous mapping and fuzzy continuous mapping.
4. ![](//:0)
-spaces and ![](//:0)
-spaces
This section is devoted to introduce the notions of -spaces and
-spaces using the notion of
-neighbourhoods at ordinary points. We will introduce different equivalent definitions,
and we show that these notions are good extensions in sense of Lowen (Citation1978]).
Definition 4.1
A fuzzy topological space is called an
-space if for all
in X, there exists
such that
;
or there exists
such that
;
.
Definition 4.2
A fuzzy topological space is called an
-space if for all
in X there exist
and
such that
and
;
.
Example 4.3
Let , and
Taking
, we get that there is
in
such that
. For all
, we can not find any f in
such that
. That is,
is an
-space.
Example 4.4
Let , and
Only there is which is a graded neighbourhood but for both of x, y. Hence, for all
,
and therefore,
is not
-space.
Proposition 4.5
Every -space is an
-space.
Proof
Clear.
Example 3.1 is an -space but not
-space.
Example 4.6
Let , and
Taking
and
, we get that there is
in
and
in
such that
and
, for some
. Hence,
is an
-space.
In the following theorems, there will be introduced some equivalent definitions for the -spaces and
-spaces.
Theorem 4.7
Let be a fuzzy topological space. Then, the following statements are equivalent.
(1) |
| ||||
(2) | For all | ||||
(3) | For all | ||||
(4) | For all |
Proof
: From (1), there is
such that
;
and then,
and
. Hence,
;
and thus, (2) holds.
: There exists
such that
;
and then, for
, we can say
;
. The other case is similar and thus, (3) is satisfied.
: From Equation 2.3, we get that
whenever
, then (4) holds.
: Since
implies that z could not be y with
for all
;
, which means that there is
such that
;
. The other case is similar and thus, (1) holds.
Theorem 4.8
Let be a fuzzy topological space. Then, the following statements are equivalent.
(1) |
| ||||
(2) | For all | ||||
(3) | For all | ||||
(4) | For all |
Proof
: Let
in X. Then,
, which means for all
, if
whenever
, then
. From (1), we get that z could not be x with
, that is,
for all
. At x, it is clear that
. Hence,
for all
, and (2) is fulfilled.
: For all
in X, we have
and
. (2) means that
, which means for all
, z could not be x with
, that is there is
and there is
such that
and then,
. The other case is similar and therefore, (3) is fulfilled.
: As in Theorem 4.1.
: As in Theorem 4.1.
The next proposition shows that the separation axioms and
are good extensions in sense of Lowen (Citation1978).
Proposition 4.9
A topological space (X, T) is a -space (
-space) if and only if the induced fuzzy topological space
is an
-space (
-space).
Proof
Let (X, T) be (
) and let
. Then, there is a neighbourhood
such that
. Taking
such that
for some
, we get
, That is,
for some
. Similarly, if there is a neighbourhood
such that
, we can find
such that
and
for some
, That is,
for some
. Hence,
is an
-space (
).
Conversely, let be an
-space (
) and
. Then, there exists
such that
for some
, which means
for some
, that is there is
such that
and
. Similarly, the other case is proved. Hence, (X, T) is a
-space (
).
Proposition 4.10
Let be an
-space (
) and let
be a fuzzy topology on X finer than
. Then,
is also
-space (
-space).
Proof
is an
-space (
) implying that there is
such that
and
or (and) there is
such that
and
, which implies that
or (and)
. Since
is finer than
, then
or (and)
, and thus,
and
or (and)
and
. Hence
is an
-space (
).
5. ![](//:0)
-spaces
Here, we introduce and study the Hausdorff separation axiom in fuzzy topological spaces.
Definition 4.11
An fuzzy topological space is called an
-space if for all
in X there exist
and
such that
;
.
Proposition 4.12
Every -space is an
-space.
Proof
Let be an
-space but not
-space. That is, for
, we get for all
that
for all
. Since for any
we have
, then
and thus,
which contradicts the axiom
. Hence,
is an
-space.
Example 4.13
Let , and
There are
and
such that, for
and
in
, we get that
and
such that
and
. That is,
is an
-space. But for all fuzzy sets
and
, we get that
and thus,
is not
-space.
Theorem 4.14
Let be an fuzzy topological space. Then, the following statements are equivalent.
(1) |
| ||||
(2) | For all fuzzy ultrafilter | ||||
(3) | For all fuzzy filter |
Proof
: Suppose that there is an fuzzy ultrafilter
on X such that
and
for all
and
. That is,
, but in common we know that
for all
, which means that for all
and
, we have
and therefore, (1) implies (2) is satisfied.
: Since for any fuzzy filter
on X we find a finer fuzzy ultrafilter
on X, that is
for all
, then (2) implies that there is
such that
;
or there is
such that
;
. Thus, (3) holds.
: Suppose for all
and
;
that
and (3) is fulfilled. Then, for all fuzzy filter
on X, we have
or
;
. Hence,
, which means a contradiction to the common result that
and therefore,
. Thus, (1) is satisfied.
Example 4.15
Let , and
There are
and
such that for
and
in
, we get that
in
and
in
such that
and thus,
is an
-space.
Proposition 4.16
A topological space (X, T) is a -space if and only if the induced fuzzy topological space
is an
-space.
Proof
Let in X. Then, there are
such that
. Taking
such that
for some
, then
and
;
, that is
and
;
and then,
and
such that
, which means that there is no element
such that
;
, which means for all
, we have
;
. Hence,
;
and then,
;
and thus,
is an
-space.
Conversely, implies that there are
and
such that
;
. That is, for
, we can say
and
, which means that
,
and moreover,
and thus, (X, T) is a
-space. (because if there is
, then
which is a contradiction).
Proposition 4.17
Let be an
-space, and let
be an fuzzy topology on X finer than
. Then,
is also an
-space.
Proof
Let . Then, there are
and
such that
;
, that is
,
and
, which means that
,
and
;
and thus,
and
in
such that
;
. Hence,
is an
-space.
6. ![](//:0)
-spaces and ![](//:0)
-spaces
In this section, we use fuzzy neighbourhood filters at ordinary sets to define the notions of -spaces and
-spaces.
Definition 5.1
A fuzzy topological space is called
regular if for all
in P(X) and
, there exist
and
such that
;
.
Definition 5.2
A fuzzy topological space is called
-space if it is regular and
.
Definition 5.3
A fuzzy topological space is called normal if for all
with
, there exist
and
such that
;
.
Definition 5.4
A fuzzy topological space is called
if it is normal and
.
Proposition 5.5
Every -space is an
-space.
Proof
Let in X.
is an
-space meaning that
for each
. Now,
,
, and
is regular implying that there are
such that
;
. Hence,
is an
-space.
Theorem 5.6
For each fuzzy topological space , the following are equivalent.
(1) |
| ||||
(2) | For all | ||||
(3) | For all | ||||
(4) | For all |
Proof
: Let
;
. Suppose that
for some
, that is, there is
with
, which means that
. Since for all
, we have
for all
;
, then
for some
for all
, and for all
for some
;
, which contradicts (1) and therefore,
for all
. Thus,
for all
. The other case is similar and hence, (2) is satisfied.
: From (2) we deduce that for all
, we have
for all
implies
. Hence, for all
,
, and all
, we get that
, which means that
, but from that
for all
and for all
, we get that
for all
and for all
and thus, (3) holds.
: Let
be a fuzzy filter on X with
for all
and
. From (3),
for all
and
and then,
for all
and
and thus, (4) is fulfilled.
: Consider
in (4), we get that
for all
and all
. Now, for
and
, we get for all
that
, which means there are
and
. Choose
and
, then we can find
and
such that
, and thus, for all
a nd
, there exist
and
such that
;
, and therefore, (1) is satisfied.
Theorem 5.7
Let be a fuzzy topological space. Then, the following are equivalent.
(1) |
| ||||
(2) | For all | ||||
(3) | For all | ||||
(4) | For all |
Proof
Similar to the Theorem 6.1.
Proposition 5.8
Every -space is an
-space.
Proof
Let in X. Since
is
, then it is
, which means that
for all
, which implies that we have
and
with
. Hence, there are
and
such that
;
and thus,
is regular and it is
. Therefore,
is
.
Example 5.9
Let , and
We notice that
is a closed set and
. Then, there are
and
such that for
and
in
, we get that
in
and
in
such that
and thus,
is an
regular space. Also, it is an
-space. Hence,
is an
-space
Example 6.2
Let , and
We see that
and
are disjoint closed subsets of X. Then, there are
and
such that for
and
in
, we get that
in
and
in
such that
and thus,
is an
normal space. Also, it is an
-space. Hence,
is an
-space
Proposition 5.11
A topological space (X, T) is if and only if the induced fuzzy topological space
is
.
Proof
(X, T) is iff
is
is proved in Proposition 4.2.
Let and
in X. Then, there are
such that
. Taking
in
, we get that
. Hence, there are
and
of x and F respectively, such that
and thus,
is an
-space.
Conversely, and
imply there are
and
for all
such that
;
. That is,
and
, which means that
,
and moreover,
, and thus, (X, T) is a
-space.
Proposition 5.12
A topological space (X, T) is iff the induced fuzzy topological space
is an
.
Proof
Similar to Proposition 6.3.
Proposition 6.5
Let be an
-space, and let
be an fuzzy topology on X finer than
. Then,
is also an
-space.
Proof
Let and F be a closed subset of X with
. Then, there are
and
such that
;
, that is
,
for all
and
, which means that
,
for all
and
;
and thus,
and
in
such that
;
. Hence,
is an
regular space. Proposition 4.3 states that
is an
-space, and thus, it is an
-space.
Proposition 6.6
Let be an
-space and let
be a fuzzy topology on X finer than
. Then,
is also an
-space.
Proof
Similar to the proof in Proposition 6.5.
Additional information
Funding
References
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