Some Mixed Soft Operations and Extremally Soft Disconnectedness via Two Soft Topologies ()
Keywords:
The aim of the present paper is to introduce and study notions of
-semi open soft (resp.
-pre
open soft,
-
-open soft,
-
-open soft,
-semi open soft,
-pre-open soft,
-
-open soft,
-
-open soft) set via two soft topologies. For this purpose, we consider two soft topologies
and
over
. Also we define
-regular open soft and ESDC on two soft topologies. Furthermore, we investigate some properties of some mixed soft operations and some characterizations of ESDC. Finally, we show
-ESDC soft topologies
.
2. Preliminaries
Soft sets and Soft Topology
Definition 2.1 [1]. The complement of a soft set (F,A) is defined as
, where
, for all
.
Theorem 2.1 [2]. Let
be a soft topological space over
,
and
are soft sets over
. Then
1)
and
.
2)
.
3)
is a closed set if and only if
.
4)
.
5)
implies
.
6)
.
7)
.
Theorem 2.2 [2]. Let
be a soft topological space over
and
and
are soft sets over
. Then
1)
and
.
2)
.
3)
.
4)
is a soft open set if and only if
.
5)
implies
.
6)
.
7)
.
Theorem 2.3 [2]. Let
be a soft set of soft topological space over
. Then
1)
.
2)
.
3)
.
Definition 2.2 [3]. Let
be a soft topological space and
. Then
is said to be
1) pre-open soft set if
;
2) semi-open soft set if
;
3)
-open soft set if
;
4)
-open soft set if
.
Definition 2.3 [4]. Let
and
be two soft sets over a common universe X. Then
is said to be a soft subset of
if
and
, for all
. This relation is denoted by
.
Definition 2.4 [4]. A soft set (F,A) over X is said to be a null soft set if
, for all
. This is denoted by
.
Definition 2.5 [4]. A soft set (F,A) over X is said to be an absolute soft set if
, for all
. This denoted by
.
Definition 2.6 [4]. The union of two soft sets
and
over the common universe
is the soft set
, where
and
if
or
if
or
if
for all
.
Definition 2.7 [4]. The intersection of two soft sets
and
over the common universe
is the soft set
, where
and for all
,
.
Definition 2.8 [5]. A pair
, where
is mapping from
to
, is called a soft set over
. The family of all soft sets on
is denoted by
.
is said to be soft equal to
if
and
. This relation is denoted
by
.
Definition 2.9 [6]. Let
be the collection of soft sets over X. Then
is said to be a soft topology on X if
1)
;
2) the intersection of any two soft sets in
belongs to
;
3) the union of any number of soft sets in
belongs to
.
The triple
is called a soft topological space over X. The members of
are said to be
soft open sets or soft open sets in X. A soft set over X is said to be soft closed in X if its complement belongs to
. The set of all soft open sets over
denoted by
or
and the set of all soft closed sets denoted by
or
.
Definition 2.10 [6]. The difference of two soft sets (F,A) and (G,A) is defined by
,
where
, for all
.
Definition 2.11 [6]. Let
be a soft topological space and
. The soft closure of
, denoted by
is the intersection of all closed soft super sets of
.
Definition 2.12 [7]. Let
be a soft topological space and
. The soft interior of
, denoted by
is the union of all open soft subsets of
.
3. Some Properties of Some Mixed Soft Operations
In this section we investigated some properties of some mixed operations such as
-semi open soft,
-pre open soft. Also we will write
for
, respectively.
Definition 3.1. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
is said to be
1)
-semi open soft if
;
2)
-pre open soft if
;
3)
-
-open soft if
;
4)
-
-open soft if
.
The complement of
-semi open (resp.
-pre open,
-
-open,
-
-open) soft set is called
-semi closed (resp.
-pre closed,
-
-closed,
-
-closed) soft (See Figure 1).
Definition 3.2. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
is said to be
1)
-semi open soft if
;
2)
-pre open soft if
;
3)
-
-open soft if
;
-soft open
-
-open soft
-semi open soft
![](https://www.scirp.org/html///html.scirp.org/file/16-7401937x225.png)
4)
-
-open soft if
.
The complement of
-semi open (resp.
-pre open,
-
-open,
-
-open)
soft set is called
-semi closed (resp.
-pre closed,
-
-closed,
-
-closed)
soft (See Figure 2).
Theorem 3.1. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If
and
,
.
2) If
and
,
.
Proof. It is seen from Definition
.
Theorem 3.2. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1)
is a
-semi open soft set if and only if
.
2)
is a
-semi open soft set if and only if
.
Proof. 1) Necessity. Let
be a
-semi open soft set. Since
, we
have
. Also
. Hence
.
Sufficieny. Let
. Therefore
and
is a
-semi open soft.
2) By a similar way.
Theorem 3.3. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If
is a
-soft open set and
is a
-pre open soft set,
is a
- pre open soft.
2) If
is a
-soft open set and
is a
-pre open soft set,
is a
-pre open soft.
Proof. (1). Let
be
-soft open and
be
-pre open soft set. Then
![](https://www.scirp.org/html///html.scirp.org/file/16-7401937x306.png)
from Theorem
Hence
is a
-pre open soft.
(2). By a similar way.
Theorem 3.4. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If either
is a
-semi open soft or
is a
-semi open soft set,
.
2) If either
is a
-semi open soft or
is a
-semi open soft set,
.
Proof. 1) Let
. We have
.
-soft open
-
-open soft
-semi open soft
We assume that
is a
-semi open soft set. Then
from Theorem 3.2. So
![]()
from Theorem 3.1. Hence we have
.
2) By a similar way.
Theorem 3.5. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If
is a
-soft open set and
is a
-semi open soft set,
is a
-semi open soft.
2) If
is a
-soft open set and
is a
-semi open soft set,
is a
-semi open soft.
Proof. 1) Let
be
-open soft and
be
-semi open soft set. Then
![]()
from Theorem
Therefore
is a
-semi open soft.
2) By a similar way.
Theorem 3.6. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Let
, then
1)
is
-
-open soft set if and only if there exists a
-soft open set such that
.
2) If
is
-
-open soft set and
, then
is
-
-open soft.
3)
is
-
-open soft set if and only if there exists a
-soft open set such that
.
4) If
is
-
-open soft set and
, then
is
-
-open soft.
Proof. 1) Necessity. Let
. Then
.
Sufficiency. Let
is
-soft open set and
. Then
. Hence
. Thus ![]()
is
-
-open soft.
2) Let
is
-
-open soft set and
. Hence
.
Thus
is
-
-open soft.
3)-4) By a similar way.
Theorem 3.7. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If
is
-
-open soft and
is
-
-open soft, then
is
-
-open soft.
2) If
is
-
-open soft and
is
-
-open soft, then ![]()
is
-
-open soft.
Proof. 1) Let
is
-
-open soft and
is
-
-open soft. Then
![]()
from Theorem
Thus
is
-
-open soft.
2) By a similar way.
Proposition 3.1. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. If
, the following statements hold:
1)
.
2)
.
Proof. It is obvious from Definition 2.1., 2.11. and
.
Theorem 3.8. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1)
is a
-pre closed soft set if and only if
.
2)
is a
-pre closed soft set if and only if
.
Proof. 1) Necessity. Let
be
-pre closed soft set. Then
is a
-pre open soft set, that is
.
Thus
.
Sufficiency. Let
, then
.
Hence
is a
-pre open soft set. Therefore
is a
-pre closed soft set.
2) By a similar way.
Theorem 3.9. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1)
is a
-
-closed soft set if and only if
.
2)
is a
-
-closed soft set if and only if
.
Proof. 1) Necessity. Let
is a
-
-closed soft set. Then
is a
-
-open
soft, that is
. Therefore,
.
Sufficiency. Let
, then
.
Hence
is a
-
-open soft set. Therefore
is a
-
-closed soft set.
2) By a similar way.
Theorem 3.10. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1)
is a
-semi closed soft set if and only if
.
2)
is a
-semi closed soft set if and only if
.
Proof. 1) Necessity. Let
is a
-semi closed soft set. Then
is a
-semi
open soft set, that is
. Therefore,
.
Sufficiency. Let
, then
.
Hence
is a
-semi open soft set. Therefore
is a
-semi closed soft set.
2) By a similar way.
Theorem 3.11. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1) If
is a
-
-open soft set and
is a
-semi open soft set,
is a
-semi open soft.
2) If
is a
-
-open soft set and
is a
-semi open soft set,
is a
-semi open soft.
Proof. 1) Let
is a
-
-open soft and
is a
-semi open soft. Then
.
Therefore
is a
-semi open soft.
2) By a similar way.
Theorem 3.12. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then
1)
is a
-
-closed soft set if and only if
.
2)
is a
-
-closed soft set if and only if
.
Proof. 1) Necessity. Let
is a
-
-closed set. Then
is a
-
-open
soft set, that is
. There-
fore,
.
Sufficiency. Let
. Then
.
Hence
is a
-
-open soft set. Therefore,
is a
-
-closed soft.
2) By a similar way.
4. Extremally Soft Disconnectedness on Two Soft Topologies
In this section we introduced extremally soft disconnectedness (briefly, ESDC) via two soft topological spaces over
and investigated some characterizations of ESDC.
Definition 4.1. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
.
is said to be
-ESDC if
implies that
.
Definition 4.2. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
.
is said to be
-regular open soft set if
.
Theorem 4.1. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then the following statements are equivalent:
1)
is
-ESDC.
2) If
,
.
3) If
,
.
4) Every
-semi open soft set is
-pre open soft.
5) If
is a
-
-open soft set,
.
6) Every
-
-open soft set is
-pre open soft.
7) Every
-semi open soft set is
-
-open soft.
8) If
is a
-regular open soft set,
.
Proof. 1) Þ 2) Let
. Then
, that is
. From
1),
. Hence
. Therefore,
.
2) Þ 3) Let
, Then
is t1-soft closed and from 2) ![]()
is
-soft closed. Therefore,
is
-soft open and since
, we have
.
3) Þ 4) Let
is a
-semi open soft set. Then ![]()
from 3). Hence,
is a
-pre open soft.
4) Þ 5) Let
is a
-
-open soft set, that is
. Then
so that
is a
-semi open soft, and from 4) it is
-
pre open soft.
. Hence,
.
5) Þ 6) Let
is a
-
-open soft set. From 5),
and
.
Hence,
. Therefore,
is a
-pre open soft.
6) Þ 7) Let
is a
-semi open soft set, then
is a
-
-open soft. From 6),
is a
-pre open soft. Hence,
is a
-
-open soft.
7) Þ 1) Let
. Then,
so that
![]()
and
is
-semi open soft. From 7),
is
-
-open soft. Hence
![]()
and
so that
.
1) Þ 8) Let
is
-regular open soft. Then
and
from 1).
Therefore,
.
8) Þ 1) Let
. Since
, the soft set ![]()
is
-regular open soft. From 8),
. Since
,
.
Hence, we have
so that
.
Theorem 4.2. Let
be an initial universe and
be a set of parameters. Let
and
be two soft topologies on
. Then the following statements are equivalent:
1)
is
-ESDC.
2) If
is
-semi open soft set,
.
3) If
is
-pre open soft set,
.
4) If
is
-regular open soft set,
.
Proof. 1) Þ 2) Every
-semi open soft set is
-
-open soft so that
from Theorem 4.1.
2) Þ 4). Every
-regular open soft set is
-semi open soft set since
.
From 2),
.
1) Þ 3) Since every
-pre open soft set is
-
-open soft, it is obvious from Theorem 4.1.
3) Þ 4) Since every
-regular open soft set is
-pre open soft,
.
4) Þ 1) If
, since
and we have ![]()
is
-regular open soft. Also we obtain from 4)
and ![]()
by
so that
,
.
Therefore
.
Lemma 4.1.
implies
and
for
.
Proof. Obvious.
Theorem 4.3. Let
be an initial universe and
be a set of parameters. Let
,
be two soft topologies on
and
. Then the following statements are equivalent:
1)
is
-ESDC.
2) If
and
,
.
3) If
,
and
,
.
4) If
,
and
,
.
Proof. 1) Þ 2) Let
and
. From (1),
. Then
.
2) Þ 3) Let
,
and
. From (2),
.
3) Þ 4) Let
,
and
. From (3),
,
and
,
. Hence
.
4) Þ 3) Let
,
and
. From (4),
.
Then we have
since
. Therefore,
.
Hence
.
3) Þ 1) Let
. Then
and
. From (3),
.
Thus
. Therefore,
.
5. t2-ESDC Soft Topologies t1
The family of all semi-open (resp. pre-open,
-open,
-open) soft sets is denoted by
(resp.
,
,
). Also the family of all
-semi open (resp.
-pre open,
-
-open,
-
-open) soft sets is denoted by
(resp.
,
,
.
Theorem 5.1. If
and
,
and
implies
.
Proof. Let
. As
,
from Lemma 4.1.
is
-soft closed for
since
.
Hence
implies
and
.
Therefore, we obtain
.
Theorem 5.2. If
and
,
and
implies
.
Proof. Let
. As
,
from Lemma 4.1.
is
-soft closed for
since
.
Hence,
implies
.
Therefore, we obtain
.
6. Conclusion
We give the definition of
-semi open soft (resp.
-pre open soft,
-
-open soft,
-
-open soft,
-semi open soft,
-pre open soft,
-
-open soft,
-
-open soft)
set via two soft topologies. Also we introduce
-regular open soft and ESDC on two soft topologies. Some properties of some mixed soft operations and characterizations of ESDC are investigated. These properties which are studied are very important for studying anymore.
References