Abstract
In this paper, a new class of generalized soft open sets in soft topological spaces, called soft b-open sets, is introduced and studied. Then discussed the relationships among soft -open sets, soft semi-open sets, soft pre-open sets and soft -open sets. We also investigated the concepts of soft b-open functions and soft b-continuous functions and discussed their relations with soft continuous and other weaker forms of soft continuous functions.
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Introduction and preliminaries
Molodtsov [1] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty. He successfully applied the soft set theory into several directions such as smoothness of functions, game theory, Riemann Integration, theory of measurement, and so on. Soft set theory and its applications have shown great development in recent years. This is because of the general nature of parametrization expressed by a soft set. Shabir and Naz [2] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. Later, Zorlutuna et al.[3], Aygunoglu and Aygun [4] and Hussain et al are continued to study the properties of soft topological space. They got many important results in soft topological spaces. Weak forms of soft open sets were first studied by Chen [5]. He investigated soft semi-open sets in soft topological spaces and studied some properties of it. Arockiarani and Arokialancy are defined soft -open sets and continued to study weak forms of soft open sets in soft topological space. Later, Akdag and Ozkan [6] defined soft -open (soft -closed) sets.
In the present paper, we introduce some new concepts in soft topological spaces such as soft b-open sets, soft b-closed sets, soft b-interior, soft b-closure, soft b-continuous functions, soft b-open functions and soft b-closed functions. We also study the relationships among soft continuity [7], soft -continuity [6], soft semi-continuity [8], soft pre-continuity [6], soft -continuous [9] and soft b-continuity of functions defined on soft topological spaces. With the help of counter examples we show the non-coincidence of these various types of mappings.
Throughout the paper, the space and stand for soft topological spaces with ( and ) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping stands for a mapping, where , and are assumed mappings unless otherwise stated.
Definition 1
[1] Let be an initial universe and be a set of parameters. Let denote the power set of and be a non-empty subset of . A pair is called a soft set over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe . For may be considered as the set of -approximate elements of the soft set .
Definition 2
[10] A soft set over is called a null soft set, denoted by , if , .
Definition 3
[10] A soft set over is called an absolute soft set, denoted by , if , .
If , then the -universal soft set is called a universal soft set, denoted by .
Definition 4
[2] Let be a non-empty subset of , then denotes the soft set over for which , for all .
Definition 5
[10] The union of two soft sets of and over the common universe is the soft set , where and for all
We write
Definition 6
[10] The intersection of two soft sets and over a common universe , denoted , is defined as , and for all .
Definition 7
[10] Let and be two soft sets over a common universe . , if , and for all .
Definition 8
[2] Let be the collection of soft sets over , then is said to be a soft topology on if satisfies the following axioms.
-
(1)
belong to ,
-
(2)
the union of any number of soft sets in belongs to ,
-
(3)
the intersection of any two soft sets in belongs to .
The triplet is called a soft topological space over . Let be a soft topological space over , then the members of are said to be soft open sets in . A soft set over is said to be a soft closed set in , if its relative complement belongs to .
Definition 9
[11] For a soft set over , the relative complement of is denoted by and is defined by , where is a mapping given by for all .
Soft b-open sets
In this section we introduce soft b-open sets in soft topological spaces and study some of their properties.
Definition 10
A soft set in a soft topological space is called
-
(i)
soft b-open (sb-open) set iff
-
(ii)
soft b-closed (sb-closed) set iff .
Theorem 1
For a soft set in a soft topological space
-
(i)
is a soft b-open set iff is a soft b-closed set.
-
(ii)
is a soft b-closed set iff is a soft b-open set.
Proof
Obvious from the Definition 10.
Definition 11
Let be a soft topological space and be a soft set over .
-
(i)
Soft b-closure of a soft set in is denoted by .
-
(ii)
Soft b-interior of a soft set in is denoted by .
Clearly is the smallest soft b-closed set over which contains and is the largest soft b-open set over which is contained in .
Theorem 2
Let be any soft set a in soft topological space . Then,
-
(i)
.
-
(ii)
.
Proof
(i) Let sb-open set and sb-closed set . Then, is a sb-closed set and .
Therefore, .
(ii) Let be sb-open set.
Then, for a sb-closed set , .
Therefore, .
Definition 12
Let be a soft topological space over and be a soft set over .
-
(i)
[3] The soft interior of is the soft set ;
-
(ii)
[2] The soft closure of is the soft set ;
-
(iii)
[5] The soft semi-interior of is a soft set .
-
(iv)
[5] The soft semi-closure of is a soft set is soft semi-closed of ;
The following concepts are used in the sequel.
Definition 13
A soft set in a soft topological space is called
-
(i)
soft regular open (soft regular closed) set [12] if ().
-
(ii)
soft -open (soft -closed) set [6] if
-
(iii)
soft pre-open (soft pre-closed) set [12] if ().
-
(iv)
soft semi-open (soft semi-closed) set [5] if ().
-
(v)
soft -open (soft -closed) set [12] if
Lemma 1
Let be a soft set in a soft topological space . Then
-
(i)
.
-
(ii)
.
Proof
(i)
Also
(ii) Similar by taking the complements.
Lemma 2
In a soft topological space we have the following
-
(i)
Every soft regular open set is soft open.
-
(ii)
Every soft open set is soft -open.
-
(iii)
Every soft -open set is both soft semi-open and soft pre-open.
-
(iv)
Every soft semi-open set and every soft pre-open set is soft -open.
Let be a soft topological space.Then, the family of all soft regular open (resp. soft -open, soft semi-open, soft pre-open, soft -open) sets in may be denoted by sr (resp. sa-open, ss-open, sp-open, -open) sets. The complement of the above represents the family of soft regular closed (resp. soft -closed, soft semi-closed, soft pre-closed, soft -closed) sets in and may be denoted by sr (resp. s-closed, ss-closed, sp-closed, s-closed) sets. Also, the family of all soft b-open (resp. soft b-closed) sets in may be denoted by (resp. ).
Theorem 3
In a soft topological space
-
(i)
Every sp-open set is sb-open set.
-
(ii)
Every ss-open set is sb-open set.
Proof
(i) Let be a sp-open set in a soft topological space .
Then, which implies
.
Thus is sb-open set.
(ii) Let be a ss-open set in a soft topological space . Then, which implies .
Thus is sb-open set.
The converses are not true as seen in the following example:
Example 1
Let , and
, where are soft sets over , defined as follows:
Then, defines a soft topology on , and thus is a soft topological space over . Clearly, the soft closed sets are .
Then, let us take then , and so ; hence, is sb-open set, but not sp-open set (since is not sp-open set).
Now, let us take ; then , and so ; thus, is sb-open set, but not ss-open set.
Remark 1
-
(i)
If is a soft set of soft topological space , then is the smallest sb-closed set containing
Thus, .
-
(ii)
If is a soft set of soft topological space then is the largest sb-open set contained in
Thus, .
Theorem 4
In a soft topological space , every sb-open (sb-closed) set is s-open (s-closed) set.
Proof
Let be a sb-open set in .
Then
.
As a result is s-open set.
The converse is not true as seen in the following example:
Example 2
Let , and let be soft topological space over . Let us consider the soft topology on given in Example 1; i.e., .
Then, let us take ; then , and so ; therefore, is s-open set but not s-open set.
Remark 2
From the above theorems, we have the following
Theorem 5
In a soft topological space
-
(i)
An arbitrary union of sb-open sets is a sb-open set.
-
(ii)
An arbitrary intersection of sb-closed sets is a sb-closed set.
Proof
(i) Let be a collection of sb-open sets. Then, for each ,
. Now
Hence is a sb-open set.
(ii) Similarly by taking complements.
Theorem 6
In a soft topological space , is sb-closed (s-open) set if and only if ().
Proof
Suppose that implies, , that implies is sb-closed set.
Conversely, suppose is a sb-closed set in .
We take and is a sb-closed. Therefore,
implies, is a sb-closed set and
For we apply soft interiors.
Theorem 7
In a soft topological space the following hold for sb-closure.
-
(i)
.
-
(ii)
.
-
(iii)
is a sb-closed set in .
-
(iv)
.
Proof
The proof is obvious.
Theorem 8
In a soft topological space the following relations hold;
-
(i)
,
-
(ii)
.
Proof
(i) or that implies or .
Thus, . (ii) Similar to that of (i).
Theorem 9
In a soft topological space the following relations hold;
-
(i)
,
-
(ii)
.
Proof
(i) or that implies or .
Thus, . (ii) Similar to that of (i).
Theorem 10
Let be a sb-open set in a soft topological space .
-
(i)
If is a sr-closed set then is a sp-open set.
-
(ii)
If is a sr-open set then is a ss-open set.
Proof
Since is sb-open set, ,
(i) Now let be sr-closed set. Therefore, .
Then . That implies
.
Hence is sp-open set.
(ii) Let be sr-open set. Therefore, .
Then .
That implies . Thus is ss-open set.
Theorem 11
Let be a sb-open set in a soft topological space .
-
(i)
If is a sr-closed set then is a ss-closed set.
-
(ii)
If is a sr-open set then is a sp-closed set.
Theorem 12
For any sb-open set in soft topological space , is sr-closed set.
Proof
Let be sb-open set in which implies . Therefore, .
Also . Thus .
So is sr-closed set.
Theorem 13
For any sb-closed set in a soft topological space is sr-open set.
Proof
Let be sb-closed set in which implies . Thus
Also
Thus from (3) and (4) . So is sr-open set.
Theorem 14
Let be a sb-open (sb-closed) set in a soft topological space , such that . Then, is a sp-open set.
Theorem 15
Let be a sb-open (sb-closed) set in a soft topological space , such that . Then, is a ss-open set.
Theorem 16
Let be a set of a soft topological space Then,
-
(i)
-
(ii)
Proof
(i) .
(ii) .
Theorem 17
Let be a soft topological space. If is an soft open set and is a sb-open set in . Then is a sb-open set in .
Proof
Hence is a sb-open set in .
Theorem 18
Let be soft topological space. If is an s-open set and is a sb-open set in . Then is a sb-open set in .
Proof
Hence is sb-open set in .
Theorem 19
is sb-open (sb-closed) set in if and only if is the union (intersection) of ss-open set and sb-open set in .
Proof
Follows from the Definitions 11.
Soft b-continuity
In this section, we introduce soft b-continuous maps, soft b-irresolute maps, soft b-closed maps and soft b-open maps and study some of their properties.
Definition 14
([13]) let and be soft classes. Let and be mappings. Then a mapping is defined as: for a soft set in , is a soft set in given by for is called a soft image of a soft set If then we shall write as
Definition 15
[13] Let be a mapping from a soft class to another soft class and a soft set in soft class where Let and be mappings. Then is a soft set in the soft classes defined as: for is called a soft inverse image of Hereafter, we shall write as
Theorem 20
[13] Let and be mappings. Then for soft sets and a family of soft sets in the soft class we have:
-
(1)
-
(2)
-
(3)
in general
-
(4)
in general
-
(5)
If then
-
(6)
-
(7)
-
(8)
in general
-
(9)
in general
-
(10)
If then .
Definition 16
A soft mapping is said to be soft b-continuous (briefly sb-continuous) if the inverse image of each soft open set of is a sb-open set in .
Theorem 21
Let be a mapping from a soft space to soft space . Then, the following statements are true;
-
(i)
is sb-continuous,
-
(ii)
the inverse image of each soft closed set in is soft b-closed in .
Proof
(i)(ii): Let be a soft closed set in . Then is soft open set. Thus, , i.e., . Hence is a sb-closed set in .
(ii)(i): Let is soft open set in . Then is soft closed set and by (ii) we have , i.e., . Hence is a sb-open set in . Therefore, is a sb-continuous function.
Definition 17
A soft mapping is called soft -continuous [9] (resp. soft -continuous [6], soft pre-continuous [6], soft semi-continuous [8]) if the inverse image of each soft open set in is s-open (resp. s-open, sp-open, ss-open) set in .
Remark 3
We have following implications, however, examples given below show that the converses of these implications are not true.
Example 3
is a soft -continuous function, but not soft continuous function given Example 4 in [6].
Example 4
is a soft pre-continuous function and soft, but not soft -continuous function given Example 5 in [6].
Example 5
is a soft semi-continuous function and soft, but not soft -continuous function given Example 6 in [6].
Example 6
Let , , , and and let and be soft topological spaces.
Define and as
, , ,
, ,
Let us consider the soft topology on given in Example 1; i.e.,
, ;
and mapping;
is a soft mapping. Then is a soft open set in is a s-open set but not s-open set in . Therefore, is a soft -continuous function but not sb-continuous function.
Example 7
Let , , , and and let and be soft topological spaces.
Define and as
, , ,
, ,
Let us consider the soft topology on given in Example 1; i.e.,
, ;
and mapping;
is a soft mapping. Then is a soft open set in ; is a s-open set but not ss-open set in . Hence, is a s-continuous function but not soft semi-continuous function.
Example 8
Let , , , and and let and be soft topological spaces.
Define and as
, , ,
, ,
Let us consider the soft topology on given in Example 1; i.e.,
, ;
and mapping;
is a soft mapping. Then is a soft open set in ; is a sb-open set but not sp-open set in . Thus, is a sb-continuous function, but not soft pre-continuous function.
Theorem 22
Every soft continuous function is sb-continuous function.
Proof
Let be a soft continuous function. Let be a soft open set in . Since is soft continuous, is soft open in . And so is sb-open set in . Therefore, is sb-continuous function.
Definition 18
A mapping is said to be soft b-irresolute (briefly sb-irresolute) if is sb-closed set in , for every sb-closed set in .
Theorem 23
A mapping is sb-irresolute mapping if and only if the inverse image of every sb-open set in is sb-open set in .
Theorem 24
Every sb-irresolute mapping is sb-continuous mapping.
Proof
Let is sb-irresolute mapping. Let be a soft closed set in , then is sb-closed set in . Since is sb-irresolute mapping, is a sb-closed set in . Hence, is sb-continuous mapping.
Theorem 25
Let , be two functions. Then
-
(i)
is sb-continuous, if is sb-continuous and is soft continuous.
-
(ii)
is sb-irresolute, if and is sb-irresolute functions.
-
(iii)
is sb-continuous if is sb-irresolute and is sb-continuous.
Proof
-
(i)
Let be soft closed set of . Since is soft continuous, by definition is soft closed set of . Now is sb-continuous and is soft closed set of , so by Definition 16, is sb-closed in . Hence is sb-continuous.
-
(ii)
Let is sb-irresolute and let be sb-closed set of . Since is sb-irresolute by Definition 18, is sb-closed set of . Also is sb-irresolute, so is sb-closed. Thus, is sb-irresolute.
-
(iii)
Let be sb-closed set of . Since is sb-continuous, is sb-closed set of . Also is sb-irresolute, so every sb-closed set of is sclosed in . Therefore, is sb-closed set of . Thus, is sb-continuous.
Theorem 26
If the bijective map is soft open and sb-irresolute, then is sb-irresolute.
Proof
Let be a sb-closed set in and let where is a soft open set in . Clearly, . Since is soft open map, by definition is soft open in and is sb-closed set in . Then , and hence . Also is sb-irresolute map and is a sb-closed set in , then is sb-closed set in . Thus, . So is sb-closed set in . Therefore, is sb-irresolute map.
Definition 19
A mapping is said to be soft b-open (briefly sb-open) map if the image of every soft open set in is sb-open set in
Definition 20
A mapping is said to be soft b-closed (briefly sb-closed) map if the image of every soft closed set in is sb-closed set in
Theorem 27
If is soft closed function and is sb-closed function, then is sb-closed function.
Proof
For a soft closed set in , is soft closed set in . Since is sb-closed function, is sb-closed set in . is sb-closed set in . Therefore, is sb-closed function.
Theorem 28
A map is sb-closed if and only if for each soft set of and for each soft open set such that , there is a sb-open set of such that and .
Proof
Suppose is sb-closed map. Let be a soft set of , and be a soft open set of , such that . Then is a sb-open set in such that and .
Conversely, suppose that is a soft closed set of . Then , and is soft open set. By hypothesis, there is a sb-open set of such that and . Thus . Hence , which implies . Since is sb-closed set, is sb-closed set. So is a sb-closed map.
Theorem 29
Let , be two maps such that is sb-closed map.
-
(i)
If is soft continuous and surjective, then is sb-closed map.
-
(ii)
If is sb-irresolute and injective, then is sb-closed map.
Proof
-
(i)
Let be a soft closed set of . Then, is soft closed set in as is soft continuous. Since is sb-closed map, is sb-closed set in . Hence sb-closed map.
-
(ii)
Let be a soft closed set in . Then, is sb-closed set in , and so is sb-closed set in . Since is sb-irresolute and injective. Hence is a sb-closed map.
Theorem 30
If is sb-closed set in and is bijective, soft continuous and sb-closed, then is sb-closed set in .
Proof
Let where is a soft open set in .
Since is soft continuous, is a soft open set containing .
Hence as is sb-closed set.
Since is sb-closed, is sb-closed set contained in the soft open set ,
which implies and hence . So is sb-closed set in .
Conclusion
In this paper, we introduce the concept of soft b-open sets and soft b-continuous functions in topological spaces and some of their properties are studied. We also introduce soft b-interior and soft b-closure and have established several interesting properties. In the end, we hope that this paper is just a beginning of a new structure, it will be necessary to carry out more theoretical research to promote a general framework for the practical application.
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Akdag, M., Ozkan, A. Soft b-open sets and soft b-continuous functions. Math Sci 8, 124 (2014). https://doi.org/10.1007/s40096-014-0124-7
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DOI: https://doi.org/10.1007/s40096-014-0124-7