Abstract
Our purpose is to present the notions of a --set and a --sets in topological space. We discuss the basic properties of --sets and --sets. Also, the achievement of the topology defined by these families of sets is obtained. Finally, these results are applied to the case of which is the digital -space (cf. Section 4).
1. Introduction and Preliminaries
In the first section, we survey some of the standard facts on the concept of near open sets in the topological spaces. A generalized open set which is called a -open set is introduced by Abd El-Monsef et al. [1]. Maki [2] developed the definition of Levin [3] and Dunham [4] to generalize closed sets, by introducing the concept of -set in topological spaces as the set that coincides with their kernel. Also, he compares the topology generated by -set and the given topology. The development continued at the hands of Calads and Dontchev [5]. They presented concepts of -sets, -sets, and -sets. The new topology deduced by the concept of pre--sets, and pre--sets were introduced by Ganster et al. [6]. Also, Caldas et al. [7, 8] set up notation of -set and -continuous function. -irresolute functions are defined as an identification of irresolute functions and -closed functions by using -sets and -sets. Notions enable us to obtain conditions under which functions and inverse functions preserve -sets and -sets. Abd El-Monsef et al. [9] introduced the notion of --sets and --sets and obtained new topologies defined by these families of sets. Also, -sets, -sets, and some of its properties were introduced and studied. The digital line is a typical example of a space proved by Khalimsky et al. [10]. The -sets and -sets concepts can be applied in a rough set theory of decreasing the boundary region of any subset of an approximation space. The notation of a , , -sets, and -sets shall be introduced. Properties of these sets and some related new separation axioms are investigated. Also, we introduce a new class of topological spaces called -spaces and as an application with the clarification that the image of -spaces under a homeomorphism is a -space and that a -space is equivalent to pre- space.
Via this paper, (or purely ) represent the topological spaces. For a subset of , , , and denote the closure of , the interior of , and the complement of , respectively. Let us recall the following definitions, which are useful in the sequel.
Definition 1. A subset of the space , which carries topology is called(a)Semiopen [11] if (b)Preopen [12] if (c)-Open [13] if (d)-Open [1] (or semi-preopen [14]) if (e)-Open [15] if The class of all semiopen sets (resp., preopen, -open, -open, and -open) is denoted by (resp., , and ).
The complement of these sets called semiclosed [16] (resp., preclosed [12], -closed [13], -closed [1], and -closed [15]), and the classes of all these sets will be denoted by (resp., , , and ).
Definition 2. A subset of a topological space is called(a)-set (resp., -set) [2] if it is an intersection (resp., union) of open supersets of (resp., closed sets contained in )(b)-set (resp., -set) [5] if it is an intersection (resp., union) of -open supersets of (resp.,-closed sets contained in )(c)-set (resp.-set) [5] if it is an intersection (resp., union) of semiopen supersets of (resp., semiclosed sets contained in )(d)Pre--set (resp., pre--set) [6] if it is an intersection (resp., union) of preopen supersets of (resp., preclosed sets contained in )(e)--set (resp.--set) [9] if it is an intersection (resp., union) of -open supersets of (resp., -closed sets contained in )
Remark 1. For some kernels in an ordinary topological space , the following different notations may be used (e.g., [4, 5, 7, 9, 10, 12, 17–20]).
For example, , where and is called the kernel of .
, where and is named the semikernel of .
, where and is named the prekernel of .
-, where and is named the -kernel of .
-, where and is named the semi-prekernel of .
-, where and is named the -kernel of .
-, where and is named the -kernel of .
Definition 3. A subset of a topological space is named(a)Generalized -set [2] if , whenever , and (b)Generalized semi--set [5] if , whenever , and (c)Generalized pre--set [6] if , whenever , and (d)Generalized --set [9] if , whenever , and
Definition 4. A space which carries topology is named a --space [21] if to each pair of distinct points of , there corresponds a -open set containing but not and a -open set containing but not . Clearly, a space is - if and only if each singleton is -closed.
Definition 5. A space which carries topology is said to be(i)-Connected [22, 23] if cannot be expressed as the union of two nonempty disjoint -open sets of (ii)- [24] if for any -open set and any point
2. -Sets and -Sets
In the second part, we introduce the concept of a --set and a --set in topological space which is denoted by respectively, and we proceed with the study of its characterization.
Definition 6. Let be a subset of a space which carries topology . We define the subsets and as follows:(a)(b)As an illustration, consider the following example.
Example 1. Let and ; then, , and , , and .
In the first result, we summarize the fundamental properties of the set --sets.
Proposition 1. Let , and be subsets of a space which carries topology . Then, the following properties are valid:(a)(b)If , then (c)(d)If , then (i.e., is an -set)(e)(f)(g)
Proof. (a)It is obvious.(b)Suppose that . Then, there exists a subset such that with since ; then , and thus, .(c)It follows from (a) and (b) that . If , then there exists such that , and . Hence, , and so, we have . Then, .(d)By Definition 6, and since , we have . By (a), we have that .(e)Suppose that there exists a point such that . Then, there exists a -open set such that and . Thus, for each , we have . This implies that . Conversely, suppose that there exists a point such that . Then, by Definition 6, there exists a subset (for all such that . Let . Then, we have that , and . This implies that . Then, the proof (e) is completed.(f).(g)Suppose that there exist a point ; then, there exists such that ; hence, there exist and such that and . Thus, . By using Proposition 1(f), one can easily verify our result.
Proposition 2. For subsets , and of a topological space , the following properties hold:(a)(b)If , then (c)(d)If is -closed in , then (e)(f)
Proof. (a), (b), and (c) are produced directly from Definition 6 and Proposition 1. To prove (d), let be -closed in ; then, . By (d) and (f) of Proposition 1, . Hence, .
(f) can be demonstrated by using (d) and Proposition 1 (f). We have .
Remark 2. is generally not true, as the following example shows.
Example 2. Let and . Let and . Then, , but .
Definition 7. In a space which carries topology , a subset is called a -set (resp., -set) of if (resp., -set).
Example 3. Let be a topological space as in Example 1. Then, is -set and is -set.
Remark 3. By Proposition 1 (d) and Proposition 2 (d), we have that(a)If is a -set or if , then is a -set(b)If is a -set or if is -closed set, then is a -setThe converse of this remark is not true in general as shown in Example 4.
Example 4. Let be a usual topology, and a singleton is not -open but -set.
Proposition 3. For a space which carries topology , the following statements hold:(a) and are -sets and -sets, where and are subsets of (b)Union of any number of -sets (resp., -sets) is -set (resp., -set)(c)Intersection of any number of -sets (resp., -sets) is -set (resp., -set)(d)A subset is a -set if and only if is a -set
Proof. We will only take a case of -sets.
(a) Obvious. To prove (b), let be a family of -sets in . If , then by Proposition 1 is -sets. To prove (c), let be a family of -set in . Then, by Proposition 1 (g) and Definition 6, we have ; hence, by Proposition 1 (a), = .
Remark 4. Let and be the family of all -sets and -sets from . Then, and are a topology on containing all -open and -closed sets, respectively. Clearly, and are Alexandroff Space [25], that is, arbitrary intersections of open sets are open.
Proposition 4. For a topological space , the following statements hold:(a)(b) and (c) and
Proof. (a)Let be a subset of and is -set or . Then, . Since every open set is -open, then , so .(b)Since [25], by similar way of (a), then . Also, since , then .(c)Since , then , and since , so .
Proposition 5. If and are two topological spaces such that , then .
Proof. Since for a two spaces and , if , then , and .
Remark 5. We note that in the above proposition if , it not necessarily leads to , as the following example.
Example 5. Let , , and ; then, and , but .
Proposition 6. If (resp., ) and (resp., , then ).
Proof. If is a preopen set and is a -open set; then, . So . If , then , and if , then . So . Then, .
Proposition 7. For a space which carries topology , the following properties are equivalents:(a) is -(b)Every subset of is set(c)Every subset of is -set
Proof. : let . Since , is a union of -closed sets; hence, is -set : clearly given by Proposition 1 : Since by (c), we have that every singleton is the union of -closed sets, that is, it is -closed; then, is a --spaceRecall that a subset of a topological space is said to be generalized closed (briefly g-closed) [3] if whenever and .
A topological space is called if every g-closed subset of is closed. Dunham and Maki [26, 27] pointed out that is if and only if for each , the singleton is open or closed.
Theorem 1. For a space which carries topology , the following properties hold:(a) and are (b)If is -, then both and are discrete spaces
Proof. (a)Let . Then, is either preclosed or open and hence is either -closed or -open. If is -open, . If is -closed in , then is -open, and hence, . Therefore, is closed in . Hence, and are spaces.(b)This follows from Proposition 7.Relationship between some types of -sets is summarized in Figure 1.
Proposition 8. For a topological space , the following statements are equivalent:(a) is a -(b)The singleton is a -set, for each (c)The singleton is a -closed, for each
Proof. : let be an arbitrary point. For any point in which , there exists such that and . Therefore, we have . This shows that . Since , we have . : let . For any , and hence, there exists such that and . Therefore, we obtain , and hence, which is -open. Thus, is -closed. : For any distinct points and of , the singletons and are -closed which implies to which is -.
Proposition 9. For topological spaces and , the following properties hold:(a)A space is - if and only if is the discrete space(b)The identity function is strongly -irresolute(c)If is connected, then is -connected
Proof. (a)Suppose that is - and be any point of . By Proposition 8, is a -set and . For any subset of , by Proposition 3, . This shows that is discrete. Conversely, for each , and hence, is a -set. Proposition 8 is -(b)Let be any -open set of . By Proposition 3, , and hence, is strongly -irresolute.(c)Suppose that is not -connected. There exist nonempty -open sets of such that and . Therefore, and are the disjoint nonempty open sets in and . This shows that is not connected.
Theorem 2. For a topological space , the following properties are equivalent:(a) is -(b) is -(c), for each
Proof. (a) (b): for any -open set , and any point , by Proposition 8, . Hence, is -. (b) (a): for each , is preopen or preclosed [21]. Let such that . In case is preopen, we have and . In case preclosed, since -, -, and hence, -. Since -, then there exists such that and ; consequently, there exists such that and . Hence, is -. (b) (c): let and containing . Then, by (a), and . Suppose that . Then, for every -open set containing , and hence, . Now, we show that implies . Assume that , then there exists such that and . Since is - and . Therefore, we have . Consequently, we obtain . (c) (b): let and . Then, we have ; by (b), , and hence, . This shows that is -.
3. Generalized -Sets and Generalized -Sets
Section 3 is devoted to the study of generalizing -set (-sets) and -set (-sets) development of Maki’s work [2].
Definition 8. In a space which carries topology , a subset is called a generalized -set of if whenever, and is -closed. A subset is -sets of if - is -set of .
Now, (resp., ) will be denoted by the set of all -sets (resp., ) in .
Proposition 10. Let be a topological space and be any index set. Then,(a)Every -set is a -set(b)Every -set is a -set(c)If for all , then (d)If for all , then
Proof. (a) and (b) are proved by Definition 6, Proposition 1, and Definition 8. (c) and (d) are proved by (e) of Proposition 1 and Definition 8.
Remark 6. The intersection of two -sets is generally not a -set. Also, the union of two -sets in generally not a -set as shown in the following example.
Example 6. Let with ; then, . If and , then and are -sets but is not -set. If and , then and are -sets but is not -set.
Remark 7. The converse of (a) and (b) of Proposition 10 is not true in general as shown by the following example.
Example 7. Let be the space in Example 6; the subset is -set but it is not a -set. Also, the subset is -set but is not -set.
Proposition 11. Let be a topological space. Then,(a)Each is a open set or is a -set of (b)Each is a open set or is a -set of
Proof. (a)Let be not a -open set; then, only -closed set containing is . Thus, and is a -set of .(b)It follows from (a) and Definition 8.The known relationships between the types of -generalized closed sets are summarized in Figure 2.
Definition 9. A function is said to be(a)Strongly -irresolute [19] if for each and each -open set of containing , there is an open set containing such that , and equivalently, if the inverse image of each -open set is open.(b)-Irresolute [10] if for each , and each -open set of containing , there is a -open set containing such that , and equivalently, if the inverse image of each -open set is -open.
Theorem 3. (a)If a function is -irresolute, then is continuous(b)The identity function is strongly -irresolute
Proof. (a)Let be any -set of , i.e., . Then, and is -open in . Since is -irresolute, is -open in for each . Hence, we have , and is -open in and is -open in . On the other hand, by the definition, . Hence, we obtain . Therefore, , and is continuous.(b)Let be any -open set of . Since is -open, by Proposition 11, and hence, is strongly -irresolute.
4. Application
The digital line or so-called Khalimsky line is the set of all integers , equipped with the topology generated by (cf.[10], [28, p. 175]), [29, p. 905, 908]. The digital line or an interval as its subspace is a topological model of a 1-dimensional computer screen and points are the pixels. In , each singleton is closed and each singleton is open (=regular open), where .The digital -space (e.g., [19], Definition 4) is the topological product of -copies of digital line and this space is denoted by . For is called the digital plane. In , singletons whose coordinates are odd and are open; those with coordinates even are closed. There are points which are neither open nor closed.
Theorem 4. For the digital -space where , we have the following properties:(1)For any point of , -(2)For any subset of , -
Proof. (i) We first note that for the case where , (disjoint union) holds, where , and for the case where , (disjoint union) and hold, where . Let . It is enough to consider the following three cases for the point .
Case 1. : since is open in , it is -open. Then, it is obvious that in . We note this result is true for the case where .
Case 2. : we put where . Note that, for the point is the smallest open set containing . Then, there exist -open sets containing the point such that and the family of . Thus, we have -; moreover, , because . We conclude that - holds for this case. We note the result above it is true for the case where .
Case 3. , where : for the point , we set ; then by definition, is an even integer ). We recall the following subsets and as follows: ; ; and (disjoint union), .
For the point is the smallest open set containing , where and . Then, there exist -open sets containing the point such that and the family of . Thus, it is shown that -, because . Then, we show that - holds for this case. Therefore, for all cases above, we have proved that - holds in .
(2) Since , by Proposition 1, it is shown that -.
5. Conclusion
Recently, the use of set and functions for topological spaces has significantly evolved in data analysis, information systems, digital topology, and others. By researching generalizations of closed sets, -sets, and -sets, some new separation axioms have been found, and they turn out to be useful in the study of digital topology. The notion of a kernel of a set has applications in computer science [10, 30]. This notion is used in most of this paper. Moreover, it also has applications in some crucial fields of science and technology.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors want to thank Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia, for their support of this work.