Research Article | Open Access
A. A. Azzam, A. A. Nasef, "Some Topological Notations via Maki’s Λ-Sets", Complexity, vol. 2020, Article ID 4237462, 8 pages, 2020. https://doi.org/10.1155/2020/4237462
Some Topological Notations via Maki’s Λ-Sets
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. . Maki  developed the definition of Levin  and Dunham  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 . 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. . 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.  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. . 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  if (b)Preopen  if (c)-Open  if (d)-Open  (or semi-preopen ) if (e)-Open  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  (resp., preclosed , -closed , -closed , and -closed ), 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)  if it is an intersection (resp., union) of open supersets of (resp., closed sets contained in )(b)-set (resp., -set)  if it is an intersection (resp., union) of -open supersets of (resp.,-closed sets contained in )(c)-set (resp.-set)  if it is an intersection (resp., union) of semiopen supersets of (resp., semiclosed sets contained in )(d)Pre--set (resp., pre--set)  if it is an intersection (resp., union) of preopen supersets of (resp., preclosed sets contained in )(e)--set (resp.--set)  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  if , whenever , and (b)Generalized semi--set  if , whenever , and (c)Generalized pre--set  if , whenever , and (d)Generalized --set  if , whenever , and
Definition 4. A space which carries topology is named a --space  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)-  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 , 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 , 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)  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 . 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 .
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
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  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  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.
The digital line or so-called Khalimsky line is the set of all integers , equipped with the topology generated by (cf., [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., , 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 .