Abstract
In this paper, some variants of strongly normal closure spaces obtained by using binary relation are introduced, and examples in support of existence of the variants are provided by using graphs. The relationships that exist between variants of strongly normal closure spaces and covering axioms in absence/presence of lower separation axioms are investigated. Further, closure subspaces and preservation of the properties studied under mapping are also discussed.
1. Introduction and Preliminaries
In 1960, the theory of digital topology arose for the study of geometric and topological properties which in turn can be used in computer graphics, image processing, etc. Digital image processing is a rapidly growing discipline and has many applications in allied branches of mathematics. Rosenfeld [1] in 1979 studied connectedness, thinning, and algorithm for border. To compare digital topology and general topology, the generalized topological structure was introduced in [2] by Smyth. The topological structure that arises from the directed graph is used in digital topology, while Šlapal [3] in 2003 studied closure operations for digital topology. He studied the closure space that arises from -ary relation and also studied connectedness in digital spaces via closure operators on graphs [4]. Closure spaces defined through binary relations were introduced in 2006 [5] by Allam et al., and in 2008 [6], the same authors generated topologies by using relations. They proved that topology generated from aftersets and foresets is dual if the relation is preorder. They also introduced lower separation axioms in terms of relation and studied these closure spaces in digital topology. Šlapal and Pfaltz [7] utilized binary relations in networks and studied closure operators associated with networks. B. M. R. Stadler and P. F. Stadler [8] studied some higher separation axioms in closure spaces, and recently, Gupta and Das [9] investigated variants of normality in closure setting by using cannonically closed sets. In 2018, Gupta and Das [10] introduced higher separation axioms such as strongly normal and strongly regular closure spaces via binary relation.
Let be any set; then, a relation on is a subset of , i.e., . The formula is abbreviated as , which means that is in relation with . In 2006, afterset of was denoted and defined as and foresets were denoted and defined as [6]. Liu [11], in 2010, represented aftersets and foresets as left relation and right relation. A set is the intersection of all aftersets containing , i.e., , if there exists such that . In this paper, we introduced variants of strongly normal closure spaces i.e., normal and almost-normal closure spaces. Some properties of newly defined notions are studied, and the relation between new notions and lower separation axioms is investigated with the help of covering axioms. As most of the practical situation involves binary relation, which can easily be expressed through graphs, an attempt in this paper is made to provide examples via graphs to support notions defined in this paper.
Definition 1 (see [5]). Let be any set and be any binary relation on . The relation gives rise to a closure operation on as follows:which satisfies the following conditions:(1)(2)(3)In addition to the above three properties, it also satisfies the idempotent condition. The set along with the operator is a closure space. In the closure space , a set is closed [5] if .
Lemma 1 (see [5]). Let be any binary relation on a nonempty set ; then, is a minimal neighborhood of for all i.e., .
Remark 1 (see [5]). The minimal neighborhood of a point in a closure space is defined as follows:
Lemma 2 (see [5]). For any binary relation on if , then .
Theorem 1. In a closure space , is the smallest closed set containing .
Definition 2. In a closure space , a set is said to be regularly closed if for a closed set , , and a set is said to be regularly open if its complement is regularly closed, i.e., .
Definition 3 (see [5]). A closure space generated from a binary relation is said to be if and only if for every two distinct points , both and hold.
Definition 4 (see [5]). Let be any binary reflexive relation; then, a closure space generated from is called as a -space if and only if for every two distinct points .
Definition 5 (see [10]). Let be a binary relation on ; then, the closure space is said to be strongly normal if for two disjoint closed sets and , there exist distinct such that , , and .
Definition 6 (see [10]). Let be a binary relation on ; then, the closure space is said to be strongly regular if for any closed set and a point , there exist disjoint and such that and .
2. Variants of Strongly Normal Closure Space
Definition 7. Let be a binary relation on ; then, the closure space is said to be normal if for two disjoint closed sets and , there exist some and in and , respectively, such that , , and .
The following example establishes that there exists a closure space generated from a relation which is normal but not strongly normal.
Example 1. Let us consider the set consisting of straight lines in a plane and define a binary relation on as if and only if line is parallel to line . Clearly, the closure space is normal but not strongly normal because for two disjoint closed sets, there do not exist disjoint and containing them.
Example 2. A closure space generated from a graph which is not normal.
From the directed graph in Figure 1, we have , and . Then, , , , and . The closure space is not normal because for two disjoint closed sets and , there does not exist some and such that , , and .
Theorem 2. For any binary relation , the closure space is normal if and only if for every closed set contained in , there exists some such that .
Proof. Let be a normal closure space and be a closed set contained in . Thus, is a closed set which is disjoint from the closed set . Since is normal, there exist and with such that and imply . Therefore, . Since is the smallest closed set by Theorem 1, . Conversely, let and be two closed sets and be contained in . By a given condition, there exists some such that . Therefore, , , and . Hence, is normal.
Theorem 3. In a normal closure space , for every pair of disjoint closed sets and , there exist and containing and such that .
Proof. Let be a normal closure space and and be two disjoint closed sets in . Since , we have , where . By the normal closure space of , there exists some such that . Thus, . Since , so again by using characterization of a normal closure space, there exists some such that which implies .
In general, normality does not imply which is evident from Example 1. For two distinct points ‘’ and ‘’ which are in the form of straight lines, there does not exist disjoint and containing ‘’ and ‘,’ respectively. But in closure space, the result is true, as shown in Theorem 4 below.
Theorem 4. In a reflexive closure space , every normal space is .
Proof. Let be a normal closure space and and be two distinct points. We have to show that is . Since and are closed, by normality of , there exist some and such that , , and . Hence, is .
Definition 8. A closure space generated from a binary relation is said to be almost normal if for a closed set and a regularly closed set disjoint from , there exist some and , respectively, such that , , and .
From the definitions, it is obvious that every normal closure space is almost normal but the converse need not be true, as shown in the following example.
Example 3. Let us consider the directed graph in Figure 2. It is clear that . Then, . The closure space generated from the graph in Figure 2 is almost normal but not normal because for two disjoint closed sets and , there does not exist some and such that , , and .
Example 4. A closure space generated from a graph which is not almost normal. The closure space generated from the binary relation in Figure 3 is not almost normal because for regularly closed set and a closed set , there does not exist some and such that , , and .
The implications in Figure 4 are obvious from the definitions. But none of these implications is reversible (see [10] and Examples 1, 3, and 4 above).
Theorem 5. A closure space generated from a binary relation is almost normal if and only if for every regularly closed set contained in , there exists some such that .
Proof. Let be an almost normal closure space, be a regularly closed set contained in , and be a closed set disjoint from . Since is almost normal, there exist some and with such that and which imply that . Therefore, . Since is the smallest closed set by Theorem 1, . Conversely, let be a regularly closed set and be a closed set. Thus, is an open set containing . By the given condition, there exists some such that . Thus, and . Hence, is almost normal.
Theorem 6. If a binary relation, then the generated closure space is almost normal if and only if for every closed set contained in a regularly open set , there exists some such that .
Proof. Let be an almost normal closure space, be a closed set contained in , and be a regularly closed set disjoint from . Since is an almost normal closure space, there exist some and with such that and which imply that . Therefore, . Since is the smallest closed set by Theorem 1, . Conversely, let be a closed set and be a regularly closed sets’ disjoint from a closed set. Thus, is a regularly open set containing . By the given condition, there exists some such that . Thus, and . Hence, is almost normal.
Definition 9. Let be any binary relation on and be the generated closure space. Then, a family of subset is said to be an R-cover of if . If elements of are of the form , then it is said to be an R-open cover of .
Definition 10. A closure space is said to be nearly compact if for every R-open cover of , there is a finite subset of such that is an R-cover of .
Theorem 7. Every regularly closed closure subspace of a nearly compact space is nearly compact.
Proof. Let be a regularly closed closure subspace of a nearly compact space . Let be an R-open cover of . We have to show that there is a finite subcollection whose closure is an R-cover of . Since is an R-open cover of , is an R-open cover of . Also, is nearly compact, so there is a finite subcollection such that is an R-cover of . If this subcollection has , then discard it. The remaining subcollection also covers . Hence, is a nearly compact closure space.
Theorem 8. The closure space generated from a reflexive relation is almost normal if it is nearly compact and .
Proof. Let be a regularly closed set which is disjoint from a closed set . We have to show that is almost normal. Since for a point and a point , . The family covers . Since is regularly closed and is nearly compact, is nearly compact by Theorem 7. Thus, there is a finite set of such that is an R-cover of . Thus, there exist some and such that , , and . Since for every point there exists , the collection covers . Thus, and as is nearly compact and . Hence, is an almost normal closure space.
Remark 2. The following example establishes that condition in Theorem 8 cannot be dropped.
Example 5. Let us consider the directed graph in Figure 5. Here, the closure space generated from the graph is nearly compact but not almost normal because for the regularly closed set and a closed set , there does not exist some and such that , , and .
3. Subspace
Definition 11 (see [5]). Let and ; then, is called a closure subspace of a closure space if for all .
Remark 3 (see [12]). Let be a subspace of a closure space . Then,(a) is closed (open) in implies that is closed (open) in (b) is closed in and is a closed set in imply that is closed in In the following example, it is shown that the normality of the closure space generated from a graph does not imply that the closure space generated from its subgraph is normal.
Example 6. Let us consider the graph in Figure 6. The closure space generated from the graph in Figure 6 is normal as it satisfies the condition of a normal closure space. But the closure space generated from its subgraph, as shown in Figure 7, is not normal.
The closure space of the subgraph in Figure 7 is not normal because for two disjoint closed sets and , there does not exist some and in such that , , and .
Theorem 9. Let be a normal closure space. Then, every closed subspace of is normal.
Proof. Let and be two disjoint closed sets in a closed subspace of . By Remark 3, and closed in implies that and are closed in . By Remark 3, and are closed in . Since is normal, there exist some and such that , , and . Thus, by Definition 11, and contain , , and . Hence, is normal.
Remark 4. A closed subspace of a normal closure space is normal. But a closed subgraph of an almost normal space need not be almost normal, as shown in Example 7 below.
Example 7. Closure space generated from a closed subgraph of an almost normal closure space need not be almost normal.
The closure space generated from the graph in Figure 8 is almost normal because for every closed set and a regularly closed set disjoint from a closed set, there exist some and satisfying the condition of the almost normal closure space.
Let us consider the closed subgraph in Figure 9 of the graph in Figure 8. The closed closure subspace generated from the subgraph in Figure 9 is not almost normal because for a closed set and a regularly closed set , there does not exist some and in satisfying the required condition.
Remark 5. In a closure space , a regularly closed closure subspace of an almost normal closure space need not be almost normal. We can say that a clopen subspace of an almost normal space is almost normal.
4. Preservation under Mapping
Definition 12 (see [5]). Let and be two closure spaces. A function is continuous at if and only if . A function from a closure space into a closure space is said to be continuous on if and only if it is continuous at each point of .
Theorem 10 (see [5]). Let be a function from a closure space into a closure space ; then, the following conditions are equivalent:(1) is continuous.(2)For every subset of , (3)The inverse image of every closed subset of is a closed subset of (4)The inverse image of every open subset of is an open subset of
Definition 13 (see [5]). A function is called open (closed) if the image of an open (closed) subset of is an open (closed) subset of .
Theorem 11. Let be a continuous closed and surjection and be normal; then, is also a normal closure space.
Proof. Let and be two disjoint closed sets in . We have to show that is normal. Since is continuous, and are disjoint closed sets in . As is normal, there exist some and with such that and . Since is closed, and are closed in . Since , implies that . Thus, . Since , there exists containing such that. Similarly, there exists containing such that . Also, implies that . Hence, is a normal closure space.
Example 8. In a closure space , the continuous image of an almost normal space need not be almost normal.
Let be a set of natural numbers and be a set. Let be a relation defined on as if and only if for all and , where is even, and be a relation defined on . Then,and is defined as in Example 4. Here, and are closure spaces. A function ,is continuous. Here, closure space is almost normal but is not almost normal because for closed set and a regularly closed set , there does not exist some and in such that , , and .
Example 9. continuous closed image of an almost normal space need not be almost normal.
Let and be two sets. Let be a relation defined on as if and only ifand relation , defined on , are two binary relations. Then, , , for , , . Here, and are closure spaces. A function is defined asis continuous. Here, is almost normal but is not almost normal because for closed set and a regularly closed set , there does not exist some and in satisfying the condition of the almost normal closure space.
Since the continuous closed image of an almost normal space need not be almost normal, so we replace a continuous function with a completely continuous.
Definition 14. Let be a binary relation and and be two closure spaces; then, a function is said to be completely continuous if inverse image of in is regularly open in .
From the above definition, we can say that if is a completely continuous onto function and is regularly closed in , then is regularly closed in .
Observation 1. In a closure space , every completely continuous function is continuous.
Example 10. The completely continuous image of an almost normal space in a closure space need not be almost normal.
Let be a set of natural numbers and be two sets. Let be a relation on , as defined in Example 8, and be the relation on given by . Then, is defined as in Example 8, and , , , and . Here, and are closure spaces. A function defined byis completely continuous. Here, is almost normal but is not almost normal because for a closed set and a regularly closed set , there does not exist some and in such that , , and .
Theorem 12. Completely continuous, closed image of an almost normal closure space, is almost normal.
Proof. Let be a closed set and be a regularly closed set. Then, is closed, and is regularly closed in . Since is almost normal, there exist some and with such that and . Thus, and are closed in . As implies , . Now, . Thus, there exists some in such that . Similarly, there exists some in such that . Also, . Hence, is an almost normal closure space.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author is thankful to Department of Science and Technology (DST), Government of India, for awarding INSPIRE fellowship (IF140967).