# Some Variants of Strong Normality in Closure Spaces Generated via Relations

Authors: Ria Gupta; Ananga Kumar Das
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 α-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.LetX be any set; then, a relation on X is a subset of X×X, i.e., R⊆X×X. The formula x,y∈R is abbreviated as xRy, which means that x is in relation R with y. In 2006, afterset of x∈X was denoted and defined as xR=y:xRy and foresets were denoted and defined as Rx=y:yRx [6]. Liu [11], in 2010, represented aftersets and foresets as left relation and right relation. A set pR is the intersection of all aftersets containing p, i.e., pR=∩p∈xRxR, if there exists x such that p∈xR. 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]).
LetX be any set and R⊆X×X be any binary relation on X. The relation R gives rise to a closure operation clR on X as follows:(1)clRA=A∪x∈X:xR∩A≠∅,which satisfies the following conditions:(1)
cl∅=∅,(2)
A⊆clA,(3)
clA∪B=clA∪clB.
In addition to the above three properties, it also satisfies the idempotent condition. The set along with the operatorclR is a closure space. In the closure space X,clR, a set A is closed [5] if clRA=A.Lemma 1 (see [5]).
LetR be any binary relation on a nonempty set X; then, x∪xR is a minimal neighborhood of x for all x∈X i.e., NRx=x∪xR.Remark 1 (see [5]).
The minimal neighborhood of a pointx in a closure space X,clR is defined as follows:(2)NRx=xR,ifxR≠∅,x,ifxR=∅.Lemma 2 (see [5]).
For any binary relationR on X if x∈yR, then xR⊆yR.Theorem 1.
In a closure spaceX,clR, clR∪NRx is the smallest closed set containing NRx.Definition 2.
In a closure spaceX,clR, a set A is said to be regularly closed if for a closed set clRA=A, clRintRA=A, and a set is said to be regularly open if its complement is regularly closed, i.e., clRintRX−clRA=X−clRA.Definition 3 (see [5]).
A closure spaceX,clR generated from a binary relation R is said to be T1 if and only if for every two distinct points xandy∈X, both x∉yR and y∉xR hold.Definition 4 (see [5]).
LetR be any binary reflexive relation; then, a closure space X,clR generated from R is called as a T2-space if and only if for every two distinct points x,y∈Xx,R∩yR=∅.Definition 5 (see [10]).
LetR be a binary relation on X; then, the closure space X,clR is said to be strongly normal if for two disjoint closed sets A=clRA and B=clRB, there exist distinct xandy such that A⊆xR, B⊆yR, and xR∩yR=∅.Definition 6 (see [10]).
LetR be a binary relation on X; then, the closure space X,clR is said to be strongly regular if for any closed set clRA=A and a point x∉clRA, there exist disjoint uR and vR such that x∈uR and clRA⊆vR.


## 2. Variants of Strongly Normal Closure Space

Definition 7.
LetR be a binary relation on X; then, the closure space X,clR is said to be normal if for two disjoint closed sets clRA=A and clRB=B, there exist some x and y in clRA and clRB, respectively, such that clRA⊆∪x∈ANRx=U, clRB⊆∪y∈BNRy=V, and U∩V=∅.
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 setX consisting of straight lines in a plane and define a binary relation R on X as aRb if and only if line a is parallel to line b. Clearly, the closure space X,clR is normal but not strongly normal because for two disjoint closed sets, there do not exist disjoint xR and yR containing them.Example 2.
A closure space generated from a graph which is not normal.
From the directed graph in Figure1, we have aR=c,d,bR=a,b,c,cR=b,c, and dR=c. Then, aR=a,b,c, bR=b,c, cR=c, and dR=c,d. The closure space X,clR is not normal because for two disjoint closed sets clRA=a=A and clRB=d=B, there does not exist some x and y such that clRA⊆∪x∈ANRx, clRB⊆∪y∈BNRy, and ∪x∈ANRx∩∪y∈BNRy=∅.Figure 1
Closure space generated from this graph is not normal.Theorem 2.
For any binary relationR, the closure space X,clR is normal if and only if for every closed set clRA=A contained in ∪x∈ANRx, there exists some y such that clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx.Proof.
LetX,clR be a normal closure space and clRA=A be a closed set contained in ∪x∈ANRx. Thus, X−∪x∈ANRx=clRB is a closed set which is disjoint from the closed set clRA. Since X,clR is normal, there exist y and z with ∪y∈ANRy∩∪z∈BNRz=∅ such that clRA⊆∪y∈ANRy and clRB⊆∪z∈BNRz imply ∪y∈ANRy⊆X−∪z∈BNRz. Therefore, clRA⊆∪y∈ANRy⊆X−∪z∈BNRz⊆∪x∈ANRx. Since clR∪y∈ANRy is the smallest closed set by Theorem 1, clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx. Conversely, let clRA=A and clRB=B be two closed sets and clRA be contained in X−clRB=∪x∈X−BNRx. By a given condition, there exists some y such that clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx. Therefore, clRA⊆∪y∈ANRy, clRB⊆X−clR∪y∈ANRy, and ∪y∈ANRy∩X−clR∪y∈ANRy=∅. Hence, X,clR is normal.Theorem 3.
In a normal closure spaceX,clR, for every pair of disjoint closed sets clRA=A and clRB=B, there exist ∪y∈ANRy and ∪z∈BNRz containing clRA and clRB such that clR∪y∈ANRy∩clR∪z∈BNRz=∅.Proof.
LetX,clR be a normal closure space and clRA=A and clRB=B be two disjoint closed sets in X. Since clRA∩clRB=∅, we have clRA⊆X−clRB, where X−clRB=∪x∈X−BNRx. By the normal closure space of X, there exists some y such that clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆X−clRB. Thus, clR∪y∈ANRy∩clRB=∅. Since clRB⊆X−clR∪y∈ANRy=∪u∈X−clR∪NRyNRy, so again by using characterization of a normal closure space, there exists some z such that clRB⊆∪z∈BNRz⊆clR∪z∈BNRz⊆X−clR∪y∈ANRy which implies clR∪y∈ANRy∩clR∪z∈BNRz=∅.
In general, normality does not implyT2 which is evident from Example 1. For two distinct points ‘c’ and ‘d’ which are in the form of straight lines, there does not exist disjoint xR and yR containing ‘c’ and ‘d,’ respectively. But in T1 closure space, the result is true, as shown in Theorem 4 below.Theorem 4.
In a reflexiveT1 closure space X,clR, every normal space is T2.Proof.
LetX,clR be a T1 normal closure space and x and y be two distinct points. We have to show that X,clR is T2. Since x and y are closed, by normality of X,clR, there exist some x and y such that x⊆∪NRx, y⊆∪NRy, and ∪NRx∩∪NRy=∅. Hence, X,clR is T2.Definition 8.
A closure spaceX,clR generated from a binary relation R is said to be almost normal if for a closed set clRA=A and a regularly closed set clRintRB=B disjoint from clRA, there exist some x and y, respectively, such that clRA⊆∪x∈ANRx=U, clRintRB⊆∪y∈BNRy=V, and U∩V=∅.
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 Figure2. It is clear that aR=b,c,d,bR=a,c,cR=∅,anddR=a,c. Then, aR=a,c,bR=b,c,d,cR=c,anddR=b,c,d. The closure space X,clR generated from the graph in Figure 2 is almost normal but not normal because for two disjoint closed sets clRA=a=A and clRB=b,d=B, there does not exist some x and y such that clRA⊆∪x∈ANRx, clRB⊆∪y∈BNRy, and ∪x∈ANRx∩∪y∈BNRy=∅.Figure 2
Closure space generated from this graph is almost normal but not normal.Example 4.
A closure space generated from a graph which is not almost normal. The closure spaceX,clR generated from the binary relation in Figure 3 is not almost normal because for regularly closed set clRintRa,b=a,b and a closed set clRd=d, there does not exist some x and y such that clRA⊆∪x∈ANRx, clRB⊆∪y∈BNRy, and ∪x∈ANRx∩∪y∈BNRy=∅.
The implications in Figure4 are obvious from the definitions. But none of these implications is reversible (see [10] and Examples 1, 3, and 4 above).Figure 3
Closure space generated from this graph is not almost normal.Figure 4
Interrelation diagram.Theorem 5.
A closure spaceX,clR generated from a binary relation R is almost normal if and only if for every regularly closed set clRintRA=A contained in ∪x∈ANRx, there exists some y such that clRintRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx.Proof.
LetX,clR be an almost normal closure space, clRintRA=A be a regularly closed set contained in ∪x∈ANRx, and X−∪x∈ANRx=clRB=B be a closed set disjoint from clRintRA=A. Since X,clR is almost normal, there exist some y and z with ∪y∈ANRy∩∪z∈BNRz=∅ such that clRintRA⊆∪y∈ANRy and clRB⊆∪z∈BNRz which imply that ∪y∈ANRy⊆X−∪z∈BNRz. Therefore, clRintRA⊆∪y∈ANRy⊆X−∪z∈BNRz⊆∪x∈ANRx. Since clR∪y∈ANRy is the smallest closed set by Theorem 1, clRintRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx. Conversely, let clRintRA=A be a regularly closed set and clRB=B be a closed set. Thus, X−clRB=∪x∈X−BNRx is an open set containing clRintRA. By the given condition, there exists some y such that clRintRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈X−BNRx. Thus, clRintRA⊆∪y∈ANRy and clRB⊆X−clR∪y∈ANRy. Hence, X,clR is almost normal.Theorem 6.
IfR a binary relation, then the generated closure space X,clR is almost normal if and only if for every closed set clRA=A contained in a regularly open set ∪x∈ANRx, there exists some y such that clRA⊆∪x∈ANRy⊆clR∪x∈ANRy⊆∪x∈ANRx.Proof.
LetX,clR be an almost normal closure space, clRA=A be a closed set contained in ∪x∈ANRx, and X−∪x∈ANRx=clRintRB=B be a regularly closed set disjoint from clRA. Since X,clR is an almost normal closure space, there exist some y and z with ∪y∈ANRy∩∪z∈BNRz=∅ such that clRA⊆∪y∈ANRy and clRintRB⊆∪z∈BNRz which imply that ∪y∈ANRy⊆X−∪z∈BNRz. Therefore, clRA⊆∪y∈ANRy⊆X−∪z∈BNRz⊆∪x∈ANRx. Since clR∪y∈ANRy is the smallest closed set by Theorem 1, clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈ANRx. Conversely, let clRA=A be a closed set and clRintRB=B be a regularly closed sets’ disjoint from a closed set. Thus, X−clRintRB=∪x∈X−BNRx is a regularly open set containing clRA. By the given condition, there exists some y such that clRA⊆∪y∈ANRy⊆clR∪y∈ANRy⊆∪x∈X−BNRx. Thus, clRA⊆∪y∈ANRy and clRintRB⊆X−clR∪y∈ANRy. Hence, X,clR is almost normal.Definition 9.
LetR be any binary relation on X and X,clR be the generated closure space. Then, a family W=Wα:α∈Λ of subset X is said to be an R-cover of X if X=∪Wα∈WWα. If elements of W are of the form ∪NRx, then it is said to be an R-open cover of X.Definition 10.
A closure spaceX,clR is said to be nearly compact if for every R-open cover Wα:α∈Λ of X, there is a finite subset P of Λ such that intclRWα:α∈P is an R-cover of X.Theorem 7.
Every regularly closed closure subspace of a nearly compact space is nearly compact.Proof.
LetY,clRY be a regularly closed closure subspace of a nearly compact space X,clR. Let U=∪∪NRxα:α∈Q be an R-open cover of Y. We have to show that there is a finite subcollection whose closure is an R-cover of Y. Since NRxα:α∈Q is an R-open cover of Y, ∪∪NRxα:α∈Q∪X−Y is an R-open cover of X. Also, X,clR is nearly compact, so there is a finite subcollection NRxα:α∈P such that intclRNRxα:α∈P is an R-cover of X. If this subcollection has X−Y, then discard it. The remaining subcollection also covers Y. Hence, Y,clR is a nearly compact closure space.Theorem 8.
The closure spaceX,clR generated from a reflexive relation R is almost normal if it is nearly compact and T2.Proof.
LetclRA=A be a regularly closed set which is disjoint from a closed set clRB=B. We have to show that X,clR is almost normal. Since for a point b∈clRB and a point a∈clRA, aR∩bR=∅. The family U=∪∪NRa:a∈clRA covers A. Since clRA is regularly closed and X,clR is nearly compact, clRA is nearly compact by Theorem 7. Thus, there is a finite set Q of clRA such that intRclR∪NRaα:α∈Q is an R-cover of clRA. Thus, there exist some a and b such that clRA⊆∪intRclR∪NRaα, b∈NRb, and ∪NRaα∩NRb=∅. Since for every point b∈clRB there exists NRb, the collection ∪∪NRbα:α∈clRb covers clRB. Thus, clRA⊆intRclR∪a∈ANRaα and clRB⊆intRclR∪b∈BNRbα as X,clR is nearly compact and ∪intRclR∪a∈ANRaα∩intRclR∪b∈BNRbα=∅. Hence, X,clR is an almost normal closure space.Remark 2.
The following example establishes thatT2 condition in Theorem 8 cannot be dropped.Example 5.
Let us consider the directed graph in Figure5. Here, the closure space X,clR generated from the graph is nearly compact but not almost normal because for the regularly closed set clRintRA=a,b and a closed set clRB=d, there does not exist some x and y such that clRintRA⊆∪x∈ANRx, clRB⊆∪x∈yNRy, and ∪x∈ANRx∩∪x∈yNRy=∅.Figure 5
Closure space generated from this relation is nearly compact but not almost normal.


## 3. Subspace

Definition 11 (see [5]).
LetY⊆X and RY⊆R; then, Y,clRY is called a closure subspace of a closure space X,clR if xRY=xR∩Y for all x∈Y.Remark 3 (see [12]).
LetY be a subspace of a closure space X. Then,(a)
A is closed (open) in X implies that Y∩A is closed (open) in Y.(b)
Y is closed in X and A is a closed set in Y imply that A is closed in X.
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 Figure6. The closure space X,clRX 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 spaceY,clRY of the subgraph in Figure 7 is not normal because for two disjoint closed sets clRYA=b=A and clRYB=c=B, there does not exist some x and y in Y such that clRYA⊆∪x∈ANRYx, clRYB⊆∪y∈BNRYy, and ∪x∈ANRYx∩∪y∈BNRYy=∅.Figure 6
Closure space generated from this graph is normal.Figure 7
Closure space generated from this subgraph is not normal.Theorem 9.
LetX,clR be a normal closure space. Then, every closed subspace Y,clRY of X,clR is normal.Proof.
LetA=clRYA and B=clRYB be two disjoint closed sets in a closed subspace Y of X. By Remark 3, clRA and clRB closed in X implies that clRA∩Y and clRB∩Y are closed in X. By Remark 3, clRYA and clRYB are closed in X. Since X,clR is normal, there exist some x and y such that clRYA⊆∪x∈ANRx, clRYB⊆∪y∈BNRy, and ∪x∈ANRx∩∪y∈BNRy=∅. Thus, by Definition 11, ∪x∈ANRx∩Y and ∪y∈BNRy∩Y contain clRYA, clRYB, and ∪x∈ANRx∩Y∩∪y∈BNRy∩Y=∅. Hence, Y,clRY 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 Example7 below.Example 7.
Closure space generated from a closed subgraph of an almost normal closure space need not be almost normal.
The closure spaceX,clRX 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 x and y satisfying the condition of the almost normal closure space.
Let us consider the closed subgraph in Figure9 of the graph in Figure 8. The closed closure subspace Y,clRY generated from the subgraph in Figure 9 is not almost normal because for a closed set A=a=clRYA and a regularly closed set B=d,e=clRYintRYB, there does not exist some x and y in Y satisfying the required condition.Figure 8
Closure space generated from this graph is almost normal.Figure 9
Closure space generated from this subgraph is not almost normal.Remark 5.
In a closure spaceX,clR, 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 X,clR is almost normal.


## 4. Preservation under Mapping

Definition 12 (see [5]).
LetX1,clR1 and X2,clR2 be two closure spaces. A function f:X1⟶X2 is continuous at x∈X1 if and only if fxR1⊆fxR2. A function from a closure space X1,clR1 into a closure space X2,clR2 is said to be continuous on X1 if and only if it is continuous at each point of X1.Theorem 10 (see [5]).
Letf be a function from a closure space X1,clR1 into a closure space X2,clR2; then, the following conditions are equivalent:(1)
f is continuous.(2)
For every subsetA of X1, fclR1A⊆clR2fA.(3)
The inverse image of every closed subset ofX2 is a closed subset of X1.(4)
The inverse image of every open subset ofX2 is an open subset of X1.Definition 13 (see [5]).
A functionf:X1,clR1⟶X2,clR2 is called open (closed) if the image of an open (closed) subset of X1 is an open (closed) subset of X2.Theorem 11.
Letf:X,clRX⟶Y,clRY be a continuous closed and surjection and X,clRX be normal; then, Y,clRY is also a normal closure space.Proof.
LetclRYA=A and clRYB=B be two disjoint closed sets in Y. We have to show that Y,clRY is normal. Since f is continuous, f−1clRYA and f−1clRYB are disjoint closed sets in X. As X,clRX is normal, there exist some x and y with ∪x∈ANRXx∩∪y∈BNRXy=∅ such that f−1clRYA⊆∪x∈ANRXx and f−1clRYB⊆∪y∈BNRXy. Since f is closed, fX−∪x∈ANRXx and fX−∪y∈BNRXy are closed in Y. Since f−1clRYA⊆∪x∈ANRXx, X−∪x∈ANRXx⊆X−f−1clRYA implies that fX−∪x∈ANRXx⊆fX−f−1clRYA. Thus, clRYA⊆Y−fX−∪x∈ANRXx=∪u∈Y−fX−∪NRXxNRYu. Since f−1∪u∈Y−fX−∪NRXxNRYu=∪x∈ANRXx, there exists ∪u∈ANRYu containing clRYA such thatf−1∪u∈ANRYu⊆∪x∈ANRXx. Similarly, there exists ∪v∈BNRYv containing clRYB such that f−1∪v∈BNRYv⊆∪y∈BNRXy. Also, f−1∪u∈ANRYu∩f−1∪v∈BNRYv⊆∪x∈ANRXx∩∪y∈BNRXy=∅ implies that ∪u∈ANRYu∩∪v∈BNRYv=∅. Hence, Y,clRY is a normal closure space.Example 8.
In a closure spaceX,clRX, the continuous image of an almost normal space need not be almost normal.
LetX be a set of natural numbers and Y=a,b,c,d be a set. Let R1 be a relation defined on X as aR1b if and only if a=b for all aandb∈X and b=a−1, where a is even, and R2=a,c,a,d,b,a,b,b,b,c,c,b be a relation defined on Y. Then,(3)iR1=i,ifiis odd,i−1,i,ifiis even,and xR2 is defined as in Example 4. Here, X,clR1 and Y,clR2 are closure spaces. A function f:X,clR1⟶Y,clR2,(4)fx=a,ifx∈ℕ−1,3,4,b,ifx=1,c,ifx=3,d,ifx=4,is continuous. Here, closure space X,clR1 is almost normal but Y,clR2 is not almost normal because for closed set clR2d=d=A and a regularly closed set clR2intR2B=a,b=B, there does not exist some x and y in Y such that clR2d⊆∪x∈ANR2x, clR2intR2a,b⊆∪y∈BNR2y, and ∪x∈ANR2x∩∪y∈BNR2y=∅.Example 9.
continuous closed image of an almost normal space need not be almost normal.
LetX=ℕ∪p,q and Y=a,b,c,d be two sets. Let R1 be a relation defined on X as aR1b if and only if(5)a=b,∀aandb∈X,b=p,fora≥4,b=a−1,fora=3,b=a+1,fora=2,b=1,fora=q,and relation R2=a,a,a,b,a,c,b,c,c,b,d,c,d,d, defined on Y, are two binary relations. Then, 1R1=1, 2R1=3R1=2,3, iR1=i,p for i≥4, pR1=p,qR1=1,q, aR2=a,b,c,bR2=b,cR2=c,anddR2=c,d. Here, X,clR1 and Y,clR2 are closure spaces. A function f:X,clR1⟶Y,clR2 is defined as(6)fx=a,ifx=q,2,3,and4,b,ifx=1,c,ifx=p,d,ifx∈ℕ−1,2,3,4,is continuous. Here, X,clR1 is almost normal but Y,clR2 is not almost normal because for closed set clR2d=d=A and a regularly closed set clR2intR2a,b=a,b=B, there does not exist some x and y in Y 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.
LetR be a binary relation and X,clRX and Y,clRY be two closure spaces; then, a function f:X,clRX⟶Y,clRY is said to be completely continuous if inverse image of ∪NRx=U in Y,clRY is regularly open in X,clRX.
From the above definition, we can say that iff:X,clRX⟶Y,clRY is a completely continuous onto function and B=clRYintRYB is regularly closed in Y,clRY, then f−1B is regularly closed in X,clRX.Observation 1.
In a closure spaceX,clRX, 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.
LetX be a set of natural numbers and Y=p,q,r,s be two sets. Let R1 be a relation on X, as defined in Example 8, and R2 be the relation on Y given by R2=p,p,q,p,q,q,q,r,r,r,s,r,s,s. Then, iR1 is defined as in Example 8, and pR2=p, qR2=p,q,r, rR2=r, and sR2=r,s. Here, X,clR1 and Y,clR2 are closure spaces. A function f:X,clR1⟶Y,clR2 defined by(7)fx=p,ifx=1and2,q,ifx=ℕ−1,2,3,4,5,6,r,ifx=3and4,s,ifx=5and6,is completely continuous. Here, X,clR1 is almost normal but Y,clR2 is not almost normal because for a closed set A=s=clR2A and a regularly closed set B=p,q=clR2intR2B, there does not exist some x and y in Y such that clR2A⊆∪x∈ANR2x, clR2intR2B⊆∪y∈BNR2y, and ∪x∈ANR2x∩∪y∈BNR2y=∅.Theorem 12.
Completely continuous, closed image of an almost normal closure space, is almost normal.Proof.
LetclRYA=A be a closed set and clRYintRYB=B be a regularly closed set. Then, f−1clRYA is closed, and f−1clRYintRYB is regularly closed in X. Since X,clRX is almost normal, there exist some x and y with ∪x∈ANRXx∩∪y∈BNRXy=∅ such that f−1clRXA⊆∪x∈ANRXx and f−1clRYintRYB⊆∪y∈BNRXy. Thus, fX−∪x∈ANRXx and fX−∪y∈BNRXy are closed in Y,clRY. As f−1clRYA⊆∪x∈ANRXx implies X−∪x∈ANRXx⊆X−f−1clRYA, clRYA⊆Y−fX−∪x∈ANRXx=∪p∈Y−fX−∪NRXxNRXp. Now, f−1∪p∈Y−fX−∪NRXxNRXp=f−1Y−fX−∪x∈ANRXx=∪x∈ANRYx. Thus, there exists some p in Y such that clRYA⊆∪p∈ANRXp. Similarly, there exists some q in Y such that clRYintRYB⊆∪q∈BNRXq. Also, ∪p∈ANRXp∪∪q∈BNRXq=∅. Hence, X,clRY is an almost normal closure space.



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*Source: 6917297-2021-07-12.xml*