# OnQp-Closed Sets in Topological Spaces

Authors: Huan Zhou; O. G. Hammad; Ahmed Mostafa Khalil
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present paper, we will propose the novel notions (e.g.,Qp-closed set, Qp-open set, Qp-continuous mapping, Qp-open mapping, and Qp-closed mapping) in topological spaces. Then, we will discuss the basic properties of the above notions in detail. The category of all Qp-closed (resp. Qp-open) sets is strictly between the class of all preclosed (resp. preopen) sets and gp-closed (resp. gp-open) sets. Also, the category of all Qp-continuity (resp. Qp-open (Qp-closed) mappings) is strictly among the class of all precontinuity (resp., preopen (preclosed) mappings) and gp-continuity (resp. gp-open (gp-closed) mappings). Furthermore, we will present the notions of Qp-closure of a set and Qp-interior of a set and explain some of their fundamental basic properties. Several relations are equivalent between two different topological spaces. The novel two separation axioms (i.e., Qp-ℝ0 and Qp-ℝ1) based on the notion of Qp-open set and Qp-closure are investigated. The space of Qp-ℝ0 (resp., Qp-ℝ1) is strictly between the spaces of pre-ℝ0 (resp., pre-ℝ1) and gp-ℝo (resp., gp-ℝ1). Finally, some relations and properties of Qp-ℝ0 and Qp-ℝ1 spaces are explained.


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## Body

## 1. Introduction

In the early eighties, the novel notions of preopen and preclosed sets (i.e., as a novel type of generalized of open sets inX,τ˜ (i.e., topological space) or a space X) and preconlinuous mappings are proposed in [1]. Consequently, many researchers turned their study to the generalizations of many different notions in X,τ˜ (for instance, semiopen sets [2], α-open sets [3], and β-open sets [4] or semi-preopen sets [5]). Furthermore, the notion of generalized closed (resp., generalized open) sets (for short, g-closed (resp., g-open) sets) in space X is presented in [6]. The relationship among g-closed (resp., g-open) sets and generalizing closedness (resp., openness) sets (i.e., generalized preclosed (resp., generalized preopen) set (for short, gp-closed (resp., gp-open) set) [7], α-generalized closed (resp., α-generalized open) set (for short, αg-closed (resp., αg-open) set) [8], pregeneralized closed (resp., pregeneralized open) set (for short, pg-closed (resp., pg-open) set) [7], and generalized α-closed (resp., generalized α-open) set (for short, gα-closed (resp., gα-open) set) [9]. The basic properties of five generalizing continuous mappings (i.e., precontinuous mapping [1], g-continuous mapping [10], gp-continuous mapping [11], αg-continuous mapping [12], pg-continuous mapping [11], and gα-continuous mapping [12]) between σ˜Y (i.e., a topology on Y) and τ˜X (i.e., a topology on X) are presented. Furthermore, the fundamental relations of generalizing open (closed) mappings (i.e., preopen (preclosed) mapping [1, 13], α-open (α-closed) mapping [14], g-open (g-closed) mapping [15], gp-open (gp-closed) mapping [11], αg-open (αg-closed) mapping [12], pg-open (pg-closed) mapping [11], gα-open (gα-closed) mapping [12]) between σ˜Y and τ˜X are studied. On the contrary, the characterizations between separation axioms classes (i.e., pre-ℝ0, pre-ℝ1, gp-ℝ0 and gp-ℝ1 spaces) (see, [16, 17]) in X,τ˜ are defined.Regarding the above discussions, as the motivation of the present paper, we will define novel sets calledQp-closed sets and Qp-open sets and investigate several of their fundamental properties. The relation between Qp-closed set (resp., Qp-open set) and other sets (for example, preclosed set (resp., preopen set), α-closed set (resp., α-pen set), g-closed set (resp., g-open set), gp-closed set (resp., gp-open set), αg-closed set (resp., αg-open set), pg-closed set (resp., pg-open set), and gα-closed set (resp., gα-open set)) in space X is introduced. Then, we define the Qp-continuous mapping and study the relations between Qp-continuous mapping and other mappings (for example, precontinuous mapping, g-continuous mapping, gp-continuous mapping, αg-continuous mapping, pg-continuous mapping, and gα-continuous mapping) between two different topological spaces. Also, we present the notion of Qp-open (Qp-closed) mapping and investigate relations between Qp-open (Qp-closed) mapping and other mappings (for example, preopen (preclosed) mapping, α-open (α-closed) mapping, g-open (g-closed) mapping, gp-open (gp-closed) mapping, αg-open (αg-closed) mapping, pg-open (pg-closed) mapping, and gα-open (gα-closed) mapping) between two different topological spaces. Finally, we propose the novel separation axioms classes (i.e., Qp-ℝ0 and Qp-ℝ1 spaces) in X,τ˜.Next, the sections of this paper are arranged as follows. In Section2, we will present many notions related to topological spaces as indicated from Definitions 1 to 4. In Section 2, we propose the novel notions of Qp-closed sets and Qp-open sets and explain the interesting properties of them. In Section 3, we give the notions of Qp-continuous mappings, Qp-open mappings, and Qp-closed mappings. In Section 4, we define Qp-ℝ0 and Qp-ℝ1 spaces. Section 5 is conclusions.In the current paper, we will use several expressions (i.e.,CA (the closure of a set A), ℐA (the interior of a set A), τ˜X (the all of open sets in X), and ℱX (the all of closed sets in X)).Next, we will present several notions which are used in this section as indicated below.Definition 1.
(Cf. [1, 3]). Assume X,τ˜ is a topological space. Then,(1)
(i)
A is preclosed set if CℐA⊆A(ii)
A is preopen set if A⊆ℐCAℂpX (resp., OpX) is the set of all preclosed (resp. preopen) sets.(2)
(i)
A is α-closed set if CℐCA⊆A(ii)
A is α-open set if A⊆ℐCℐA
ℂαX (resp., OαX) is the set of all α-closed (resp. α-open) sets.Definition 2.
(Cf. [6–9]). Assume X,τ˜ is a topological space. Then,(1)
(i)
A is g-closed set if CA⊆A whenever A⊆L and L∈τ˜X, where CA is a closure of A, i.e.,(1)CA=∩F|A⊆FandF∈ℱX.GℂX be the set of all g-closed sets.(ii)
A is g-open set if X\A∈GℂX and GOX be the set of all g-open sets.(2)
(i)
A is gp-closed set if CpA⊆A whenever A⊆L and L∈τ˜X, where CpA is a preclosure of A, i.e.,(2)CpA=∩F|A⊆FandF∈ℂpX.GℂpX be the set of all gp-closed sets.(ii)
A is gp-open set if X\A∈GℂpX and GOpX be the set of all gp-open sets.(3)
(i)
A is αg-closed set if CαA⊆A whenever A⊆L and L∈τ˜X, where CαA is a α-closure ofA, i.e.,(3)CαA=∩F|A⊆FandF∈ℂαX.ℂαGX be the set of all αg-closed sets.(ii)
A is αg-open set if X\A∈ℂαGX and OαGX be the set of all αg-open sets.(4)
(i)
A is pg-closed set if CpA⊆A whenever A⊆L and L∈OpX and ℂpGX be the set of all pg-closed sets.(ii)
A is pg-open set if X\A∈ℂpGX and OpGX be the set of all gp-open sets.(5)
(i)
A is gα-closed set if CαA⊆A whenever A⊆L and L∈OαX, and GℂαX be the set of all gα-closed sets.(ii)
A is gα-open set if X\A∈GℂαX, and GOαX be the set of all gα-open sets.Definition 3.
(Cf. [1, 10, 13–15]). Let ψ:X⟶Y be mapping and σ˜Y be a topology on Y and τ˜X is a topology on X. Then,(1)
ψ is precontinuous mapping (resp., α-continuous mapping and g-continuous mapping) if L∈σ˜Y,ψ−1L∈OpXresp.,ψ−1L∈OαX,ψ−1L∈GOX.(2)
(i)
ψ is preopen mapping (resp., α-open mapping and g-open mapping) if L∈τ˜X,ψL∈QpOYresp.,ψL∈OαY,ψL∈GOY.(ii)
ψ is preclosed mapping (resp., α-closed mapping and g-closed mapping) if L∈τ˜X,ψL∈QpℂYresp.,ψL∈ℂαY,ψL∈GℂY.Definition 4.
(Cf. [16, 17]). A topological space X,τ˜ is said to be(1)
Pre-ℝ0 space (resp., gp-ℝ0 space) if ∀x∈L∈OpX s.t. Cpx⊂L (resp., ∀x∈L∈GOpX s.t., Cgpx⊂L) and Cgpx is a gp-closure of x, defined as(4)Cgpx=∩F|x⊆FandF∈GℂpX.(2)
Pre-ℝ1 space (resp., gp-ℝ1 space) if ∀x,y∈X, with Cpx≠Cpy (resp., Cgpx≠Cgpy), there exist disjoint preopen sets (resp., gp-open sets) L and M s.t. Cpx⊆L (resp., Cgpx⊆L) and Cpy⊆M (resp., Cgpy⊆M).


## 2.Qp-Closed Sets and Qp-Open Sets

In the following section, we propose novel sets (i.e.,Qp-closed sets and Qp-open sets) and discuss several interesting theorems and examples.Definition 5.
We callA is Qp-closed set in X,τ˜ if(5)CgpℐA⊆A,where CgpℐA is a gp-closure of ℐA, i.e.,(6)CgpℐA=∩F|ℐA⊆FandF∈GℂpX.
QpℂX is the set of all Qp-closed sets in X.Lemma 1.
LetF∈GℂpX s.t. CgpℐF⊆A⊆F. Then, A∈QpℂX.Proof.
AsF∈GℂpX implies CgpF=F, thus, CgpℐA⊆CgpℐF⊆A. Therefore, A∈QpℂX.The converse of Lemma1 (i.e., CgpℐF⊈⊆F) does not hold by the following example.Example 1.
Assume thatX=1,2,3 (i.e., X,τ˜ be topological space) and τ˜=X,ϕ,1. Then,(7)ℱX=X,ϕ,2,3,ℂpX=X,ϕ,2,3,2,3,GℂpX=X,ϕ,2,3,1,2,1,3,2,3,(8)QpℂX=2X.
LetA=3∈QpℂX and F=X∈GℂpX. Then, CgpℐF=CgpX=⊈⊆F.Theorem 1.
The following two properties are holding inX,τ˜.(1)
IfA∈ℂpX, then A∈QpℂX(2)
IfA∈GℂpX, then A∈QpℂXProof.
(1)
AsCgpA⊆CpA and CpA⊆CA imply that CgpA⊆CA, since A is preclosed set (i.e., CℐA⊆A), then CgpℐA⊆CℐA⊆A. Thus, A∈QpℂX.(2)
LetA∈GℂpX (i.e., CgpA=A). Then, CgpℐA⊆CgpA=A. Thus, A∈QpℂX.The converse of Theorem1 (i.e., A∈QpℂX but A∉ℂpX and A∉GℂpX) does not hold by the following example.Example 2.
(continued from Example1). As 1,3∈QpℂX, 1,3∉ℂpX and 1∈QpℂX but 1∉GℂpX.Theorem 2.
Arbitrary intersection ofQp-closed sets is Qp-closed set.Proof.
Suppose thatFk|k∈Λ be a collection of Qp-closed sets in X. Then, CgpℐFk⊆Fk, for every k. As ∩Fk⊆Fk, for every k, Cgp∩Fk⊆CgpFk for every k. Thus, Cgp∩Fk⊆∩CgpFk,k∈Λ. Hence, Cgpℐ∩Fk⊆Cgp∩ℐFk⊆∩CgpℐFk⊆∩Fk. Therefore, ∩Fk is Qp-closed set.Remark 1.
The union of twoQp-closed sets need not be Qp-closed set (i.e., A,B∈QpℂX, but A∪B∉QpℂX) as the next example; let X=1,2,3 (i.e., X,τ˜ be topological space) and τ˜=X,ϕ,1,2. Then,(9)ℱX=X,ϕ,3,ℂpX=GℂpX=QpℂX=X,ϕ,1,2,3,1,3,2,3.
LetA=1∈QpℂX and B=2∈QpℂX. Then, A∪B=1,2∉QpℂX.Corollary 1.
The following two properties are holding inX,τ˜.(1)
LetA∈QpℂX and B∈ℂpX. Then, A∩B∈QpℂX.(2)
LetA∈QpℂX and B∈GℂpX. Then, A∩B∈QpℂX.Proof.
From Theorems1 and 2, the proof is clear.Definition 6.
CQpA is called Qp-closure of A in X,τ˜ if(10)CQpA=∩F|A⊆FandF∈QpℂX.Theorem 3.
The following seven properties are holding inX,τ˜.(1)
A∈QpℂX⇔CQpA=A(2)
A⊆CQpA⊆CpAandCQpA⊆CgpA(3)
CQpϕ=ϕ and CQpX=X(4)
IfA⊆B, then CQpA⊆CQpB(5)
CQpCQpA=CQpA(6)
CQpA∪CQpB⊆CQpA∪B(7)
CQpA∩B⊆CQpA∩CQpBProof.
(1)
LetA∈QpℂX, and from Definition 6, we have CQpA=A. Conversely, let CQpA=A. Then, from Theorem 2, we have A∈QpℂXwhich follows from Theorem 1 (1) and (2), respectively, and are obvious.(2)
LetA⊆F such that F∈QpℂX; then, from (1) above, we have CQpA⊆CQpF=F. Again CQpCQpA⊆CQpF=F. Thus, CQpCQpA⊆∩F|A⊆FandF∈QpℂX=CQpA and follows from (4).The equality of Theorem3 (6) and (7) (i.e., CQpA∪CQpB≠CQpA∪B and CQpA∩B≠CQpA∩CQpB) does not hold by the following example.Example 3.
(continued from Remark1). As A=1,B=2, and A∪B=1,2, then CQp1=1,CQp2=2,andCQp1,2=X, and hence, CQpA∪CQpB=1,2≠X=CQpA∪B.Example 4.
Assume thatX=1,2,3 (i.e., X,τ˜ be topological space) and τ˜=X,ϕ,2,3,2,3. Then,(11)ℱX=ℂpX=GℂpX=QpℂX=X,ϕ,1,1,2,1,3.
LetA=1, B=2, and A∩B=ϕ. Then, CQp1=1,CQp2=1,2,andCQpϕ=ϕ, and hence, CQpA∩CQpB=1≠ϕ=CQpA∩B.The relationship among theQp-closed sets and other sets (i.e., closed sets, α-closed sets, g-closed sets, gα-closed sets, αg-closed sets, and pg-closed sets) is presented by the following theorem.Theorem 4.
The following six properties is holding inX,τ˜.(1)
IfA∈ℱX, then A∈QpℂX(2)
IfA∈ℂαX, then A∈QpℂX(3)
IfA∈GℂX, then A∈QpℂX(4)
IfA∈GℂαX, then A∈QpℂX(5)
IfA∈ℂαGX, then A∈QpℂX(6)
IfA∈ℂpGX, then A∈QpℂXProof.
(1)
AsCpA⊆CA and by Theorem 1 (1), we have CgpℐA⊆CpℐA⊆CℐA⊆A. Thus, A∈QpℂX.(2)
AsCpA⊆CαA and by Theorem 1 (1), we have CgpℐA⊆CpℐA⊆CℐA⊆CℐCA⊆A. Thus, A∈QpℂX.(3)
AsCgpA⊆CgA and by Theorem 1 (2), we have CQpA⊆CgpA⊆CgA⊆A. Thus, A∈QpℂX.(4)
AsCgpA⊆CgA and by Theorem 1 (2), we have CQpA⊆CgpA⊆CgA⊆A. Thus, A∈QpℂX.The converse of Theorem 4 (i.e.,A∈QpℂX, but A∉ℱX,A∉ℂαX,A∉GℂX,A∉GℂαX,A∉OαGX, and A∉ℂpGX) does not hold by the following example.Example 5.
(continued from Example1). Clearly,(12)GℂαX=ℂpGX=ℂαGX=ℂαX=ℂpXand GℂX=GℂpX. Thus, GℂX=GℂpX, but 1∉ℱX,1∉ℂαX,1∉GℂX,1∉GℂαX,1∉ℂαGX, and 1∉ℂpGX.Definition 7.
A is called Qp-open set if X\A∈QpℂX and QpOX is the set of all Qp-open sets in X.Lemma 2.
The following properties are holding inX,τ˜.(1)
X\CgpX\A=ℐgpA(2)
X\ℐgpXA=CgpAProof.
It is clear.Theorem 5.
The following properties are holding inX,τ˜:(13)A∈QpOX⇔A⊆ℐgpCA.Proof.
Suppose thatA∈QpOX. Then, X\A∈QpℂX and CgpℐX\A⊆X\A. From Lemma 2, we have A⊆ℐgpCA. Conversely, A⊆ℐgpCA. Then, X\ℐgpCA⊆X\A. Thus, ℐgpCXA⊆X\A. Thus, X\A∈QpℂX, and hence, A∈QpOX.Lemma 3.
LetL∈GOpX such that L⊆A⊆ℐgpCL. Then, A∈QpOX.Proof.
AsL∈GOpX, X\L∈GℂpX, and X\ℐgpCL\\⊆X\A⊆X\L, thus, by Lemma 2, we have CgpℐX\L⊆X\A⊆X\L. By Lemma 1, we get X\A∈QpℂX, and hence, A∈QpOX.The converse of Lemma3 (i.e., L⊆⊈ℐgpCL) does not hold by the following example.Example 6.
(continued from Example1). As(14)OpX=X,ϕ,1,1,2,1,3,GOpX=X,ϕ,1,2,3,1,2,1,3,(15)QpOX=2X,let A=1,2∈QpOX and L=ϕ∈GOpX. Then, L⊆A⊈ϕ=ℐgpCL.Theorem 6.
The following two properties are holding inX,τ˜.(1)
IfA∈OpX, then A∈QpOX(2)
IfA∈GOpX, then A∈QpOXProof.
From Theorem1 and Lemma 2, the proof is clear.The converse of Theorem6 (i.e., A∈QpOX, but A∉OpX and A∉GOpX) does not hold by the following example.Example 7.
(continued from Examples1 and 6). 2\\∈QpOX, but 2∉OpX, and 2,3∈QpOX, but 2,3\\∉GOpX.Theorem 7.
Arbitrary union ofQp-open sets is Qp-open set.Proof.
From Theorem2 and Lemma 2, the proof is clear.Remark 2.
The intersection of twoQp-open sets need not be Qp-open set (i.e., A,B∈QpOX, but A∩B∉QpOX) as given in Remark 1. As(16)OpX=GOpX=QpOX=X,ϕ,1,2,1,2,1,3,2,3,let A=2,3∈QpOX and B=1,3∈QpOX. Then, A∩B=3∉QpOX.Corollary 2.
The following two properties are holding inX,τ˜.(1)
LetA∈QpOX and B∈OpX. Then A∪B∈QpOX.(2)
LetA∈QpOX and B∈GOpX. Then A∪B∈QpOX.Proof.
From Theorem6 and Lemma 7, the proof is clear. □Definition 8.
ℐQpA is called Qp-interior of A in X,τ˜ if(17)ℐQpA=∪L|L⊆AandL∈QpOX.Lemma 4.
The following two properties are holding inX,τ˜:(1)
X\CQpA=ℐQpX\A(2)
X\ℐQpA=CQpX\AProof.
It is clear.Theorem 8.
The following properties are holding inX,τ˜.(1)
A∈QpOX⇔ℐQpA=A(2)
ℐpA⊆ℐQpA⊆A,ℐgpA⊆ℐQpA(3)
ℐQpϕ=ϕ and ℐQpX=X(4)
IfA⊆B, then ℐQpA⊆ℐQpB(5)
ℐQpℐQpA=ℐQpA(6)
ℐQpA∪ℐQpB⊆ℐQpA∪B(7)
ℐQpA∩B⊆ℐQpA∩ℐQpBProof.
It is similar to Theorem3.The equality of Theorem8 (6) and (7) (i.e., ℐQpA∪ℐQpB\\≠ℐQpA∪B and ℐQpA∩B≠ℐQpA∩ℐQpB) does not hold by the following examples.Example 8.
(continued from Remarks1 and 2). As A=2,B=3, and A∪B=2,3, then ℐQp2=2,ℐQp3=ϕ,andℐQp2,3=2,3, and hence, ℐQpA∪ℐQpB=2≠2,3=ℐQpA∪B.Example 9.
(continued from Remarks1 and 2). As A=1,3,B=2,3, and A∩B=3, then ℐQp1,3=1,3,ℐQp2,3=2,3,andℐQp3=ϕ, and hence, ℐQpA∩ℐQpB=3≠ϕ=ℐQpA∩B.Theorem 9.
The following properties is holding inX,τ˜. Then,(18)x∈CQpA⇔L∩A≠ϕ,∀x∈L∈QpOX.Proof.
The proof is clear.Lemma 5.
The following properties are holding inX,τ˜. Then,(1)
A∩ℐgpCℐgpA∈QpOX(2)
A∪CgpℐCgpA∈QpℂXProof.
ℐgpCℐgpA∩ℐgpCℐgpA=ℐgpCℐgpA∩ℐgpCℐgpA=ℐgpCℐgpA.
Then, we haveA∩ℐgpCℐgpA=A∩ℐgpCℐgpA∩ℐgpCℐgpA⊆∩ℐgpCℐgpA∩ℐgpCℐgpA.
Hence,A∩ℐgpCℐgpA∈QpOX.
In (2), by (1), we haveX\A∪CgpℐCgpA=X\A\\∩ℐgpCℐgpXA∈QpOX. Therefore, A∪CgpℐCgpA\\∈QpℂX.Theorem 10.
The following two properties are holding inX,τ˜. Then,(1)
ℐQpA=A∩ℐgpCℐgpA(2)
CQpA=A∪CgpℐCgpAProof.
(1)
SupposeB=ℐQpA. Since B∈QpOX and B⊆A, then, we have B⊆ℐgpCℐgpB⊆ℐgpCℐgpA. Thus, B⊆A∩ℐgpCℐgpA. By Lemma 5, we get A∩ℐgpCℐgpA∈QpOX. From Definition 8, we have A∩ℐgpCℐgpA⊆B. Therefore, B=A∩ℐgpCℐgpA and hence, ℐQpA=A∩ℐgpCℐgpA.(2)
By Lemma4, we have CQpA=X\ℐQpX\A=X\X\A∩ℐgpCℐgpX\A=A∪CgpℐCgpA.The relationship among theQp-open sets and other sets (i.e., open sets, α-open sets, g-open sets, gα-open sets, αg-open sets, and pg-open sets) is presented by the following theorem.Theorem 11.
The following properties is holding inX,τ˜.(1)
IfA∈τ˜X, then A∈QpOX(2)
IfA∈OαX, then A∈QpOX(3)
IfA∈GOX, then A∈QpOX(4)
IfA∈GOαX, then A∈QpOX(5)
IfA∈OαGX, then A∈QpOX(6)
IfA∈OpGX, then A∈QpOXProof.
It is similar to Theorem4.The converse of Theorem11 (i.e., A∈QpOX, but A∉τ˜X,A∉OαX,A∉GOX,A∉GOαX,A∉OαGX, and A∉OpGX) does not hold by the following example.Example 10.
(continued from Examples1, 5, and 6). Clearly,(19)GOαX=OpGX=OαGX=OαX=OpX.and GOX=GOpX. Thus, 2,3∈QpOX, but 2,3∈τ˜,2,3∈XOαX,2,3∉GOX,2,3∉GOαX,\\2,3∉OαGX, and 2,3∉OpGX.


## 3.Qp-Continuous Mappings, Qp-Open Mappings, and Qp-Closed Mappings

Definition 9.
A mappingψ:X⟶Y is called Qp-continuous if(20)L∈σ˜Y,ψ−1L∈QpOX,where σ˜Y is a topology on Y and X is defined on a topology τ˜Theorem 12.
The following two properties are holding inX,τ˜ and Y,σ˜.(1)
Every precontinuous mapping isQp-continuous mapping(2)
Every gp-continuous mapping isQp-continuous mappingProof.
From Theorem6, the proof is clear.The converse of Theorem12 (i.e., ψ is Qp-continuous mapping, but ψ not precontinuous mapping and ψ is Qp-continuous mapping but ψ not gp-continuous mapping) does not hold by the following example.Example 11.
Assume thatX=1,2,3 (i.e., X,τ˜ be topological space, τ˜=X,ϕ,1) and Y=u,v,w (i.e., Y,σ˜ be topological space, σ˜=Y,ϕ,u,v,w).(1)
Supposeψ:X⟶Y be a mapping defined by(21)ψ1=ψ2=u,andψ3=w.Then,(22)ℱX=X,ϕ,2,3,ℂpX=X,ϕ,2,3,2,3,GℂpX=X,ϕ,2,3,1,2,2,3,1,3,OpX=X,ϕ,1,1,2,1,3,GOpX=X,ϕ,1,2,3,1,2,1,3,QpℂX=QpOX=2X.Asw∈σ˜Y,ψ−1w=3∈QpOX, but 3∉OpX.Thus,ψ is Qp-continuous mapping but ψ not precontinuous mapping.(2)
Supposeψ:X⟶Y be a mapping defined by(23)ψ1=uandψ2=ψ3=w.
Asw∈σ˜Y,ψ−1w=2,3∈QpOX, but 2,3∉GOpX.
Thus,ψ is Qp-continuous mapping but ψ not gp-continuous mapping.Theorem 13.
Assume thatψ:X⟶Y be a mapping. Then, the following six properties are equivalent:(1)
ψ is Qp-continuous(2)
For everyx∈X and every open set L⊂Y containing ψx, there exists Qp-open set P⊂X containing x such that ψP⊂L(3)
ψ−1F∈QpℂX, F⊂Y, be a closed set(4)
ψCQpA⊆CψAA⊆X(5)
CQpψ−1B⊆ψ−1CBB⊆Y(6)
ψ−1ℐB⊆ℐQpψ−1BB⊆YProof.
(1)⇒ (2) Since L⊂Y containing ψx is the open set, then ψ−1L∈QpOX. Put P=ψ−1L which contains x; hence, ψP⊂L.(2)⇒ (1) Suppose L⊂Y be the open set, and let x∈ψ−1L; then, ψx∈L and hence, there exists Px∈QpOX such that x∈Px and ψPx⊂L. Thus, x∈Px⊂ψ−1L, so ψ−1L=∪x∈ψ−1LPx, but ∪x∈ψ−1LPx∈QpOX. Therefore, ψ−1L∈QpOX, and thus, ψ is Qp-continuous.(1)⇒ (3) Suppose F⊂Y be closed set. Thus, Y\F is the open set and ψ−1Y\F∈QpOX, i.e., X\ψ−1F∈QpOX. Hence, ψ−1F∈QpℂX.(3)⇒ (4) Suppose A⊆X and F be a closed set in Y containing ψA. By (3), we have ψ−1F is Qp-closed set containing A. Then, CQpA⊆CQpψ−1F=ψ−1F, and thus, ψCQpA⊆F. Hence, ψCQpA⊆CψA.(4)⇒ (5) Suppose B⊆Y and A=ψ−1B. By assumption, we have ψCQpA⊆CψA⊆CB. Then, CQpA⊆ψ−1CB. Therefore, CQpψ−1B⊆ψ−1CB.(5)⇒ (6) Suppose B⊆Y. By assumption, we have CQpψ−1Y\B⊆ψ−1CYB. Then, CQpX\ψ−1B\\⊆ψ−1Y\ℐB, and thus, X\ℐQpψ−1B⊆X\ψ−1ℐB. By taking complement, we obtain ψ−1ℐB⊆ℐQpψ−1B.(6)⇒ (1) Suppose L be any open set in Y. Then, ℐL=L. By assumption, ψ−1ℐL⊆ℐQpψ−1L, and hence, ψ−1L\\⊆ℐQpψ−1L. Thus, ψ−1L=ℐQpψ−1L, and we have ψ−1L∈QpOX. Therefore, ψ is Qp-continuous.Remark 3.
Composition of twoQp-continuous mappings does not need to be Qp-continuous mapping, as shown by the following example.Example 12.
Assume thatX=1,2,3 (i.e., X,τ˜ be topological space, τ˜=X,ϕ,2,1,2), Y=u,v,w (i.e., Y,σ˜ be topological space, σ˜=Y,ϕ,u), and Z=s,r,t (i.e., Z,θ˜ be topological space, θ˜=Z,ϕ,t).(1)
Supposeψ:X⟶Y be a mapping defined by(24)ψ1=v,ψ2=u,andψ3=w.Then,(25)ℱX=X,ϕ,31,3,ℂpX=X,ϕ,1,3,1,3,GℂpX=QpℂX=X,ϕ,1,3,1,3,2,3,OpX=X,ϕ,2,1,2,2,3,GOpX=QpOX=X,ϕ,1,2,1,2,2,3.Thus,ψ is Qp-continuous mapping.(2)
Supposeφ:Y⟶Z be a mapping defined by(26)φu=φv=s,andφw=t.
Then,(27)ℱY=Y,ϕ,v,w,ℂpY=Y,ϕ,v,w,v,w,GℂpY=Y,ϕ,v,w,u,v,u,w,v,w,OpY=Y,ϕ,u,u,v,u,w,GOpY=Y,ϕ,u,v,w,u,v,u,w,QpℂY=QpOY=2Y.
Thus,φ is Qp-continuous mapping.
From (1) and (2),t∈θ˜Z, but φ°ψ−1t=ψ−1φ−1t=ψ−1w=3∉QpOX. Thus, φ°ψ is not Qp-continuous mapping.Definition 10.
A mappingψ:X⟶Y is called(1)
Qp-open if(28)L∈τ˜X,ψL∈QpOY,whereτ˜X is a topology on X and Y is defined on a topology σ˜.(2)
Qp-closed if(29)F∈ℱX,ψF∈QpℂY,where ℱX is the closed sets of X.Theorem 14.
The following two properties are holding inX,τ˜ and Y,σ˜.(1)
Every preopen (resp. preclosed) mapping isQp-open (resp. Qp-closed) mapping(2)
Every gp-open (resp. gp-closed) mapping isQp-open (resp. Qp-closed) mappingProof.
It follows from Theorem6.Example 13.
(continued from Example11).(1)
Supposeψ:X⟶Y be a mapping defined by(30)ψ1=ψ2=v,andψ3=u.(i)
As2∈τ˜X,ψ2=v∈QpOY, but v∉OpY. Thus, ψ is Qp-open mapping but ψ not preopen mapping.(ii)
As3∈ℱX,ψ3=u∈QpOY, but u∉ℂpY. Thus, ψ is Qp-closed mapping but ψ not preclosed mapping.(2)
Supposeφ:X⟶Y be a mapping defined by(31)φ1=v,φ2=w,andφw=u.(i)
As1,2∈τ˜X, φ1,2=v,w∈QpOY, but v,w∉GOpY. Thus, φ is Qp-open mapping but φ not gp-open mapping.(ii)
As3∈ℱX, u∉GℂpY. Thus, φ is Qp-closed mapping but φ not gp-closed mapping.Theorem 15.
Assume thatψ:X⟶Y is a mapping. Then, the following two properties are equivalent:(1)
ψ is Qp-open.=(2)
For everyx∈X and U is a neighborhood of x, there exists Qp-open set B⊂Y containing ψx such that B⊂ψUProof.
The proof is clear.Theorem 16.
Assume thatψ:X⟶Y is Qp-open (resp. Qp-closed) mapping and B⊂Y. If A⊂X is a closed (resp. open) set containing ψ−1B, then there exists Qp-open (resp. Qp-closed) set ℌ⊂Y containing B such that ψ−1ℌ⊂A.Proof.
Obvious.Corollary 3.
For every setB⊆Y, if ψ:X⟶Y is Qp-open, then ψ−1CQpB⊆Cψ−1B.Proof.
Obvious.Theorem 17.
For any subsetA of X, a mapping ψ:X⟶Y is Qp-open ⇔ψℐA⊆ℐQpψA.Proof.
Assume thatψ:X⟶Y is Qp-open mapping and A⊆X. Then, ℐA∈τ˜X and ψℐA is Qp-open set contained in ψA. Hence, we have ψℐA⊆ℐQpψA. Conversely, for every A of X, ψℐA⊆ℐQpψA and L∈τ˜X. Then, ℐL=L, ψL⊆ℐQpψL. Therefore, ψL=ℐQpψL, and we have ψL is Qp-open mapping.Theorem 18.
Assume thatψ:X⟶Y is a bijective mapping. Then, the following three properties are equivalent:(1)
ψ−1 is Qp-continuous(2)
ψ is Qp-open(3)
ψ is Qp-closedProof.
(1)⇒ (2) Let U∈τ˜X. Then, X\U∈ℱX. Since ψ−1 is Qp-continuous, ψ−1−1X\U=ψX\U=Y\ψU∈QpℂY. So, ψU∈QpOY. Hence, ψ is Qp-open mapping.(2)⇒ (3) Let F∈ℱX. Then, X\F∈τ˜X. Since ψ is Qp-open, ψX\F=Y\ψF∈QpOY. So, ψU∈QpℂY. Hence, ψ is Qp-closed mapping.(3)⇒ (1) Let F∈ℱX. Since ψ is Qp-closed, ψ−1−1F=ψF∈QpℂY. Hence, ψ−1 is Qp-continuous mapping.Remark 4.
Composition of twoQp-open (Qp-closed) mappings do not need to be Qp-open (Qp-closed) as shown by the following example.Example 14.
Assume thatX=1,2,3 (i.e., X,τ˜ be topological space, τ˜=X,ϕ,1,2,3), Y=u,v,w (i.e., Y,σ˜ be topological space, σ˜=Y,ϕ,u), and Z=r,s,t (i.e., Z,θ˜ be topological space, θ˜=Z,ϕ,s,r,s).(1)
ConsiderℱY,ℂpY,OpY,GℂpY,GOpY,QpℂY, and QpOY are computing in Example 12. Suppose ψ:X⟶Y be a mapping defined by(32)ψ1=u,ψ2=v,andψ3=w.Thus,ψ is Qp-open (Qp-closed) mapping.(2)
Supposeφ:Y⟶Z be a mapping defined by(33)φu=s,φv=r,andφw=t.
Then,(34)ℱZ=Z,ϕ,t,r,t,ℂpZ=Z,ϕ,t,r,r,t,GℂpZ=QpℂZ=Z,ϕ,r,t,s,t,r,t,OpZ=Z,ϕ,s,s,t,r,s,GOpZ=QpOZ=Z,ϕ,r,s,r,s,s,t.
Thus,φ is Qp-open (Qp-closed) mapping.
From (1) and (2), as3 is open in X, φψ3=φw=t∉QpOZ. Therefore, φ°ψ is not Qp-open mapping. Also, as 1,2 is closed in X, φψ1,2=φu,v\\=r,s∉QpℂZ. Therefore, φ°ψ is not Qp-closed mapping.


## 4.Qp-ℝ0 and Qp-ℝ1 Spaces

Definition 11.
(1)
KQpA is called Qp-kernel of A (i.e., A be subset of a space X) if(35)KQpA=∩L∈QpOX|A⊂L.(2)
KQpx is called Qp-kernel of x (i.e., x be a point of a space X) if(36)KQpx=∩L∈QpOX|x∈L.Lemma 6.
The following properties are holding inX,τ˜. Then,(1)
y∈KQpx⇔x∈CQpyx∈X(2)
KQpA=∩x∈X|CQpx∩A≠ϕProof.
It is similar to Lemmas 3.1 and 3.2 of [16].Lemma 7.
For any elementsx and y in X,τ˜, then the following two properties are equivalent:(1)
KQpx≠KQpy(2)
CQpx≠CQpyProof.
It is similar to Lemma 3.6 of [16].Definition 12.
We callQp-ℝ0 space in X,τ˜ if ∀x∈L∈QpOX such that CQpx⊂L.Theorem 19.
The following two properties are holding inX,τ˜:(1)
Every pre-ℝ0 space is Qp-ℝ0 space(2)
Everygp-ℝ0 space is Qp-ℝ0 spaceProof.
It follows from Theorem6.Theorem 20.
Letx,y∈X. Then, Qp-ℝ0 space in X,τ˜⇔CQpx≠CQpy implies CQpx∩CQpy=ϕ.Proof.
From Definition12, the proof is clear.Theorem 21.
Letx,y∈X. Then, Qp-ℝ0 space in X,τ˜⇔KQpx≠KQpy implies KQpx∩KQpy=ϕ.Proof.
From Lemmas6 (1), Lemma 7, Definition 12, and Theorem 20, the proof is clear.Theorem 22.
The following five properties are equivalent inX,τ˜:(1)
X,τ˜ is an Qp-ℝ0 space(2)
For anyA≠ϕ and L∈QpOX such that A∩L≠ϕ, there exists F∈QpℂX such that A∩F≠ϕ and F⊂L(3)
For anyL∈QpOX,L=∪F∈QpℂX|F⊂L(4)
For anyF∈QpOX,F=∩L∈QpOX|F⊂L(5)
For anyx∈X, CQpx⊂KQpxProof.
It is similar to Theorem 3.8 of [16].Corollary 4.
The following two properties are equivalent inX,τ˜:(1)
X,τ˜ is an Qp-ℝ0 space(2)
CQpx=KQpx∀x∈XProof.
From Definition12 and Theorem 22, the proof is clear.Theorem 23.
The following two properties are equivalent inX,τ˜:(1)
X,τ˜ is an Qp-ℝ0 space(2)
x∈CQpy⇔y∈CQpx∀x,y∈XProof.
The proof is clear.Theorem 24.
The following four properties are equivalent inX,τ˜:(1)
X,τ˜ is an Qp-ℝ0 space(2)
IfF∈QpℂX, then F=KQpF(3)
IfF∈QpℂX and x∈F, then KQpx⊂F(4)
Ifx∈X, then KQpx⊂CQpxProof.
From Lemma6, Theorem 23, and Definition 12, the proof is clear.Definition 13.
We callX,τ˜ is Qp-ℝ1 space if(i)
∀x,y∈X with CQpx≠CQpy(ii)
There exist disjointQp-open sets L and M s.t. CQpx⊆L and CQpy⊆MTheorem 25.
LetX,τ˜ be a topological space. Then,(1)
Every pre-ℝ1 space is Qp-ℝ1 space(2)
Everygp-ℝ1 space is Qp-ℝ1 spaceProof.
It is obvious.Theorem 26.
IfX,τ˜ is an Qp-ℝ1 space, then X,τ˜ is Qp-ℝ0.Proof.
It is from Definitions12 and 13.


## 5. Conclusion

We proposed novel notions (i.e.,Qp-closed set, Qp-open set, Qp-continuous mapping, Qp-open mapping, and Qp-closed mapping) and explained the basic interesting relations and properties of above notions. The relationship among the Qp-closed sets (resp., Qp-open sets) and other sets are given in Figure 1 (resp., Figure 2). Finally, a novel two separation axioms (i.e., Qp-ℝ0 and Qp-ℝ1) based on the notion of Qp-open set and Qp-closure are discussed. In the future, we will add new works (i.e., weakly Qp-ℝ0 space) and also extend several results from [18, 19] to Qp-closed sets.Figure 1
The relationship among theQp-closed sets and other sets.Figure 2
The relationship among theQp-open sets and other sets.



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