# A Note on Quotient Reflective Subcategories of O-REL

Authors: Muhammad Qasim; Ch. Muhammad Afaq Aslam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we examine the category of ordered-RELspaces. We show that it is a normalized and geometric topological category and give the characterization of localT¯0, local T0′, and local T1 ordered-RELspaces. Furthermore, we characterize explicitly several notions of T0’s and T1 objects in O-REL and study their mutual relationship. Finally, it is shown that the category of T0’s (resp. T1) ordered-RELspaces are quotient reflective subcategories of O-REL.


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## Body

## 1. Introduction

Many mathematical concepts were developed to describe certain structures of topology. The concepts of uniform convergences, uniform continuity, Cartesian closedness, completeness, and total boundedness do not exist in general topology. As a remedy, several approaches have been made to define these concepts in topology by mathematicians. For example, the concepts of uniform convergence in the sense of Kent [1] and Preuss [2], of set-convergence in the sense of Wyler [3], Tozzi [4] (which scrutinize filter convergence to bounded subset and generalizes classical point-convergence and supertopologies), of nearness by Bentely [5] and Herrlich [6] (particularly containing proximities and contiguities), and that of hullness by Čech [7] and Leseberg [8] containing the concepts of b-topologies and closures, respectively. In 2018, Leseberg [9] introduced a global concept which embeds the category of the above mentioned concepts into the category of RELspaces and RELmaps as subcategories. This construct, denoted by REL, forms thereby a topological category [9].Classical separation axioms are very common and important ideas in general topology, and have many applications in all fields of mathematics. With the help ofT0 reflection [10], characterizations of locally semi-simple morphisms are obtained in algebraic topology. Furthermore, lower separation axioms can be used in digital topology where they describe digital lines, and in image processing and computer graphs to construct cellular complexes [11–13]. With having the understanding of T0 and T1 separation properties, several mathematicians have extended this idea to arbitrary topological categories [14–18].Classical separation axioms at some pointp (locally) were generalized and have been inspected in [14], where the purpose was to describe the notion of strongly closed sets (resp., closed) in arbitrary set based topological categories [19]. Moreover, the notions of compactness [20], Hausdorffness [14], regular and normal objects [21], perfectness [20], and soberness [22] have been generalized by using the closed and strongly closed sets in some well-defined topological categories over sets [20, 23–26]. Furthermore, the notion of closedness is suitable for the formation of closure operators [27] in several well-known topological categories [28–30].The salient objectives of this study are stated as follows:(i)
To define initial, final, discrete, and indiscrete objects inO-REL(ii)
To characterize localT¯0, local T0′, and local T1 objects in O-REL and examine their mutual relationship(iii)
To give the characterization ofT¯0, T0′, and T1 objects in O-REL and examine their mutual relationship(iv)
To define several structures using ordered-RELspaces and discuss each of theT0 and T1 axioms there and examine their mutual relationship(v)
To examine the quotient-reflective properties of ordered-RELspaces.


## 2. Preliminaries

Recall [31, 32], a functor U:C⟶Set (the category of sets and functions) is called topological if
(i)
U is concrete(ii)
U consists of small fibers(iii)
EveryU-source has a unique initial lift or every U-sink has an unique final lift, i.e., if for every source fj:X⟶Xj,ηjj∈I there exists an unique structure η on X such that g:Y,ζ⟶X,η is a morphism iff for each j∈I, fj∘g:Y,ζ⟶Xj,ηj is a morphism.Moreover, a topological functor is called discrete (respectively, indiscrete) if it has a left (respectively, right) adjoint. In addition, a functor is called a normalized topological functor if constant objects, i.e., subterminals, have an unique structure, and said to be geometric functor if the discrete functor is left exact, i.e., it preserves finite limits [31, 32].LetX be a non-empty, then R⊂P¯X×X is called a relative system for X, and it is denoted by REL(X). Moreover, REL(X) can be ordered by settingR¯<<R iff for each R¯∈R¯, there exists R∈R such that R⊂R¯.Furthermore, we denote bysecR≔R¯⊂X×X:∀R∈R,R∩R¯≠ϕ and by stackR=R¯⊂X×X:∃R∈R,R⊂R¯.Definition 1 (cf. [33]).
LetX≠ϕ, then βX⊂P¯X is called boundedness or B-set on X, if βX satisfies the following axioms:
(i)
ϕ∈βX(ii)
B2⊂B1∈βXimpliesB2∈βX(iii)
a∈Ximpliesa∈βX.
And for B-setsβX and βY a function g:X⟶Y is called bounded iff it satisfies;
(1)gB:B∈βX⊂βY.
ByBOUND we denote the corresponding defined category.Definition 2 (cf. [33]).
The tripleX,βX,r is called RELative space (shortly RELspace) if for the boundedness βX the function r:βX⟶P¯RELX satisfies the following conditions:
(i)
B∈βX and R¯<<R∈rB implies R¯∈rB(ii)
ϕ∉rB for B∈βX(iii)
R∈rϕ iff R=ϕ(iv)
a∈X implies a×a∈ra.
The RELspaceX,βX,r is called ordered-RELspace provided that the following axiom holds:
(v)
ϕ≠B1⊂B∈βX implies rB1⊂rB.Definition 3 (cf. [33]).
LetX,βX,r and Y,βY,v be two RELspaces, then a bounded function g:X⟶Y is called RELative map (shortly RELmap) iff it satisfies the following condition:
(2)B∈βX\ϕandR∈rBimplyinggXR∈vgB,where gXR=g×gR:R∈R with g×gR=g×ga,c:a,c∈R=ga,gc:a,c∈R. By O-REL, we denote the full subcategory of REL, whose objects are the ordered RELspaces. Note that O−REL is a bireflective subcategory of REL [34].Example 4.
LetX,TX be a preuniform convergence space; then, the associated RELspace X,P¯X,rTX can be defined as follows:
(3)rTXϕ=ϕandforB∈P¯X\ϕ,rTXB=R∈RELX:∃N∈TX,N⊂secR.
LetPU-REL denotes the category, whose objects are triples X,P¯X,rTX and morphisms are RELmaps. Note that PUCONV≅PU-REL [9], where PUCONV is the category of preuniform convergence spaces and uniformly continuous maps as defined in [2].Example 5.
LetX,βX,t be a set-convergence space; then, the associated RELspace X,βX,rt can be defined by
(4)rtϕ=ϕandforB∈βX\ϕ,
rtB=R∈RELX:∃E∈FILXE,B∈tandR⊂secE⊗E, where E⊗E=R⊂X×X:∃E1,E∈EsuchthatE1×E⊂R and FIL(X) is the collection of all filters defined on X.
LetSET-REL denotes the category, whose objects are triples X,βX,rt and morphisms are RELmaps. Note that SETCONV≅SET-REL [9], where SETCONV is the category of set-convergence spaces and morphisms are b-continuous maps as defined in [3].Example 6.
Letx,ζ be prenearness space; then, the associated RELspace X,P¯X,rζ can be described as
(5)rζϕ=ϕandforB∈P¯X\ϕ,rζB=R∈RELX:∃Q⊂P¯XB∪Q∈ζandR<<Q×Q,whereQ×Q≔D×D:D∈Q.
Note thatPNEAR≅PN-REL [6, 9], where PNEAR is the category, whose objects are prenearness spaces and morphisms are nearness preserving maps as defined in [6], and PN-REL is the category of triples X,P¯X,rζ and morphisms are RELmaps.Example 7.
For a B-setβX, we put rbϕ≔ϕ, and for B∈βX\ϕ, we set rbB≔R∈RELX:∃x∈B,R⊂ẋ×ẋ; hence, X,βX,rb defines a RELspace, which is diagonal, meaning that for B∈βX\ϕ and R∈sB, we can find x∈B such that ∀R∈R,x,x∈R.
LetΔ-REL be denote the corresponding defined full subcategory of REL; then, Δ-REL≅BOUND.Remark 8.
In this context, note thatBORN, the full subcategory of BOUND, whose objects are the bornological spaces, then also has evidently a corresponding counterpart in REL.Example 9.
LetX,βX,q be b-topological space; then, the associated RELspace X,βX,rq is defined by
(6)rqϕ≔ϕandforB∈βX\ϕ,rqB≔R∈RELX:∃ω⊂βX,∃a∈BR<<ω×ωanda∈∩qE:E∈ω.Note thatb-TOP≅bTOP-REL [9], where bTOP-REL denotes the full subcategory of REL, whose objects are triples X,βX,rq, and b-TOP denotes the category of b-topological spaces and b-continuous maps as defined in [9].


## 3.O−REL as a Normalized and Geometric Topological Category

Note that the forgetful functorU:C⟶Set, where C=REL is topological in the following sense:Lemma 10.
LetXj,βXj,rj be a collection of RELspaces. A source fi:X,βIX,rIX⟶Xj,βXj,rjj∈I is initial in REL iff
(7)βIX≔B⊂X:gjB∈βXj,∀j∈I,and for all B∈βIX,
(8)rIXB≔R∈RELX:gXjR∈rjgjB,∀j∈I.Proof.
It is given in [34]. Consequently, since O-REL is a full and isomorphism-closed subcategory which is bireflective in REL, it is topological, too.Lemma 11.
LetXj,βXj,rj be a collection of ordered-RELspaces. A sink fi:Xj,βXj,rj⟶X,βfinX,rfinj∈I is final in O−REL iff
(9)βfinX≔B⊂X:∃j∈I,∃Bj∈βjX∣B⊂gjBj∪DX,where DX=∅∪a:a∈X, and for B∈βfinX\ϕ,
(10)rfinB≔R∈RELX:∃j∈I,∃Bj∈βXj,∃Rj∈rjBj∣R<<gjXRj∪,R∈RELX:∃a∈B∣a,a∈∩R:R∈Rwithrfinϕ≔ϕ.Proof.
It is easy to observe thatX,βfinX,rfin is an ordered-RELspace and fi:Xj,βXj,rjj∈I⟶X,βfinX,rfin is a RELmap. Suppose that g:X,βfinX,rfin⟶Y,βY,rY is a mapping. We show that g is a RELmap iff g∘fj is a RELmap. Necessity is obvious since the composition of two RELmaps is RELmap again.
Conversely, letg∘fj:Xj,βXj,rj⟶Y,βY,rY be a RELmap.
Then, first, we show thatg is a bounded map. Let Bi∈βjX; it implies that gfjBj=g∘fjBj∈βY. For our own convenience, take fjBj=B′, and since fj is a RELmap, then B′∈βfinX, and consequently, g is bounded.
Now, letBj∈βXj\ϕ and Ri∈rjBj. By the Definition 3, we have gfjBj=g∘fjBj∈rYgfjBj. On the other hand, fj is a RELmap; it follows that fjRj∈rfinfjBj. Take fjRj=R′. Then, we have R′∈rfinB′, and subsequently, gR′∈rYgB′ which shows g is a RELmap.Lemma 12.
LetX≠ϕ, and X,βX,r be an ordered-RELspace.
(i)
A RELstructureβX,r is discrete iff βX,r≔DX,rdis, where DX=∅∪a:a∈X and rdisa=R∈RELX:a,a∈∩R:R∈R=R∈RELX:R<<a,a with rdisϕ≔ϕ(ii)
A RELstructureβX,r is indiscrete iff βX,r≔P¯X,rid, where ridB=R∈RELX:ϕ∉R if βX≠ϕ with ridϕ≔ϕ.Proof.
By applying Lemma11, we get the desired result.Remark 13.
The topological functorU:C⟶Set, where C=O−REL is normalized since an unique RELstructure βX=∅, and r∅=∅ exists whenever X=∅ and a unique RELstructure βX=∅,a, r∅=∅ and ra=∅,a,a exists whenever X=a. Furthermore, the topological functor U:O−REL⟶Set is geometric since the regular sub-object of a discrete RELspace is discrete, and finite product of discrete RELstructures is discrete again.


## 4. LocalT0 and Local T1 Ordered-RELspaces

In this section, we define notions forT0 and T1 ordered-RELspaces at some point.LetX be any set and p∈X. We define the wedge product ofXatp as the two disjoint copies of X at p and denote it as X∨pX. For a point a∈X∨pX, we write it as a1 if a belongs to the first component of the wedge product; otherwise, we write a2 that is in the second component. Moreover, X2 is the cartesian product of X.Definition 14 (cf. [14]).
(i)
A mappingAp:X∨pX⟶X2 is said to be principalp-axis mapping provided that(11)Apaj≔a,p;j=1,p,a;j=2,(ii)
A mappingSp:X∨pX⟶X2 is said to be skewedp-axis mapping provided that(12)Spaj≔≔a,a;j=1,p,a;j=2,(iii)
A mapping∇p:X∨pX⟶X is said to be fold mapping atp provided that(13)∇paj≔a,j=1,2.
Assume thatU:C⟶Set is a topological functor, X∈ObjC with UX=Z and p∈Z.Definition 15 (cf. [14]).
(i)
X is T¯0 at p provided that the initial lift of the U-source Z∨pZ⟶ApUX2=Z2andZ∨pZ⟶∇pUDZ=Z is discrete(ii)
X is T0′ at p provided that the initial lift of the U-source Z∨pZ⟶idUX∨pX=Z∨pZandZ∨pZ⟶∇pUDZ=Z is discrete, where X∨pX is the wedge product in C, i.e., the final lift of the U-sink UX=Z⟶i1,i2Z∨pZ, where i1,i2 represent the canonical injections(iii)
X is T1 at p provided that the initial lift of the U-source Z∨pZ⟶SpUX2=Z2andZ∨pZ⟶∇pUDZ=Z is discrete.Remark 16.
(i)
InTOP, T¯0 and T′0 at p (respectively, T1 at p) are equivalent to the classical T0 at p (respectively, the classical T1 at p), i.e., for each a∈X with a≠p, there exists a neighborhood Na of ``a" not containing ``p" or (respectively, and); there exists a neighborhood Np of ``p" not containing ``a" [35](ii)
A topological spaceX is T0 (respectively T1) iff X is T0 (respectively T1) at p for each p∈X [35](iii)
LetU:C⟶Set be a topological functor, X∈ObjC and p∈UX be a retract of X. Then, if X is T¯0 or T1 at p, then X is T0′ at p but not conversely in general [36].Theorem 17.
LetX,βX,r be ordered-RELspace and p∈X. Then, X,βX,r is T¯0 at p if and only if for each a∈X with a≠p, the following holds:
(i)
a,p∉βX(ii)
R∈RELX:R<<a,p∉raorR∈RELX:R<<p,a∉rp(iii)
R∈RELX:R<<a,p∉rp or R∈RELX:R<<p,a∉ra(iv)
R∈RELX:R<<a,a,p,p∉ra or R∈RELX:R<<p,p,a,a∉rp.Proof.
LetX,βX,r be T¯0 at p; we show the conditions i to iv are holding:
(i)
Suppose thata,p∈βX for all a∈X with a≠p. Let U=a1,a2∈X∨pX, then since ∇pU=∇pa1,a2=∇pa1,∇pa2=a∈DX and for j=1,2, πjApU=a,p∈βX (by the assumption), where πj:X2⟶X for j =1,2 are projection maps. By Definitions 1 and 15 and Lemma 10, a contradiction, it follows a,p∉βX(ii)
Assume thatR∈RELX:R<<a,p∈ra and R∈RELX:R<<p,a∈rp. Particularly, let R1=a1,a2∈RELX∨pX and B=a1∈DX∨pX\∅; then, ∇pR1=∇pa1,a2=a,a∈rdisa. By the assumption, π1ApR1=π1Apa1,π1Apa2=a,p∈ra and π2ApR1=π2Apa1,π2Apa2=p,a∈rp. Since X,βX,r is T¯0 at p, it follows that R1∈r¯disa1, where r¯dis is the discrete structure on X∨pX.
Similarly, forB=a2∈DX∨pX\∅, we get R1∈r¯disa2, a contradiction to the discreteness of r¯disB.
Thus,R∈RELX:R<<a,p∉ra or R∈RELX:R<<p,a∉rp.
(iii)
Suppose thatR∈RELX:R<<a,p∈rp and R∈RELX:R<<p,a∈ra. In particular, let R2=a2,a1∈RELX∨pX and B=a1∈DX∨pX\∅; then, ∇pR1=∇pa2,a1=a,a∈rdisa, and by the assumption π1ApR2=p,a∈ra and π2ApR2=a,p∈rp. Since X,βX,r is T¯0 at p, we get that R2∈r¯disa1, where r¯dis is the discrete structure on X∨pX
Similarly, forB=a2∈DX∨pX\∅, we get R2∈r¯disa2, a contradiction.
Therefore,R∈RELX:R<<a,p∉rp or R∈RELX:R<<p,a∉ra.
(iv)
Assume thatR∈RELX:R<<a,a,p,p∈ra and R∈RELX:R<<p,p,a,a∈rp. Let R3=a1,a1,a2,a2∈RELX∨pX and B=a1∈DX∨pX\∅; then, ∇pR3=∇pa1,a1,a2,a2=a,a∈rdisa, π1ApR3=a,a,p,p∈ra, π2ApR3=p,p,a,a∈rp (by the assumption). Since X,βX,r is T¯0 at p, it follows that R3∈r¯disa1, where r¯dis is the discrete structure on X∨pX.
Similarly, forB=a2∈DX∨pX\∅, we get R3∈r¯disa2, a contradiction.
Hence,R∈RELX:R<<a,a,p,p∉ra or R∈RELX:R<<p,p,a,a∉rp.
Conversely, supposei to iv are holding.
LetβX∨pX,r¯ be the initial structure induced by Ap:X∨pX⟶X2,βX2,r2 and ∇p:X∨pX⟶X,DX,rdis, where BX2,r2 is the product RELstructure on X2 and DX,rdis the discrete RELstructure on X.
We show thatβX∨pX,r¯ is the discrete REL structure on X∨pX, i.e., we show that βX∨pX=DX∨pX=∅∪aj;j=1,2andaj∈X∨pX and for B∈DX∨pX, r¯B=R¯∈RELX∨pX:R¯<<aj,aj;j=1,2.
LetU∈βX∨pX and ∇pU∈DX; if ∇pU=∅, then U=∅. Suppose ∇pU≠∅. Then, we have ∇pU=a for some a∈X, and if a=p, then U=p; let a≠p; then, it further implies that U=a1 or U=a2 and U=a1,a2. By the assumption, πjApU=πjApa1,a2=a,p∉βX (for j =1,2). Thus, U=a1 and U=a2; subsequently, βX∨pX=DX∨pX.
Now,B∈DX∨pX\∅ implies B=a1 and B=a2, and by Lemma 10, r¯B = R¯∈RELX∨pX:πjApR¯∈rπjApBand∇pR¯∈rdis∇pB,wherej=1,2.
SupposeB=a1, then
r¯a1=R¯∈RELX∨pX:πjApR¯∈rπjApa1and∇pR¯∈rdis∇pa1,wherej=1,2; it follows that r¯a1=R¯∈RELX∨pX:π1ApR¯∈raandπ2ApR¯∈rpand∇pR¯∈rdisa.
Since∇pR¯∈rdisa=R∈RELX:R<<a,a, we have the following possibilities of R:
R¯∈RELX∨pX:R¯<<a1,a1,
R¯∈RELX∨pX:R¯<<a2,a2,
R¯∈RELX∨pX:R¯<<a1,a2,
R¯∈RELX∨pX:R¯<<a2,a1,
R¯∈RELX∨pX:R¯≪a1,a1,a2,a2.
Case (i). SupposeR¯∈RELX∨pX:R¯<<a1,a1. It follows that for all R¯∈R¯ such that a1,a1⊆R¯,andπ1Apa1,a1⊆π1ApR¯, π1ApR¯<<π1Apa1,a1=π1Apa1,π2Apa1=a,a. By Definition 2, π1ApR¯<<a,a∈ra. Similarly, π2ApR¯<<p,p∈rp. Therefore, R¯∈RELX∨pX:R<<a1,a1 holdsCase (ii).R¯∈RELX∨pX:R¯<<a2,a2 holds. The proof is similar to Case (i)Case (iii). LetR¯∈RELX∨pX:R¯<<a1,a2. It follows that for all R¯∈R¯ such that a1,a2⊆R¯, and π1Apa1,a2⊆π1ApR¯, π1ApR¯<<π1Apa1,a2=a,p. By the assumption, we get π1ApR¯<<a,p∉ra. Similarly, π2ApR¯<<p,a∉rp. Thus, R¯∈RELX∨pX:R¯<<a1,a2 cannot be possibleCase (iv). Similar to Case (iii), we conclude thatR¯∈RELX∨pX:R¯<<a2,a1 is not possibleCase (v). IfR¯∈RELX∨pX:R¯<<a1,a1,a2,a2. It follows that for all R¯∈R¯ such that a1,a1,a2,a2⊆R¯, and π1Apa1,a1,a2,a2⊆π1ApR¯, for all R¯∈R¯ implying π1ApR¯<<π1Apa1,a1,a2,a2=a,a,p,p. By the assumption, π1ApR¯<<a,a,p,p∉ra. Similarly, π2ApR¯<<p,p,a,a∉rp. Hence, R¯∈RELX∨pX:R¯<<a1,a1,a2,a2 is not possible.Similarly, ifB=a2, only Case (i) and Case (ii) are holding. By Lemma 12, r¯B=R¯∈RELX∨pX:R¯<<aj,aj;j=1,2 is discrete.Therefore, by Definition15, X,βX,r is T¯0 at p.Theorem 18.
LetX,βX,r be an ordered-RELspace and p∈X.
X,βX,r is T1 at p if and only if for any a∈X with a≠p, the following holds:
(i)
a,p∉βX(ii)
R∈RELX:R<<a,p∉ra and R∈RELX:R<<p,a∉rp(iii)
R∈RELX:R<<a,p∉rp and R∈RELX:R<<p,a∉ra(iv)
R∈RELX:R<<a,a,p,p∉ra and R∈RELX:R<<p,p,a,a∉rp.Proof.
By following the same technique used in Theorem17, and replacing the mapping Ap by the mapping Sp, we get the proof.Theorem 19.
All ordered-RELspaces areT′0 at p.Proof.
LetX,βX,r be ordered-RELspace and p∈X. By Definition 15, we show that for each U∈βX∨pX,U⊂ikV (where k=1,2) for some V∈βX and ∇pU∈DX. ∇pU=ϕ implying U=ϕ. Suppose ∇pU≠ϕ, it implies that ∇pU=a for some a∈X. If a=p, then ∇pU=p implying U=p.
Supposea≠p, it follows that U=a1, a2 or a1,a2. If U=a1,a2, then a1,a2⊂i1V for some V∈βX which shows that a2 should be in the first component of the wedge product X∨pX, a contradiction. In similar manner, a1,a2⊂i2V for some V∈βX. Hence, U=a1,a2. Thus, we must have U=aj for j=1,2 only and consequently, βX∨pX=DX∨pX, the discrete RELstructure on X∨pX.
Now, forB∈DX∨pX\ϕ, by Lemma 10, r¯B = R¯∈RELX∨pX:R¯<<i1sforsomes∈rB,R¯<<i2sforsomes∈rBand∇pR¯∈rdis∇pB. Since ∇pR¯∈rdisB=R∈RELX:R<<aj,ajwherej=1,2, we have the following possibilities of R:
R¯∈RELX∨pX:R¯<<a1,a1,
R¯∈RELX∨pX:R¯<<a2,a2,
R¯∈RELX∨pX:R¯<<a1,a2,
R¯∈RELX∨pX:R¯<<a2,a1,
R¯∈RELX∨pX:R¯<<a1,a1,a2,a2.
In particular, forR¯∈RELX∨pX:R¯<<a1,a2. It follows that, for all R¯∈R¯ such that a1,a2⊂R¯, and (for k =1,2), ika1,a2⊂iKR¯ implying ikR¯<<ika1,a2. It follows a2 (respectively, a1) in the first (respectively, second) component of the wedge product X∨pX, a contradiction. Similarly, for R¯∈RELX∨pX:R¯<<a2,a1 and R¯∈RELX∨pX:R¯<<a1,a1,a2,a2, we get a contradiction.
Therefore,r¯B=R¯∈RELX∨pX:R<<aj,aj;j=1,2. Consequently, by Definition 15(i) and Lemma 10, X,βX,r is T′0 at p.


## 5.T0 and T1 Ordered-RELspaces

In this section, we define generically notions ofT0 and T1 in ordered-RELspaces.The characterization ofT0 objects in categorical topology has been an important idea in a topological universe. Therefore, several attempts has been made such as in 1971 Brümmer [15], in 1973 Marny [18], in 1974 Hoffman [17], in 1977 Harvey [16], and in 1991 Baran [14] to discuss various approaches to generalize classical T0 object and examined the relationship between different forms of generalized T0 objects. One of the main purposes of generalization is to define Hausdorff objects in arbitrary topological categories. In 1991, Baran [14, 37] also generalizes the classical T1 objects of topology to topological categories [14, 37]. In abstract topological categories [21], T1 objects are used to define T3,T4, normal objects, regular, and completely regular. To characterize separation axioms, Baran’s approach was to use initial and final lifts and discreteness.In 1991, Baran [14] used the generic element method of topos theory introduced by Johnstone [38], to define generic separation axioms, due to the fact that points does not make sense in topos theory. In general, the wedge product X∨pX at p can be replaced by X2∨ΔX2 at diagonal Δ. Any element a,b∈X2∨ΔX2 is written as a,b1 (resp., a,b1) if it lies in the first (resp., second) component of X2∨ΔX2. Clearly, a,b1=a,b2, if and only if a=b.Definition 20 (cf. [14]).
(i)
A mappingA:X2∨ΔX2⟶X3 is called principal axis mapping provided that(14)Aa,bj≔a,b,a;j=1,a,a,b;j=2,(ii)
A mappingS:X2∨ΔX2⟶X3 is called skewed axis mapping provided that(15)Sa,bj≔a,b,b;j=1,a,a,b;j=2,(iii)
A mapping∇:X2∨ΔX2⟶X2 is called fold mapping provided that(16)∇a,bj≔a,b,j=1,2.Any elementa,b∈X2∨ΔX2 is written as a,b1 (resp., a,b1) if it lies in the first (resp., second) component of X2∨ΔX2. Clearly, a,b1=a,b2 if and only if a=b.Now, we replace the pointp by any generic point δ and define the following separation axioms.Definition 21.
LetU:C⟶Set be a topological functor, X∈ObjC with UX=Z.
(i)
X is T0¯ provided that the initial lift of the U-source Z2∨ΔZ2⟶AUX3=Z3andZ2∨ΔZ2⟶∇UDZ2=Z2 is discrete [14](ii)
X is T′0 provided that the initial lift of the U-source Z2∨ΔZ2⟶idUZ2∨ΔZ2′=Z2∨ΔZ2andZ2∨ΔZ2⟶∇UDZ2=Z2 is discrete, where Z2∨ΔZ2′ is the final lift of the U-sink UX2=Z2⟶i1,i2Z2∨ΔZ2 [14, 39](iii)
X is called T0 provided that Xdoesn’t contain an indiscrete subspace with at least two points [18, 40](iv)
X is T1 provided that the initial lift of the U-source Z2∨ΔZ2⟶SUX3=Z3andZ2∨ΔZ2⟶∇UDZ2=Z2 is discrete [14].Remark 22.
(i)
InTOP, all the properties of being T0, T¯0 and T0′ (respectively, T1) are equivalent to those classical ones which are T0 (respectively, T1), i.e., for each a,b∈X with a≠b, there exists a neighbourhood Na of ``a" not containing ``b" or (respectively and), there exists a neighbourhood Nb of ``b" not containing ``a" [14, 18, 40](ii)
In any topological category,T¯0 implies is T′0 but not conversely in general. Also, each of the T¯0 and T′0 has no relation to a T0 [39](iii)
LetU:C⟶Set be a topological functor, X∈ObjC and p∈UX be a retract of X. Then, if X is T¯0 (respectively T1), then X is T¯0 at p (respectively T1 at p) but not conversely in general [36].Theorem 23.
LetX,βX,r be an ordered-RELspace.
X,βX,r is T¯0 iff for each a,b∈X with a≠b, the following holds:
(i)
a,b∉βX(ii)
R∈RELX:R<<a,b∉ra or R∈RELX:R<<b,a∉rb(iii)
R∈RELX:R<<a,b∉rb or R∈RELX:R<<b,a∉ra(iv)
R∈RELX:R<<a,a,b,b∉ra or R∈RELX:R<<b,b,a,a∉rb.Proof.
SupposeX,βX,r is T¯0, we show that conditions i to iv are holding.
(i)
Suppose thata,b∈βX for each a,b∈ X, a≠b. Let U=a,b1,a,b2∈X2∨ΔX2. Note that ∇U=∇a,b1,a,b2=a,b∈DX2 and π1AU=a∈βX. By the assumption, πkAU=πkAa,b1,a,b2=a,b∈βX, where πk:X3⟶X2 (for k =2,3) are projection maps. By Definitions 1 and 15 and Lemma 10, it leads to a contradiction, it follows that a,b∉βX(ii)
Suppose thatR∈RELX:R<<a,b∈ra and R∈RELX:R<<b,a∈rb. Let R1=a,b1,a,b2∈RELX2∨ΔX2 and B=a,b1∈DX2∨ΔX2\ϕ, then ∇R1=∇a,b1,a,b2=∇a,b1,∇a,b2=a,b∈rdis2a,b. By Definition 2, π1Aa,b1,a,b2=π1Aa,b1,π1Aa,b2=a,a∈ra and by the assumption, π2Aa,b1,a,b2=b,a∈rb and π3Aa,b1,a,b2=a,b∈ra. Since X,βX,r is T¯0, we conclude R1∈r¯dis2a,b1, where r¯dis2 is the discrete structure on X2∨ΔX2
Similarly, forB=a,b2∈DX2∨ΔX2\ϕ, we get R1∈r¯dis2a,b2, a contradiction.
Therefore,R∈RELX:R<<a,b∉ra or R∈RELX:R<<b,a∉rb.
(iii)
Suppose thatR∈RELX:R<<a,b∈rb and R∈RELX:R<<b,a∈ra. In particular, let R2=a,b2,a,b1∈RELX2∨ΔX2 and B=a,b1∈DX2∨ΔX2\ϕ, then ∇R2=∇a,b2,a,b1=a,b∈rdis2a,b. By Definition 2, π1Aa,b2,a,b1=a,a∈ra and by the assumption, π2Aa,b2,a,b1=π2Aa,b2,π2Aa,b1=a,b∈rb and π3Aa,b2,a,b1=π3Aa,b2,π3Aa,b1=b,a∈ra. Since X,βX,r is T¯0 it follows that R2∈r¯dis2a,b2, where r¯dis2 is the discrete structure on X2∨ΔX2
Similarly, forB=a,b2∈DX2∨ΔX2\ϕ, we get R2∈r¯dis2a,b1, a contradiction.
Thus,R∈RELX:R<<a,b∉rb or R∈RELX:R<<b,a∉ra.
(iv)
Suppose thatR∈RELX:R<<a,a,b,b∈ra and R∈RELX:R<<b,b,a,a∈rb. Let R3=a,b1,a,b1,a,b2,a,b2∈RELX2∨ΔX2, and B=a,b1∈DX2∨ΔX2\ϕ; then, ∇R3=∇a,b1,a,b1,a,b2,a,b2=a,b∈rdis2a,b, and by Definition 2, π1Aa,b1,a,b1,a,b2,a,b2=a,a∈ra. By the assumption, π2Aa,b1,a,b1,a,b2,a,b2=b,b,a,a∈rb and π3Aa,b1,a,b1,a,b2,a,b2=a,a,b,b∈ra. Since X,βX,r is T¯0, we conclude R3∈r¯dis2a,b1, where r¯dis2 is the discrete structure on X2∨ΔX2.
Similarly, forB=a,b2∈DX2∨ΔX2\ϕ, we get R3∈r¯dis2a,b2, a contradiction to the discreteness of r¯dis2B.
Hence,R∈RELX:R<<a,a,b,b∉ra or R∈RELX:R<<b,b,a,a∉rb.Conversely, supposei to iv are holding.LetβX2∨ΔX2,r¯2 be the initial structure induced by A:X2∨ΔX2⟶X3,βX3,r3 and ∇:X2∨ΔX2⟶X2,DX2,rdis2, where βX3,r3 is the product RELstructure on X3 and DX2,rdis2 the discrete RELstructure on X2.We show thatβX2∨ΔX2,r¯2 is the discrete RELstructure on X2∨ΔX2, i.e, βX2∨ΔX2=DX2∨ΔX2=ϕ∪a,bj:a,bj∈X2∨ΔX2forj=1,2 and for B∈DX2∨ΔX2\ϕ,r¯2B=R¯∈RELX2∨ΔX2:R¯<<a,bj,a,bj,j=1,2.LetU∈βX2∨ΔX2 and ∇U∈BX2. If ∇U=ϕ, then U=ϕ. Suppose ∇U≠ϕ, then it follows that ∇U=a,b for some a,b∈X2. If a=b, then U=b,b. Next, let a≠b; then, we have U=a,b1 or U=a,b2 or U=a,b1,a,b2 and π1AU=π1Aa,b1,a,b2=π1Aa,b1,π1Aa,b2=a,a, and by the assumption, we get πkAa,b1,a,b2=a,b∉βX, (for k =2,3). Thus, U=a,b1 or U=a,b2, and subsequently, βX2∨ΔX2=DX2∨ΔX2.Now,B∈βX2∨ΔX2\ϕ implies B=a,b1 and B=a,b2, and by Lemma 10, r¯2B=R¯∈RELX2∨ΔX2:πjAR¯∈rπjABand∇R¯∈rdis2∇B,wherej=1,2,3.SupposeB=U=a,b1, thensince ∇R∈rdis2a,b=R∈RELX2:R<<a,b,a,b, we have the following possibilities:R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b2,R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b2,R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b1,R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,a,b2,a,b2.
Case (i). IfR¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1. It follows that for all R¯∈R¯a,b1,a,b1⊆R¯ and π1Aa,b1,a,b1⊆π1AR¯, π1AR¯<<π1Aa,b1,a,b1=π1Aa,b1,π1Aa,b1=a,a, and by the Definition 2, we get π1AR¯<<a,a∈ra.In a similar way,π2AR¯<<b,b∈rb and π3AR¯<<a,a∈ra.Thus,R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1 holdsCase (ii).R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b2 holds. The proof is similar to Case (i)Case (iii).R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b2. It follows that, for all R¯∈R¯ such that a,b1,a,b2⊆R¯. And π1Aa,b1,a,b2⊆π1AR¯, π1AR¯<<π1Aa,b1,a,b2=a,a, and by Definition 2π1AR¯<<a,a∈raSimilarly, by the assumptionπ2AR¯<<b,a∉rb and π3AR¯<<a,b∉ra.Therefore,R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b2 is not possible.Case (iv). Similar to Case (iii), we conclude thatR¯∈RELX2∨ΔX2:R¯<<a,b2,a,b1 is not possible.
Case (v). IfR¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,a,b2,a,b2. It follows that, for all R¯∈R¯ such that a,b1,a,b1,a,b2,a,b2⊆R¯ and π1Aa,b1,a,b1,a,b2,a,b2⊆π1AR¯ implies π1AR¯<<π1Aa,b1,a,b1,a,b2,a,b2=a,a. By Definition 2, π1AR¯<<a,a∈ra.Similarly, by the assumption,π2AR¯<<b,b,a,a∉rb and π3AR¯<<a,a,b,b∉rb.Hence,R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,a,b2,a,b2 is not possible.Similarly, if B=a,b2 only Case (i) and Case (ii) are holding. By Lemma 12, R¯∈RELX2∨ΔX2:R¯<<a,bj,a,bj,j=1,2 is discrete. Therefore, by Definition 21 (i), X,βX,r is T¯0.Theorem 24.
LetX,βX,r be an ordered-RELspace.
X,βX,r is T0 iff for each a,b∈X with a≠b, each of the following conditions are satisfied:
(i)
a,b∉βX(ii)
R∈RELX:R<<a,b∉ra or R∈RELX:R<<a,b∉rb(iii)
R∈RELX:R<<b,a∉ra or R∈RELX:R<<b,a∉rb(iv)
R∈RELX:R<<a,a∉rb or R∈RELX:R<<b,b∉ra.Proof.
LetX,βX,r be T0, a,b∈βX and R∈RELX:R<<a,b∈ra and R∈RELX:R<<a,b∈rb and R∈RELX:R<<b,a∈ra and R∈RELX:R<<b,a∈rb and R∈RELX:R<<a,a∈ra and R∈RELX:R<<b,b∈rb.
LetU=a,b. Note that U,βU,rU is the subspace of X,βX,r, where βU,rU is the initial lift of the ordered-RELsystem induced by the inclusion map i:S⟶U and for any S⊂U, S∈βU, whenever iS=S∈βU and for any R∈RELU,R∈rS, whenever iR=R∈rB.
By the assumption,iU=U=a,b∈βU and by Definition 1, we get βU=P¯U.
Now, for anyR∈RELU let R=a,a∈RELU. By Definition 2, ia,a=a,a∈ra. By the assumption, R=a,a∈rb implying that R∈RELX:R<<a,a∈ra and R∈RELX:R<<a,a∈rb.
Similarly, forR=b,b∈RELU, it follows that R∈RELX:R<<b,b∈ra and R∈RELX:R<<b,b∈rb.
Now, ifR=a,b∈RELU then by the assumption, R∈RELX:R<<a,b∈ra and R∈RELX:R<<a,b∈rb.
And forR=b,a∈RELU then by the assumption, R∈RELX:R<<b,a∈ra and R∈RELX:R<<b,a∈rb.
Therefore,rU=R∈RELU:ϕ∈R and βU,rU=P¯U,rid, which is a contradiction by Lemma 12. Thus i−iv are holding.
Conversely, suppose that for alla,b∈X with a≠b, conditions i−iv are holding. We show that the initial structure βU,rU is not an indiscrete ordered-RELstructure on U. Let U=a,b⊂X. By the assumption, a,b∉βX and R∈RELX:R<<a,b∉ra or R∈RELX:R<<a,b∉rb and R∈RELX:R<<b,a∉ra or R∈RELX:R<<b,a∉rb and R∈RELX:R<<a,a∉rb or R∈RELX:R<<b,b∉ra. Thus, U,βU,r is not an indiscrete ordered-RELsubspace of X,βX,r. Hence, by Definition 21 (iii), X,βX,r is T0.Theorem 25.
LetX,βX,r be an ordered-RELspace. Then, X,βX,r is T1 iff for all a,b∈X with a≠b, the following holds:
(i)
a,b∉βX(ii)
R∈RELX:R<<a,b∉ra and R∈RELX:R<<b,a∉rb(iii)
R∈RELX:R<<a,b∉rb and R∈RELX:R<<b,a∉ra(iv)
R∈RELX:R<<a,a,b,b∉ra and R∈RELX:R<<b,b,a,a∉rb.Proof.
Similarly, using Theorem23, and replacing mapping A by the mapping S, we obtain the proof.Theorem 26.
All ordered-RELspaces areT′0.Proof.
LetX,βX,r be an ordered-RELspace. By Definition 21, we show that for any U∈βX2∨ΔX2, U⊂ikV (where k=1,2) for some V∈βX2 and ∇U∈DX2. If ∇U=ϕ implies U=ϕ. Suppose ∇U≠ϕ, hence ∇U=a,b for some a,b∈X2.
Supposea≠b, it follows that U=a,b1ora,b2ora,b1,a,b2. If U=a,b1,a,b2, then a,b1,a,b2⊂i1V for some V∈βX2, which shows that a,b2 must be in the first component of X2∨ΔX2, a contradiction. Similarly, a,b1,a,b2⊂i2V, for V∈βX2. Hence, U=a,bj for j=1,2. Consequently, βX2∨ΔX2=DX2∨ΔX2, the discrete ordered-RELstructure on X2∨ΔX2.
Now, forB∈DX2∨ΔX2\ϕ, and by Lemma 10, r¯2B=R¯∈RELX2∨ΔX2:R¯<<i1sforsomes∈rB,R¯<<i2sforsomes∈rBand∇R¯∈rdis2∇B. But ∇R¯∈rdis2∇B gives the following possibilities:
R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,
R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b2,
R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b2,
R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b1,
R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,a,b2,a,b2.
In particular, forR¯∈RELX2∨ΔX2:R¯<<a,b1,a,b2. Then, it follows, for all R¯∈R¯a,b1,a,b2⊂R¯, and consequently, ika,b1,a,b2⊂R¯ (for k =1,2). As a result, a,b2 (respectively, a,b1) is in the first (respectively, second) component of the wedge product X2∨ΔX2 which leads to a contradiction. Similarly, for R¯∈RELX2∨ΔX2:R¯<<a,b2,a,b1 and R¯∈RELX2∨ΔX2:R¯<<a,b1,a,b1,a,b2,a,b2, we get a contradiction.
Hence,r¯2B=R¯∈RELX2∨ΔX2:R¯<<a,bj,a,bj;j=1,2. Thus, by Lemma 10 and Definition 21, X,βX,r is T′0.Remark 27.
LetX be an ordered-RELspace.
(i)
By Theorems17 and 23, X is T¯0 iff X is T¯0 at p, for each p∈X(ii)
By Theorems18 and 25, X is T1 iff X is T1 at p, for each p∈X(iii)
By Theorems19 and 26, X is T′0 iff X is T′0 at p, for each p∈X(iv)
By Theorems23–26, T1⟹T¯0⟹T0⟹T′0 but the converse does not hold in general.Corollary 28.
LetX,P¯X,r be in PU−REL. Then the following statements are equivalent.
(i)
X,P¯X,r is T0¯(ii)
X,P¯X,r is T0¯PUCONV, where T0¯PUCONV is the category of T0¯ pre-uniform convergence spaces and uniformly continuous maps(iii)
For eacha,b∈X with a=b, and for all B∈P¯X, R∈RELX:R<<a,b∉rB or R∈RELX:R<<b,a∉rB, and R∈RELX:R<<a,a,b,b∉rB.Proof.
By applying Example4, Theorem 23, and Theorem 3.1.10 of [41].Corollary 29.
LetX,P¯X,r be in PU−REL. Then, the following statements are equivalent:
(i)
X,P¯X,r is T1(ii)
X,P¯X,r is T1PUCONV, where T1PUCONV is the category of T1 pre-uniform convergence spaces and uniformly continuous maps(iii)
For alla,b∈X with a=b, and for all B∈P¯X, R∈RELX:R<<a,b∉rB and R∈RELX:R<<b,a∉rB, and R∈RELX:R<<a,a,b,b∉rB.Proof.
This follows from Example4, Theorem 25, and Theorem 3.2.4 of [41].


## 6. Quotient-Reflective Subcategories of the Category of Ordered-RELspaces

Definition 30 (cf. [42]).
Given a topological functorU:C⟶Set, and a full and isomorphism-closed subcategory H of C, we say that H is
(i)
Epireflective inC and closed if and only if H is closed under the formation of products and extremal subobjects (i.e., subspaces)(ii)
Quotient-reflective inC if and only if H is epireflective and is closed under finer structures (i.e., if A∈H, B∈C, UA=UB, and id:A⟶B is a C-morphism, then B∈H).Theorem 31.
(i)
AnyT¯0O-REL, T0O-REL and T1O-REL is a quotient-reflective subcategory of O-REL(ii)
T0′O-REL is a normalized topological constructProof.
i Suppose C=T¯0O−REL and X,βX,r∈C. It can be easily verified that C is a full and isomorphism-closed subcategory of O-REL and closed under finer structures. It remains to show that X is closed under extremal subobjects and closed under the formation of products.
LetA⊂X and βA,rA denotes the sub O-REL structure on A, induced by the inclusion map i:A⟶X. We show that A,βA,rA is T¯0O-REL space. Suppose that for any a,b ∈ A with a≠b, a,b∈βA, then by the inclusion map ia,b=ia,ib=a,b∈βX, a contradiction by Theorem 23.Thus, a,b∉βA.
Now, supposeR∈RELA:R<<a,b∈rAa and R∈RELA:R<<b,a∈rAb. It follows that, for all R∈R such that a,b⊂R, and by the inclusion map ia,b⊂iR implying a,b⊂R. It follows that R∈RELX:R<<a,b∈rXa, a contradiction by Theorem 23. Similarly, by the same argument R∈RELA:R<<b,a∈rXb, a contradiction. Therefore, R∈RELAR<<a,b∉rAa or R∈RELA:R<<b,a∉rAb.
In similar way,R∈RELA:R<<a,b∉rAb or R∈RELA:R<<b,a∉rAa, and R∈RELA:R<<a,a,b,b∉rAa or R∈RELA:R<<b,b,a,a∉rAb. Hence, X is closed under extremal subobjects.
Next, suppose thatX=∏k∈IXk, where βXk,rXk are the T¯0O-REL structures on Xk induced by projection map πk:Xk⟶X for all k∈I, i.e., Xk,βXk,rXk∈C. We show that X,βX,rX is a T¯0O-REL space. Let a,b∈βX for any a,b∈X with a≠b. Then, πka,b=πka,πkb=ak,bk∈βXk, a contradiction by Theorem 23. Thus, a,b∉βX.
Now, supposeR∈RELX:R<<a,b∈rXa and R∈RELX:R<<b,a∈rXb. It follows R∈R implies a,b⊂R. Then, there is k∈I for which ak≠bk∈Xk, and πka,b⊂πkR implying πka,πkb=ak,bk⊂πkR. It follows that R∈RELX:πkR<<ak,bk∈rXkak, a contradiction by Theorem 23. By the same process, R∈RELX:πkR<<bk,ak∈rXkbk, a contradiction. Hence, R∈RELX:R<<a,b∉rXa or R∈RELX:R<<b,a∉rXb. In similar way, R∈RELX:R<<a,b∉rXb or R∈RELX:R<<b,a∉rXa, and R∈RELX:R<<a,a,b,b∉rXa or R∈RELX:R<<b,b,a,a∉rXb. Hence, X is closed under the formation of products.
Therefore, the categoryT¯0O-REL is a quotient-reflective subcategory of O-REL.
Analogous to the above argument, settingC=T0O−REL or T1O−REL, the proof can be easily followed by using Theorem 24 or Theorem 25, respectively.
(ii) By the Theorem26 and Remark 13, T0′O-REL and O-REL are isomorphic categories and thus T0′O-REL is normalized



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