#### 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 local , local , and local ordered-RELspaces. Furthermore, we characterize explicitly several notions of ’s and objects in **O-REL** and study their mutual relationship. Finally, it is shown that the category of ’s (resp. ) ordered-RELspaces are quotient reflective subcategories of **O-REL**.

#### 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 of 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 and separation properties, several mathematicians have extended this idea to arbitrary topological categories [14–18].

Classical separation axioms at some point (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 in **O-REL**(ii)To characterize local , local , and local objects in **O-REL** and examine their mutual relationship(iii)To give the characterization of , , and objects in **O-REL** and examine their mutual relationship(iv)To define several structures using ordered-RELspaces and discuss each of the and axioms there and examine their mutual relationship(v)To examine the quotient-reflective properties of ordered-RELspaces.

#### 2. Preliminaries

Recall [31, 32], a functor (the category of sets and functions) is called topological if (i) is concrete(ii) consists of small fibers(iii)Every -source has a unique initial lift or every -sink has an unique final lift, i.e., if for every source there exists an unique structure on such that is a morphism iff for each , 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].

Let be a non-empty, then is called a relative system for , and it is denoted by (X). Moreover, (X) can be ordered by setting

iff for each , there exists such that .

Furthermore, we denote by and by .

*Definition 1 (cf. [33]). *Let , then is called boundedness or -set on , if satisfies the following axioms:
(i)(ii)(iii).And for B-sets and a function is called bounded iff it satisfies;
By **BOUND** we denote the corresponding defined category.

*Definition 2 (cf. [33]). *The triple is called RELative space (shortly RELspace) if for the boundedness the function r: satisfies the following conditions:
(i) and implies (ii) for (iii) iff (iv) implies .The RELspace is called ordered-RELspace provided that the following axiom holds:
(v) implies .

*Definition 3 (cf. [33]). *Let and be two RELspaces, then a bounded function is called RELative map (shortly RELmap) iff it satisfies the following condition:
where with . By **O-REL,** we denote the full subcategory of **REL**, whose objects are the ordered RELspaces. Note that is a bireflective subcategory of [34].

*Example 4. *Let be a preuniform convergence space; then, the associated RELspace can be defined as follows:
Let **PU-REL** denotes the category, whose objects are triples 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. *Let be a set-convergence space; then, the associated RELspace can be defined by
, where and FIL(X) is the collection of all filters defined on .

Let **SET-REL** denotes the category, whose objects are triples 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. *Let be prenearness space; then, the associated RELspace can be described as
Note that **PNEAR****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 and morphisms are RELmaps.

*Example 7. *For a B-set , we put , and for we set ; hence, defines a RELspace, which is diagonal, meaning that for and , we can find such that .

Let **-REL** be denote the corresponding defined full subcategory of **REL**; then, **-REL****BOUND**.

*Remark 8. *In this context, note that **BORN**, the full subcategory of **BOUND**, whose objects are the bornological spaces, then also has evidently a corresponding counterpart in **REL**.

*Example 9. *Let be b-topological space; then, the associated RELspace is defined by

Note that **b-TOP****bTOP-REL** [9], where **bTOP-REL** denotes the full subcategory of **REL**, whose objects are triples , and **b-TOP** denotes the category of b-topological spaces and b-continuous maps as defined in [9].

#### 3. as a Normalized and Geometric Topological Category

Note that the forgetful functor , where is topological in the following sense:

Lemma 10. *Let be a collection of RELspaces. A source is initial in iff
and for all ,
*

*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. *Let be a collection of ordered-RELspaces. A sink is final in iff
where , and for ,
*

*Proof. *It is easy to observe that is an ordered-RELspace and is a RELmap. Suppose that is a mapping. We show that is a RELmap iff is a RELmap. Necessity is obvious since the composition of two RELmaps is RELmap again.

Conversely, let be a RELmap.

Then, first, we show that is a bounded map. Let ; it implies that . For our own convenience, take , and since is a RELmap, then and consequently, is bounded.

Now, let and . By the Definition 3, we have . On the other hand, is a RELmap; it follows that Take . Then, we have and subsequently, which shows is a RELmap.

Lemma 12. *Let , and be an ordered-RELspace.
*(i)*A RELstructure is discrete iff , where and with *(ii)*A RELstructure is indiscrete iff , where if with .*

*Proof. *By applying Lemma 11, we get the desired result.

*Remark 13. *The topological functor , where is normalized since an unique RELstructure , and exists whenever and a unique RELstructure , and exists whenever . Furthermore, the topological functor is geometric since the regular sub-object of a discrete RELspace is discrete, and finite product of discrete RELstructures is discrete again.

#### 4. Local and Local Ordered-RELspaces

In this section, we define notions for and ordered-RELspaces at some point.

Let be any set and . We define the *wedge product of**at* as the two disjoint copies of at and denote it as . For a point we write it as if belongs to the first component of the wedge product; otherwise, we write that is in the second component. Moreover, is the cartesian product of .

*Definition 14 (cf. [14]). *(i)A mapping is said to be **principal****-axis mapping** provided that(ii)A mapping is said to be **skewed****-axis mapping** provided that(iii)A mapping is said to be **fold mapping at** provided thatAssume that is a topological functor, with and .

*Definition 15 (cf. [14]). *(i) is at provided that the initial lift of the -source is discrete(ii) is at provided that the initial lift of the -source is discrete, where is the wedge product in , i.e., the final lift of the -sink , where represent the canonical injections(iii) is at provided that the initial lift of the -source is discrete.

*Remark 16. *(i)In **TOP**, and at (respectively, at ) are equivalent to the classical at (respectively, the classical at ), i.e., for each with , there exists a neighborhood of not containing or (respectively, and); there exists a neighborhood of not containing [35](ii)A topological space is (respectively ) iff is (respectively ) at for each [35](iii)Let be a topological functor, and be a retract of . Then, if is or at , then is at but not conversely in general [36].

Theorem 17. *Let be ordered-RELspace and pX. Then, is at p if and only if for each with , the following holds:
*(i)

*(ii)*

*(iii)*

*or*(iv)

*or*

*Proof. *Let be at ; we show the conditions to are holding:
(i)Suppose that for all with . Let , then since and for , (by the assumption), where for j =1,2 are projection maps. By Definitions 1 and 15 and Lemma 10, a contradiction, it follows (ii)Assume that and . Particularly, let and ; then, . By the assumption, and . Since is at , it follows that , where is the discrete structure on .Similarly, for , we get , a contradiction to the discreteness of .

Thus, or .
(iii)Suppose that and . In particular, let and ; then, , and by the assumption and . Since is at , we get that , where is the discrete structure on Similarly, for , we get , a contradiction.

Therefore, or .
(iv)Assume that and . Let and ; then, , , (by the assumption). Since is at , it follows that , where is the discrete structure on .Similarly, for , we get , a contradiction.

Hence, or .

Conversely, suppose to are holding.

Let be the initial structure induced by and , where is the product RELstructure on and the discrete RELstructure on X.

We show that is the discrete REL structure on , i.e., we show that and for , .

Let and ; if , then . Suppose . Then, we have for some and if , then ; let ; then, it further implies that or and . By the assumption, (for j =1,2). Thus, and ; subsequently, .

Now, implies and , and by Lemma 10, =

Suppose , then

; it follows that .

Since , we have the following possibilities of :

,

,

,

,

.

Case (i). Suppose . It follows that for all such that , . By Definition 2, . Similarly, . Therefore, holds Case (ii). holds. The proof is similar to Case (i) Case (iii). Let . It follows that for all such that , and , . By the assumption, we get . Similarly, . Thus, cannot be possible Case (iv). Similar to Case (iii), we conclude that is not possible Case (v). If . It follows that for all such that , and , for all implying . By the assumption, . Similarly, . Hence, is not possible.

Similarly, if , only Case (i) and Case (ii) are holding. By Lemma 12, is discrete.

Therefore, by Definition 15, is at .

Theorem 18. *Let be an ordered-RELspace and .** is at if and only if for any with , the following holds:
*(i)*(ii)** and *(iii)* and *(iv)* and .*

*Proof. *By following the same technique used in Theorem 17, and replacing the mapping by the mapping , we get the proof.

Theorem 19. *All ordered-RELspaces are at .*

*Proof. *Let be ordered-RELspace and . By Definition 15, we show that for each (where ) for some and . implying . Suppose , it implies that for some . If , then implying .

Suppose , it follows that , or . If , then for some which shows that should be in the first component of the wedge product , a contradiction. In similar manner, for some . Hence, . Thus, we must have for only and consequently, , the discrete RELstructure on .

Now, for , by Lemma 10, = . Since , we have the following possibilities of :

,

,

,

,

.

In particular, for . It follows that, for all such that , and (for k =1,2), implying . It follows (respectively, ) in the first (respectively, second) component of the wedge product , a contradiction. Similarly, for and , we get a contradiction.

Therefore, . Consequently, by Definition 15(i) and Lemma 10, is at .

#### 5. and Ordered-RELspaces

In this section, we define generically notions of and in ordered-RELspaces.

The characterization of 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 object and examined the relationship between different forms of generalized 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 objects of topology to topological categories [14, 37]. In abstract topological categories [21], objects are used to define , 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 at can be replaced by at diagonal . Any element is written as (resp., ) if it lies in the first (resp., second) component of . Clearly, , if and only if .

*Definition 20 (cf. [14]). *(i)A mapping is called **principal axis mapping** provided that(ii)A mapping is called **skewed axis mapping** provided that(iii)A mapping is called **fold mapping** provided that

Any element is written as (resp., ) if it lies in the first (resp., second) component of . Clearly, if and only if .

Now, we replace the point by any generic point and define the following separation axioms.

*Definition 21. *Let