# Approach Merotopological Spaces and their Completion

Authors: Mona Khare; Surabhi Tiwari
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

This paper introduces the concept of an approach merotopological space and
studies its category-theoretic properties. Various topological categories are shown to be embedded into the category whose objects are approach merotopological spaces. The order structure of the family of all approach merotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch completion of an approach merotopological space is constructed. The present study is a unified look at the completion of metric spaces, approach spaces, nearness spaces, merotopological spaces, and approach merotopological spaces.


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## Body

## 1. Introduction

Some of the applications of nearness-like structures within topology are unification, extensions, homology, and connectedness. The categories ofR0-topological spaces, uniform spaces [1, 2], proximity spaces [2, 3], and contiguity spaces [4, 5] are embedded into the category of nearness spaces. The study of proximity, contiguity, and merotopic spaces in the more generalized setting of L-fuzzy theory can be seen in [6–13]. In [14], the notion of an approach space was introduced via different equivalent set of axioms to measure the degree of nearness between a set and a point. While developing the theory of approach spaces, Lowen et al. many a time employed tools from nearness-like structures. The notion of “distance” in approach spaces is closely related to the notion of nearness; further proximity and nearness concepts arise naturally in the context of approach spaces as can be seen in [15–18]. Hence it became mandatory looking into the nearness-like concepts in approach theory, more clearly. With the same spirit, Lowen and Lee [19] made an attempt to measure how near a collection of sets is and, in the process, axiomatized the two equivalent concepts: approach merotopic and approach seminearness structures, respectively, to measure the degree of smallness and nearness of an arbitrary collection of sets, and therefore generalized approach spaces in a sense. In 2004, Bentley and Herrlich [20] gave the idea of merotopological spaces as a supercategory of many of the above mentioned categories. They also constructed the functorial completion of merotopological spaces employing the theory of bunches in merotopological spaces. In [21], we axiomatized the notion of approach nearness by adding to the axioms of an approach merotopy the axiom relating a collection of sets and the closure induced by the respective approach merotopy; and we analogously obtained cluster completion of an approach nearness space.Prerequisites for the paper are collected in Section2. In Section 3, we axiomatize approach merotopological spaces. The category AMT of approach merotopological spaces and their respective morphisms is shown to be a topological construct and a supercategory of some of the known topological categories, including the category of topological spaces and continuous maps. Order structure of the family of all approach merotopologies on X is also discussed. In Section 4, bunch completion of an approach merotopological space is constructed, employing the theory of bunches. The concept of regularity in an approach merotopic space is introduced to obtain a relationship between cluster and bunch completion of an approach nearness space. Indeed, it is shown that cluster completion is a retract of the bunch completion of a regular approach nearness space (X,ν).


## 2. Preliminaries and Basic Results

LetX be a nonempty ordinary set. The power set of X is denoted by 𝒫(X) and the family of all subsets of 𝒫(X) is denoted by 𝒫2(X). We denote by ℵ0 the first infinite cardinal number, by |A| the cardinality of A where A⊆X, and by J an arbitrary index set. For 𝒜, ℬ subsets of 𝒫(X), 𝒜∨ℬ≡{A∪B:A∈𝒜,B∈ℬ}; 𝒜corefinesℬ (written as 𝒜≺ℬ) if and only if for all A∈𝒜 there exists B∈ℬ such that B⊆A. For 𝒜⊆𝒫(X), stack(𝒜)={A⊆X:B⊆A for some B∈𝒜} and sec 𝒜={B⊆X:A∩B≠∅, for all A∈𝒜}={B⊆X:X-B∉ stack(𝒜)}. Observe that sec2𝒜=stack(𝒜), for all 𝒜∈𝒫2(X). A grill on X is a subset 𝒢 of 𝒫(X) satisfying ∅∉𝒢; if A∈𝒢 and A⊆B, then B∈𝒢; and if A∪B∈𝒢, then A∈𝒢 or B∈𝒢. For basic definitions and results of merotopic spaces and nearness spaces, we refer to [1].Definition 2.1 (see [20]).
Amerotopological space is the triple (X,ξ,cl), where ξ is a merotopy and cl is a Kuratowski closure operator on X such that {cl(A):A∈𝒜}∈ξ⇒𝒜∈ξ, for all 𝒜∈𝒫2(X).Definition 2.2 (see [19, 21]).
A functionν:𝒫2(X)→[0,∞] is called an approach merotopy on X if for any 𝒜,ℬ∈𝒫2(X) the following conditions are satisfied:(AM1)
𝒜≺ℬ⇒ν(𝒜)≤ν(ℬ),(AM2)
⋂𝒜≠∅⇒ν(𝒜)=0,(AM3)
∅∈𝒜⇒ν(𝒜)=∞,(AM4)
ν(𝒜∨ℬ)≥ν(𝒜)∧ν(ℬ).The pair(X,ν) is called an approach merotopic space. For an approach merotopic space (X,ν), we define clν(A)={x∈X:ν({{x},A})=0}, for all A⊆X. Then clν is a Čech closure operator on X.An approach merotopyν on X is called an approach nearness on X [21] if the following condition is satisfied:(AN5)
ν({clν(A):A∈𝒜})≥ν(𝒜).In this case,clν is a Kuratowski closure operator on X. Denote cl(𝒜)={cl(A):A∈𝒜}.Definition 2.3 (see [21]).
Let𝒜∈𝒫2(X) and ν be an approach merotopy on X. Then we say that 𝒜 is(i)
near in ν if ν(𝒜)=0,(ii)
ν-clan if 𝒜 is a near grill,(iii)
ν-closed if ν({{A}∪ℬ})=0, for all ℬ⊆𝒜⇒A∈𝒜,(iv)
ν-cluster if 𝒜 is a ν-closed ν-clan.For any approach merotopic spaces(X,νX) and (Y,νY), a map f:X→Y is called a contraction if νY(f(𝒜))≤νX(𝒜), for all 𝒜∈𝒫2(X), or equivalently νX(f-1(ℬ))≥νY(ℬ), for all ℬ∈𝒫2(Y). For any approach merotopies ν1 and ν2 on X,ν2≤ν1 (ν1 is finer than ν2, or ν2 is coarser than ν1) if the identity mapping 1X:(X,ν1)→(X,ν2) is a contraction (see [19]). For standard definitions in the theory of categories we refer to [22], for approach spaces we refer to [14], and for lattices see [23].


## 3. Approach Merotopological Spaces

In this section, we introduce approach merotopological spaces and establish some category-theoretic results for them. Lattice structure of the family of all approach merotopologies onX is also discussed.Definition 3.1.
Anapproach merotopological space is a triple (X,ν,cl), where ν is an approach merotopy on X and cl is a Kuratowski closure operator on X such that the following condition is satisfied:(AM5)
ν(cl(𝒜))≥ν(𝒜), for all 𝒜∈𝒫2(X). We call ν to be an approach merotopology with respect to the closure operator cl onX.Example 3.2.
(i) Let(X,ν,cl) be an approach merotopological space and let 𝔊(X) be the family of all grills on X. Define νg:𝒫2(X)→[0,∞] as follows:
(3.1)νg(𝒜)=inf{ν(𝒢):𝒜≺𝒢,𝒢∈𝔊(X)},∀𝒜∈𝒫2(X).
Then νg is an approach merotopology with respect to cl on X (see [19, 24]).
(ii) Let(X,cl) be a topological space. Then (X,νg,cl) is an approach merotopological space, where νg is defined as in the above example in which ν:𝔊(X)→[0,∞] can be defined in the following ways for 𝒢∈𝔊(X).
(a)
ν(𝒢)=0, if ⋂cl(𝒢)≠∅; and ν(𝒢)=sup{|cl(G)|:G∈𝒢and|cl(G)|<ℵ0}, otherwise.(b)
ν(𝒢)=0, if ⋂cl(𝒢)≠∅; and ν(𝒢)=supG∈𝒢|stack(cl(G))|, otherwise.(c)
ν(𝒢)=0, if ⋂cl(𝒢)≠∅; and ν(𝒢)=|cl(𝒢)|, otherwise.(d)
ν(𝒢)=0, if ⋂cl(𝒢)≠∅; and ν(𝒢)=infG∈seccl(𝒢)|G|, otherwise.(e)
ν(𝒢)=0, if ⋂cl(ℬ)≠∅ for every finite subset ℬ of 𝒢; and ν(𝒢)=inf{|G|:G∈seccl(𝒢)}, otherwise.(f)
ν(𝒢)=0, if ⋂cl(𝒢)≠∅ or each element of 𝒢 is infinite; and ν(𝒢)=sup{|G|:G∈𝒢) and |G|<ℵ0}, otherwise. (In this case, cl is a T1-closure operator on X.)
Observe that if(X,cl) is a symmetrical topological space (i.e., x∈cl({y})⇒y∈cl({x}), for all x,y∈X), then clν=cl in all of the above cases (note that a T1-space is already a symmetrical topological space).
(iii) Letf:(X,clX)→(Y,clY) be a closed and continuous map. Define ν:𝔊(X)→[0,∞] as follows: for 𝒢∈𝔊(X),ν(𝒢)=0, if ⋂{clY(f(G)):G∈𝒢})≠∅; and ν(𝒢)=infG∈seccl(𝒢)|G|, otherwise. Then (X,νg,clX) is an approach merotopological space. Further if y∈clY(f(x))⇒f(x)∈clY({y}) for all y∈Y and x∈X, then clν=clX.
(iv) Let(X,ν,cl) be an approach merotopological space. Define νc:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),(3.2)νc(𝒜)=sup{ν(ℬ):ℬ⊆𝒜,|ℬ|<ℵ0}.
Then νc is an approach merotopology with respect to the closure cl on X.
(v) Let(X,cl) be a topological space and r∈(0,∞]. Define νr:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),νr(𝒜)=0, if ⋂cl(𝒜)≠∅;νr(𝒜)=∞, if ∅∈𝒜; and νr(𝒜)=r, otherwise. Then νr is an approach merotopology with respect to the closure cl on X.
(vi) The functionνd:𝒫2(X)→[0,∞] this is defined as follows: for 𝒜∈𝒫2(X),νd(𝒜)=0, if ⋂𝒜≠∅, and νd(𝒜)=∞, otherwise, is an approach merotopology with respect to the discrete closure operator cld on X and clν=cld. We call (X,νd,cld) the discrete approach merotopological space.
(vii) The functionνi:𝒫2(X)→[0,∞] this is defined as follows: for 𝒜∈𝒫2(X),νi(𝒜)=0, if ∅∉𝒜, and νi(𝒜)=∞, otherwise, is an approach merotopology with respect to the indiscrete closure operator cli on X and clν=cli. We call (X,νi,cli) the indiscrete approach merotopological space.Having established the existence of an adequate number of approach merotopologies onX (taking several values in [0,∞]), it is now relevant to study the category AMT having objects as approach merotopological spaces. A morphism in AMT is a map f:X→Y which is both continuous (with respect to the topologies) and contraction (with respect to the approach merotopies). We denote by TOP the category of topological spaces and continuous maps. Also let T:AMT→TOP and M:AMT→AMER denote the forgetful functors that keep the topology but forget the approach merotopy, and keep the approach merotopy but forget the topology, respectively.Proposition 3.3.
Let(X,ν,cl) be an approach merotopological space. Then cl(A)⊆clv(A) for all A⊆X, where clv denotes the closure operator induced by the approach merotopy ν on X.Proof.
LetA⊆X and x∈cl(A). Then ν({cl({x}),cl(A)})=0 which yields ν({{x},A})=0 and hence x∈clν(A).Theorem 3.4.
The categoryAMT is a topological construct.Proof.
Let((Yj,νj,clj))j∈J be a family of approach merotopological spaces indexed by J, and let (fj:T(X)→T(Yj))j∈J be a source in TOP. Define ν:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),(3.3)ν(𝒜)=sup{infi=1nsupj∈Jνj(fj(𝒜i)):(𝒜)i=1n∈ℭ(𝒜)},
where ℭ(𝒜) is the collection of all finite families (𝒜i)i=1n⊆𝒫2(X) such that 𝒜1∨𝒜2∨⋯∨𝒜n≺𝒜. Then ν is an approach merotopy on X (see [19, Theorem 3.8]). Let cl denote the closure induced by the initial topology on X and (𝒜i)i=1n∈ℭ(𝒜). Then (cl(𝒜))i=1n∈ℭ(cl(𝒜)). Also since fj is continuous for all j∈J, therefore νj(fj(𝒜i))≤νj(fj(cl(𝒜i))), for all (𝒜i)i=1n∈ℭ(𝒜) which in turn yields ν(𝒜)≤ν(cl(𝒜)). Hence (X,ν,cl) is an approach merotopological space. To show that the source is initial, let (Z,νZ,clZ) be an approach merotopological space and g:T(Z)→T(X) be a continuous map such that, for every j∈J, the map fj∘g:Z→Yj is an AMT-morphism. Then g:M(Z)→M(X) is a contraction since (fj:M(X)→M(Yj))j∈J is initial in AMER (see [19, Theorem 3.8]). Thus (X,ν,cl) is the initial source in AMT and hence AMT is a topological construct.Corollary 3.5.
Let((Yj,νj,clj))j∈J be a family of approach merotopological spaces. Then (fj:X→Yj)j∈J is an initial source in AMT if and only if (fj:T(X)→T(Yj))j∈J is initial in TOP and (fj:M(X)→M(Yj))j∈J is initial in AMER.Proposition 3.6.
Let((Xj,νj,clj))j∈J be a family of approach merotopological spaces and (fj:Xj→X)j∈J be a sink in AMT. Then (fj:Xj→X)j∈J is a final sink in AMT if and only if both of the following conditions hold:(i)
(fj:T(Xj)→T(X))j∈J   is final in  TOP,(ii)
the approach merotopology with respect to the closure operatorcl induced by the final topology on X is defined as follows: for 𝒜∈𝒫2(X),(3.4)ν(𝒜)={0,if⋂cl(𝒜)≠∅,infj∈Jνj(fj-1(cl(𝒜))),otherwise.Proof.
To prove thatν is an approach merotopy with respect to cl on X, we will prove only (AM5). For this, let ⋂cl(𝒜)≠∅. Then there exists x∈X such that 𝒜⊆{A⊆X:x∈cl(A)}. Hence ν(cl(𝒜))=0=ν(𝒜). To show that (fj:Xj→X)j∈J is final, let (Z,νZ,clZ) be an approach merotopological space and let g:X→Z be a map such that, for every j∈J,g∘fj:Xj→Z is an AMT-morphism. Then g:X→Z is continuous since (fj:T(Xj)→T(X))j∈J is final in TOP. Also for 𝒜⊆𝒫(X) with ⋂cl(𝒜)=∅,(3.5)ν(𝒜)=infj∈Jνj(fj-1(cl(𝒜)))≥infj∈JνZ(g(fj(fj-1(cl(𝒜)))))≥νZ(g(cl(𝒜)))≥νZ(g(𝒜)).
On the other hand, if ⋂cl(𝒜)≠∅, then there exists x∈X such that g(x)∈g(cl(A))⊆clZ(g(A)), for all A∈𝒜. Hence νZ(g(𝒜))=0. Thus, g:(X,ν)→(Z,νZ) is a contraction implying that (X,ν,cl) is the final approach merotopological space.Letρ be a metric on X. Define νρ:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),(3.6)νρ(𝒜)=infx∈XsupA∈𝒜infy∈Aρ(x,y).
Then νρ is an approach merotopology with respect to the closure cl induced by ρ on X and clνρ=cl. Also for any metric spaces (X,ρ) and (Y,ρ'),f:(X,ρ)→(Y,ρ') is a nonexpansive map if and only if f:(X,νρ,clνρ)→(Y,νρ′,clνρ') is an AMT-morphism. Thus the category MET of metric spaces and nonexpansive maps is embedded as a full subcategory into AMT by the functor F:MET→AMT defined as follows: F((X,ρ))=(X,νρ,clνρ) and F(f)=f.Proposition 3.7.
The categoryTOP is a bireflective full subcategory of AMT.Proof.
Let(X,ν,cl) be any approach merotopological space. Set r=sup{ν(𝒜):𝒜∈𝒫2(X),ν(𝒜)≠0 and ∅∉𝒜} and νr is defined as in Example 3.2 (v). Then νr∧ν is an approach merotopology on X with respect to the closure operator cl, which is coarser than ν. Therefore for any approach merotopological space (X,ν,cl), the identity mapping 1X:(X,ν,cl)→(X,νr∧ν,cl) is the TOP-bireflection of (X,ν,cl).Remark 3.8.
(i) There is a full embedding ofTOP into AMT that is defined by associating to a topological space X, the approach merotopy νr as defined in Example 3.2 (v). Thus a topological space can be viewed as an approach merotopological space having the approach merotopy νr. Thus following this convention, the forgetful functor T:AMT→TOP shall be viewed as keeping the topology but replacing the approach merotopy with that of νr.(ii) It is known that an approach nearness space always induces a Kuratowski closure operator on the underlying space. Thus by adjoining to an approach nearness space(X,ν), the topology given by the closure operator clν induced by ν, we can obtain a full embedding of ANEAR (the category of approach nearness spaces and contractions) into AMT. (Observe that every contraction is a continuous map with respect to the closures induced by the approach merotopies.) Therefore it is clear that an approach nearness space (X,ν) can be regarded as a special approach merotopological space (X,ν,cl) for which cl=clν. Hence approach merotopological spaces are generalization of approach nearness spaces. Also since the underlying topology of an approach nearness space is always symmetric, therefore if we restrict the category TOP to the category STOP having objects symmetrical topological spaces, then the full embedding of TOP into AMT is actually the full embedding of STOP into ANEAR. Moreover for every approach merotopic space (X,ν), if we associate the discrete closure operator cl on X, then we obtain an approach merotopological space (X,ν,cl). Thus the category AMER can also be embedded into AMT.(iii) Letδ be an approach space. Then the function νδ:𝒫2(X)→[0,∞] that is defined as follows: for 𝒜∈𝒫2(X),(3.7)νδ(𝒜)=infx∈XsupA∈𝒜δ(x,A)
is an approach merotopy on X. Also for any approach spaces (X,δ) and (Y,δ'),f:(X,δ)→(Y,δ′) is a contraction map if and only if f:(X,νδ)→(Y,νδ') is a contraction. So we get the functor F:AP→AMER defined as follows: F((X,δ))=(X,νδ) and F(f)=f. Therefore AP is a full subcategory of AMER (cf. [17, 19]). As a consequence, AMT is also a supercategory of AP.In Remark3.8, we have established that a topological space can be regarded as an approach merotopological space by associating the approach merotopy νr with the topology on X. It should be clarified here that, in general for an approach merotopological space (X,ν,cl), the topology on X is not determined by its approach merotopy ν; it happens so for a topological space only if (X,cl) is a symmetrical topological space. Thus, in general, for an approach merotopological space (X,ν,cl),cl need not be equal to clν. This fact can be supported by the following example: consider ℝ with {∅,ℝ}∪{[0,n]:n∈ℕ} as the collection of all closed sets. Then (ℝ,cl) is a non-symmetric and T0-space, where cl is the closure operator associated with the topology on ℝ defined as above. Further let the approach merotopy νi on ℝ be indiscrete. Then (ℝ,νi,cl) is an approach merotopological space. Also observe that (ℝ,νi) is an approach nearness space but clνi≠cl.Remark 3.9.
LetMERTOP denote the category of all merotopological spaces and their respective morphisms (see [20]). It is easy to verify that, for any merotopological space (X,ξ,cl), the pair (X,νξ,cl) is an approach merotopological space (where νξ is the induced approach merotopy on X defined as follows: for 𝒜∈𝒫2(X),ν(𝒜)=0, if 𝒜∈ξ; and ν(𝒜)=∞, otherwise. Also for any merotopic spaces (X,ξ) and (Y,ξ'),f:(X,ξ)→(Y,ξ') is a merotopic map if and only if f:(X,νξ)→(Y,νξ') is a contraction. Thus MERTOP is embedded as a full subcategory in AMT by the functor F:MERTOP→AMT such that F((X,ξ,cl))=(X,νξ,cl) and F(f)=f.
An approach merotopological space(X,ν,cl) is induced by a merotopological space (X,ξ,cl) if and only if ν(𝒫2(X))⊆{0,∞}. Also for any approach merotopological space (X,ν,cl), the triples (X,ξν,cl) and (X,ξν,cl), where(3.8)ξν={𝒜∈𝒫2(X):ν(𝒜)=0},ξν={𝒜∈𝒫2(X):ν(𝒜)<∞},
are merotopological spaces; and if f:(X,ν)→(Y,ν′) is a contraction, then f:(X,ξν)→(Y,ξν') and f:(X,ξν)→(Y,ξν′) are merotopic maps. So this defines the functors G:AMT→MERTOP by G((X,ν,cl))=(X,ξν,cl) and G(f)=f and G':AMT→MERTOP by G'((X,ν,cl))=(X,ξν,cl) and G'(f)=f. Hence we can conclude that the category MERTOP is a bicoreflective and bireflective subcategory of AMT: for any approach merotopological space (X,ν,cl), the identity mappings 1X:(X,νξν,cl)→(X,ν,cl) and 1X:(X,ν,cl)→(X,νξν,cl) are the MERTOP-bicoreflection and MERTOP-bireflection of (X,ν,cl), respectively. Further in [20], Bentley and Herrlich embedded TOP in MERTOP as its full bicoreflective subcategory. Therefore TOP can be embedded as a full subcategory in AMT both bicoreflectively and bireflectively.Next, let us introduce ordering between approach merotopological spaces and discuss their order structure. The exact meet and join of a family of merotopologies onX are constructed.Definition 3.10.
Letν and ν' be approach merotopologies with respect to the closures cl and cl' on X, respectively. Define the ordering “≤” as follows: (X,ν,cl)≤(X,ν',cl′) if ν≤ν' and the topology induced by cl is weaker than the topology induced by cl' on X.Theorem 3.11.
The family of all approach merotopologies onX forms a completely distributive complete lattice with respect to the partial order “≤”. The zero of this lattice is the indiscrete approach merotopology νi with respect to the closure cli induced by the indiscrete topology (called indiscrete closure) on X and the unit is the discrete approach merotopology νd with respect to the discrete closure cld induced by the discrete topology (called discrete closure) on X.Proof.
Let{(Xj,νj,clj):j∈J} (where J is an arbitrary index set) be a family of approach merotopological spaces. If τj denotes the topology induced by clj for all j∈J, then supj∈Jτj=⋂{Vα:Vα is a topology on X and τj⊆Vα, for all j∈J}. Let cl denote the closure operator induced by supj∈Jτj. Then for A⊆X,cl(A)=⋂{B⊆X:A⊆B and B is supj∈Jτj-closed}. Define νsup:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),(3.9)νsup(𝒜)=sup{infi=1nsupj∈Jνj(𝒜i):(𝒜i)i=1n∈ℭ(𝒜)},
where ℭ(𝒜) is the collection of all finite families (𝒜i)i=1n⊆𝒫2(X) such that 𝒜1∨𝒜2∨⋯∨𝒜n≺𝒜. Then νsup with respect to the closure operator cl is an approach merotopology on X and is the supremum of the family of merotopologies {νj:j∈J} with respect to the closures {clj:j∈J} on X, respectively (techniques of the proof are similar to those of Theorem 3.4). Now we construct the infimum of the given family. Define νinf:𝒫2(X)→[0,∞] as follows: for 𝒜∈𝒫2(X),(3.10)vinf(A)=infj∈Jvj(cl(A)),
where cl:𝒫(X)→𝒫(X) is defined as follows: for A⊆X,cl(A)=⋂{B⊆X:A⊆B and clj(B)=B, for all j∈J}. Then νinf is an approach merotopology with respect to the closure cl on X which is also the infimum of the merotopologies {νj:j∈J} with respect to the closures {clj:j∈J} on X, respectively (proof follows similarly as in Proposition 3.6).


## 4. Completion of Approach Merotopological Spaces

In this section, we construct bunch completion of any approach merotopological space. The concept of regularity in an approach merotopic space is introduced to establish a relationship between the bunch completion and the cluster completion (constructed in [22]) of an approach nearness space in the classical theory. For regularity in a merotopic space, we refer to [1]. It is noteworthy to discuss here that clusters and bunches of an approach merotopological space are essentially the clusters and bunches of the associated merotopology ξν={𝒜∈𝒫(LX):ν(𝒜)=0}. Similar is the case with the definition of bunch completeness of an approach merotopological space. But in this section, using these definitions, we have constructed the completion (X+,ν+,clX+) of an approach merotopological space (X,ν,cl). Observe that ν+ takes several values in [0,∞] if ν does so, and the completion of ξν, that is, ξν+, is not, in general, equal to ν+. A similar type of study can also be seen in [17], where Lowen et al. had constructed the completion of a T0-approach space (X,δ) using the clusters of δ which are precisely the clusters of the associated merotopic space.Definition 4.1.
Let(X,ν,cl) be an approach merotopological space. A nonempty grill 𝒢 on X is said to be a ν-bunch if ν(𝒢)=0 and cl(G)∈𝒢⇒G∈𝒢.By aν-cluster 𝒜 in an approach merotopological space (X,ν,cl), we mean that 𝒜 is a ν-cluster of the approach merotopic space (X,ν).Proposition 4.2.
Let(X,ν,cl) be an approach merotopological space. Then every ν-cluster is a maximal near ν-bunch.Proof.
Let𝒞 be a ν-cluster. Then clearly 𝒞 is a near element in ν. Let cl(G)∈𝒞. Then ν({cl(C),cl(G)})=0, for all C∈𝒞 which by (AM5) yields that ν({C,G})=0, for all C∈𝒞. Consequently G∈𝒞 as 𝒞 is ν-closed.For any approach merotopological space(X,ν,cl), let X+ denote the family of all ν-bunches on X. It must be clarified here that a ν-bunch need not be a ν-cluster. For example, let 𝒢(𝔪)={A⊆X:|cl(A)|≥𝔪} where 𝔪 is an infinite cardinal number. Consider the approach merotopological space (X,ν,cl) of the last case of Example 3.2 (ii) (f). Then for 𝔪>c (where 𝔠 is the cardinality of the set ℝ of all real numbers), 𝒢(𝔪) is a ν-bunch which is not a ν-cluster because 𝒢(𝔪) is a proper subset of 𝒢(ℵ0) and ν(𝒢(ℵ0))=0.Theorem 4.3.
Let(X,ν,cl) be an approach merotopological space and let clX+ denote the operator defined by clX+(ω)={ℬ∈X+:⋂ω⊆ℬ}, for all ω⊆X+. Then clX+ is a Kuratowski closure operator on X+.Proof.
ClearlyclX+(∅)=∅ and ω⊆clX+(ω), where ω⊆X+. Let ℬ∈clX+(ω1∪ω2) but ℬ∉clX+(ω1)∪clX+(ω2). Then ⋂ω1∩⋂ω2⊆ℬ but ⋂ω1⊈ℬ and ⋂ω2⊈ℬ. Therefore there exist A∈⋂ω1 and B∈⋂ω2 such that A∉ℬ and B∉ℬ, which gives A∪B∉ℬ. Also A∈𝒜 and B∈ℬ, for every 𝒜∈ω1 and ℬ∈ω2, and consequently A∪B∈⋂(ω1∪ω2), which contradicts that ⋂(ω1∪ω2)⊆ℬ. Thus ℬ∈clX+(ω1)∪clX+(ω2). Hence clX+(ω1∪ω2)⊆clX+(ω1)∪clX+(ω2). For the reverse inclusion, let ℬ∉clX+(ω1∪ω2). Then ⋂ω1⊈ℬ and ⋂ω2⊈ℬ, giving that ℬ∉clX+(ω1)∪clX+(ω2). Finally let ℬ∈clX+(clX+(ω)) but ℬ∉clX+(ω). Then ⋂clX+(ω)⊆ℬ but ⋂ω⊈ℬ. Therefore there exists A∈⋂ω such that A∉ℬ which implies that A∉⋂clX+(ω) and thus ⋂ω⊈𝒩, for some 𝒩∈ω. Consequently 𝒩∉ω, which is not possible. Hence clX+(clX+(ω))=clX+(ω).It can be easily verified that, for any approach merotopological space(X,ν,cl), the family {A⊆X:x∈cl(A)} is a ν-bunch. Therefore we can define the map 𝔢X:X→X+ as follows: for x∈X,𝔢X(x)={A⊆X:x∈cl(A)}.Theorem 4.4.
The mapF:AMT→AMT defined by F((X,ν,cl))=(X+,ν+,clX+) and F(f)=f+, where ν+:𝒫2(X+)→[0,∞] is defined as follows: for Ω∈𝒫2(X+),ν+(Ω)=ν(⋃{⋂ω:ω∈Ω}), and f+:X+→Y+  (when f:X→Y) is defined as follows: for ℬ∈X+,f+(ℬ)={A⊆Y:f(B)⊆cl(A), for some B∈ℬ}, is a functor. Moreover, e:1→F is a natural transformation from the identity functor.Proof.
First we will show that(X+,ν+,clX+) is an approach merotopological space. Let Ω,ℑ∈𝒫2(X+) and Ω≺ℑ. Then ⋃{⋂ω:ω∈Ω}≺⋃{⋂λ:λ∈ℑ}. Thus ν+(Ω)≤ν+(ℑ). If ⋂Ω≠∅, then there exists 𝒜∈⋂Ω and therefore ⋂ω⊆𝒜, for every ω∈Ω. Consequently ⋃{⋂ω:ω∈Ω}⊆𝒜 which yields ν+(Ω)=0. Condition (AM3) follows by the convention ∅∈⋂∅. Now
(4.1)ν+(Ω∨ℑ)=ν(⋃{⋂ω:ω∈Ω∨ℑ})≥ν(⋃{⋂ω:ω∈Ω}∨⋃{⋂ω:ω∈ℑ})≥ν(⋃{⋂ω:ω∈Ω})∧ν(⋃{⋂ω:ω∈ℑ})=ν+(Ω)∧ν+(ℑ).
For (AM5) we have to show that ⋃{⋂ω:ω∈Ω}⊆⋃{⋂clX+(ω):clX+(ω)∈clX+(Ω)}, where clX+(Ω)={clX+(ω):ω∈Ω}. Let A∈⋃{⋂ω:ω∈Ω}. Then A∈⋂ω, for some ω∈Ω. So, A∈𝒜 for every 𝒜∈clX+(ω) as ⋂ω⊆𝒜, for each 𝒜∈clX+(ω). Thus A∈⋃{⋂clX+(ω):clX+(ω)∈clX+(Ω)}. Hence (X+,ν+,clX+) is an approach merotopological space.
Next letf:X→Y be an AMT-morphism. Then we will show that f+:X+→Y+ is an AMT-morphism. That f+ is a map follows from the fact that cl(f+(ℬ))≺f(ℬ), for all ℬ∈𝒫2(X). For the continuity of f+:X+→Y+, let ω⊆X+ and ℬ∈clX+(ω) and A∈⋂f+(ω). Then A∈f+(𝒞), for every 𝒞∈ω. Consequently f(B)⊆cl(A), for some B∈ℬ and hence ⋂f+(ω)⊆f+(ℬ) which in turn gives that f+:X+→Y+ is continuous. Now to show that f+:X+→Y+ is a contraction, let Ω∈𝒫2(X+) and A∈⋃{⋂f+(ω):f+(ω)∈f+(Ω)}. Then there is an A∈⋂f+(ω0), for some ω0∈Ω. Thus for every ℬ∈ω0, there is a B∈ℬ such that f(B)⊆cl(A). Since ⋂f(ℬ)⊆f(B), therefore cl(⋃{⋂f+(ω):f+(ω)∈f+(Ω)})≺f(⋃{⋂ω:ω∈Ω}). Thus νY+(f+(Ω))≤νX+(Ω) and f+:(X+,νX+,clX+)→(Y+,νY+,clY+) is an AMT-morphism. Hence F is well defined. Clearly F preserves identity. To show that F is a functor, we finally show that, for AMT-morphisms f:X→Y and g:Y→Z,(g∘f)+=g+∘f+. Let ℬ∈X+. Then(4.2)(g∘f)+(ℬ)={G⊆Z:thereisaB∈ℬsuchthat(g∘f)(ℬ)⊆clZ(G)}={G⊆Z:(g∘f)-1(clZ(G))∈ℬ}=f+({G⊆Z:g-1(clZ(G))∈ℬ})=(g+∘f+)(ℬ).
Now to prove thate:1→F is a natural transformation, we will first show that 𝔢X:X→X+ is an AMT-morphism. For this, let A⊆X and 𝒜∈𝔢X(cl(A)). Then ⋂𝔢X(cl(A))⊆𝒜 and thus A∈𝒜 because 𝒜∈X+. So, 𝒜∈clX+𝔢X(A). Therefore 𝔢X(cl(A))⊆clX+(𝔢X(A)) and hence 𝔢X:X→X+ is continuous. Also since ⋂𝔢X(A)⊆𝒜 for all A∈𝒜∈X+, therefore ⋃{⋂𝔢X(A):A∈𝒜}⊆𝒜 which in turn yields that ν+(𝔢X(𝒜))≤ν(𝒜). Hence 𝔢X:X→X+ is an AMT-morphism. Finally let x∈X and B∈𝔢Y(f(x)). Then f(x)∈clY(B) and therefore f-1(clY(B))∈𝔢X(x) which gives B∈(f+∘𝔢X)(x). On the other hand if B∈f+(𝔢X(x)), then there exists C∈𝔢X(x) such that f(C)⊆clY(B). Consequently f(x)∈clY(B) and B∈𝔢Y(f(x)). Hence f+∘𝔢X=𝔢Y∘f.Definition 4.5.
(i) Let(X,νX,clX) and (Y,νY,clY) be approach merotopological spaces. A mapping f:(X,νX,clX)→(Y,νY,clY) is said to be strict in AMT if {clY(f(A)):A⊆X} is a base for closed subsets of Y (i.e., strict in TOP (see [25])), and νY(ℬ)≤sup{νX(𝒜):cl(f(𝒜))≺ℬ}, for all ℬ∈𝒫2(X) (i.e., strict in AMER).
(ii) Further,f said to be initial in AMT if f-1(clY(f(A)))⊆clX(A) for all A⊆X (called initial in TOP), and νX(𝒜)≤νY(f(𝒜)) for all 𝒜∈𝒫2(X) (called initial in AMER).Observe that an initial mapf:X→Y is strict in AMER if and only if the approach merotopy νY on Y is defined as: νY(ℬ)=sup{νX(𝒜):clY(f(𝒜))≺ℬ}, for all B∈𝒫2(Y). Also for any initial map f:(X,νX,clX)→(Y,νY,clY) in AMER, there is a strict approach merotopology μY with respect to the closure operator clY on Y defined as: μY(ℬ)=sup{νX(𝒜):clY(f(𝒜))≺ℬ}, for all ℬ∈𝒫2(Y).Proposition 4.6.
Let(X,ν,cl) be an approach merotopological space. Then (i)
the map𝔢X:X→X+ is initial and strict;(ii)
clX+(𝔢X(X))=X+;(iii)
𝔢X+:X+→X++  is an injective map.Thus  X+ can be regarded as a dense subspace of X++.Proof.
(i) LetA⊆X and x∈𝔢X-1(clX+(𝔢X(A))). Then we have 𝔢X(x)∈clX+(𝔢X(A)), that is ⋂𝔢X(A)⊆𝔢X(x) which yields x∈cl(A). Therefore 𝔢X:X→X+ is initial in TOP. That 𝔢X is initial in AMER, follows immediately by the relation 𝒜⊆⋃{⋂ω:ω∈𝔢X(𝒜)}, for all 𝒜∈𝒫2(X). Thus 𝔢X:X→X+ is initial in AMT. For strictness in TOP, we refer to [20]. For strictness in AMER, let Ω⊆𝒫(X+) and 𝒜=⋃{⋂ω:ω∈Ω}. Then clX+(𝔢X(𝒜))≺Ω. Thus ν+(Ω)=sup{ν(𝒜):clX+(𝔢X(𝒜))≺Ω} as 𝔢X is initial in AMER.
(ii) ClearlyclX+(𝔢X(X))⊆X+. Let A∈⋂𝔢X(X). Then X=cl(A) implying that A∈ℬ, for all ℬ∈X+. Hence clX+(𝔢X(X))=X+.
(iii) See [20, Proposition 22].Definition 4.7.
An approach merotopological space(X,ν,cl) is said to be bunch complete if ⋂cl(ℬ)≠∅, for all ℬ∈X+.Proposition 4.8.
An approach merotopological space(X,ν,cl) is bunch complete if and only if for every grill 𝒢 on one X such that one ν(𝒢)=0, has ⋂cl(𝒢)≠∅.Proof.
Let(X,ν,cl) be bunch complete and let 𝒢 be a grill on X such that ν(𝒢)=0. Then 𝒢⊆{G⊆X:cl(G)∈𝒢}. Thus ⋂cl(𝒢)≠∅. Converse is obvious.Proposition 4.9.
Let(X,ν,cl) be an approach merotopological space. Then (X+,ν+,clX+) is a bunch complete approach merotopological space and (X+,clX+) is a T0-space.Proof.
That(X+,ν+,clX+) is an approach merotopological space follows from Theorem 4.4. Let Ω be a ν+-bunch and ℬ=⋃{⋂ω:ω∈Ω}. Then ℬ is a ν-bunch: let A∪B∈ℬ. Then there exists ω0∈Ω such that A∪B∈⋂ω0. Consequently ω0⊆clX+(𝔢X(A∪B))=clX+(𝔢X(A))∪clX+(𝔢X(B)). Since Ω is a grill on X+, therefore clX+(𝔢X(A))∪clX+(𝔢X(B))∈Ω which in turn gives that clX+(𝔢X(A))∈Ω or clX+(𝔢X(B))∈Ω. Also A∈⋂clX+(𝔢X(A))∈Ω and B∈⋂clX+(𝔢X(B))∈Ω yield that A∈ℬ or B∈ℬ. Now we will show that ⋂clX+(Ω)≠∅, for which it is sufficient to note that ⋂ω⊆ℬ, for all ω∈Ω. Thus ν+ is bunch complete. Now let ℬ1,ℬ2∈X+ and ℬ1∈clX+({ℬ2}) and ℬ2∈clX+({ℬ1}). Then ℬ1⊆ℬ2 and ℬ2⊆ℬ1. Hence clX+ is a T0-closure operator on X+.Remark 4.10.
Since any topological space(X,cl) can be viewed as an approach merotopological space (X,νr,cl), following the convention established previously, therefore each topological space (X,cl) is bunch complete.The following theorem shows that the bunch completionX+ of X possesses a universal mapping property that shows it to be very large.Theorem 4.11.
Let(X,νX,clX) be an approach merotopological space and let 𝔢X:X→X+ be the canonical mapping into the bunch completion. If f:X→W is an initial AMT-morphism into the approach merotopological space (W,νW,clW) such that clW(f(X))=W, then the map v:W→X+ that is defined as follows: for w∈W,v(w)={A⊆X:w∈clW(f(A))}, is such that v∘f=𝔢X. Moreover,(i)
v:W→X+ is an AMT-morphism(ii)
iff:X→W is strict, then v:W→X+ is initial, strict and clX+(v(w))=X+.Proof.
Thatv:W→X+ is a map follows from the assumption that f:X→W is initial. Let A∈𝔢X(x). Then x∈clX(A) and therefore f(x)∈clW(f(A)), that is, A∈v(f(x)). Hence v∘f=𝔢X.(i)
For the continuity ofv, let B⊆W and w∈clW(B) but v(w)∉clX+(v(B)). Then ⋂v(B)⊈v(w), that is, there exists A∈⋂v(B) such that A∉v(w). Consequently B⊈clW(f(A)), and thus there is b∈B such that A∉v(b), a contradiction. Therefore v(w)∈clX+(v(B)) and v(clW(B))⊆clX+(v(B)), for all B⊆W. Hence v:W→X+ is continuous. Now we will show that v is a contraction. Let 𝒞∈𝒫2(W) and ℐ=⋃{⋂v(C):C∈𝒞}. Then clW(f(ℐ))≺𝒞. By the initiality of f,ν+(v(𝒞))≤νW(𝒞). Hence v is a contraction.(ii)
It can be verified that ifclW(f(𝒜))≺ℬ, then clX+(𝔢X(𝒜))≺v(ℬ), for all 𝒜∈𝒫2(X) and ℬ∈𝒫2(W). So, νX(𝒜)≤ν+(v(ℬ)), for all 𝒜∈𝒫2(X) such that clW(f(𝒜))≺ℬ, since 𝔢X:X→X+ is initial. Thus sup{νX(𝒜):clW(f(𝒜))≺ℬ}≤ν+(v(ℬ)) implying that νW(ℬ)≤ν+(v(ℬ)), since f is strict in AMER. Hence v:M(W)→M(X+) is initial in AMER. To show that v:T(X)→T(X+) is initial in TOP, we refer to [20].
Now to show thatv is strict, first we will show that clX+(𝔢X(ℐ))≺Ω, where ℐ=⋃{⋂ω:ω∈Ω}. On the contrary, suppose that clX+(𝔢X(ℐ))⊀Ω. Then there exists B∈ℐ such that ω⊈clX+(𝔢X(B)), for all ω∈Ω, that is, for every ω∈Ω, there is ℰ∈ω such that ℰ∉clX+(𝔢X(B)), that is, for every ω∈Ω, there is ℰ∈ω such that B∉ℰ. Consequently, B∉⋂ω for all ω∈Ω, and thus B∉ℐ, a contradiction. Hence clX+(𝔢X(ℐ))≺Ω. Since v∘f=𝔢X, therefore νW(f(ℐ))≤sup{νW(𝒞):clX+(v(𝒞))≺Ω}. But by the initiality of f,ν+(Ω)≤sup{νW(𝒞):clX+(v(𝒞))≺Ω} giving that v is initial.
Finally for denseness ofv, we know that clW(f(X))=W, which gives v(clW(f(X)))=v(W). But v(clW(f(X)))=clX+((v∘f)(X))=clX+(𝔢X(X))=X+. Hence v(W)=X+ and therefore clX+(v(W))=X+.We now concentrate on approach nearness spaces. In [21], we have constructed the cluster completion (X*,ν*) of an approach nearness space (X,ν) (where X* is the family of all ν-clusters) by defining ν* in the same way as ν+ and the map 𝔢X:X→X* similarly as in the case of bunch completion. Since each approach nearness space can be viewed as an approach merotopological space, therefore, by Proposition 4.9, we can obtain a bunch completion of an approach nearness space. Also X*⊆X+ by Proposition 4.2. We introduce the property of regularity in approach merotopic spaces and prove that X* is a retract of X+ for a regular approach nearness space (X,ν). For this, we first consider the following.Lowen and Lee [19] also gave an equivalent description of an approach merotopy on X (by generalizing the concept of micromeric collections of a nonempty set) as a function γ:𝒫2(X)→[0,∞] such that the following conditions are satisfied for any 𝒜,ℬ∈𝒫2(X):(AM1′)
𝒜≺ℬ⇒γ(𝒜)≥γ(ℬ),(AM2′)
there existsA∈𝒜 such that |A|≤1⇒γ(𝒜)=0,(AM3′)
γ(∅)=∞,(AM4′)
γ(𝒜∪ℬ)≥γ(𝒜)∧γ(ℬ).The functionγ is called an approach merotopy on X and (X,γ) is an approach merotopic space. The relation between an approach merotopy γ (generalizing micromeric collections) and an approach merotopy ν (generalizing near collections) on X is given by γ↦νγ and ν↦γν, where, for any 𝒜∈𝒫2(X),νγ(𝒜)=γ(sec𝒜) and γν(𝒜)=ν(sec𝒜).Definition 4.12.
An approach merotopic space(X,ν) is called regular if one of the following equivalent conditions holds:(i)
ν(𝒜)=ν({B⊆X:thereisA∈𝒜satisfyingν({A,X-B})>0}), for all 𝒜∈𝒫2(X);(ii)
γ(𝒜)=ν({B⊆X:forallA∈𝒜,ν({A,B})=0}), for all 𝒜∈𝒫2(X);(iii)
ν(𝒜)=γ({B⊆X:forallA∈𝒜,ν({A,B})=0}), for all 𝒜∈𝒫2(X);(iv)
γ(𝒜)=γ({B⊆X:thereisA∈𝒜satisfyingν({A,X-B})>0}), for all 𝒜∈𝒫2(X).The above equivalences hold by noting thatsec{B⊆X: there is A∈𝒜 satisfying ν({A,X-B})>0}={B⊆X: for all A∈𝒜,ν({A,B})=0} and by the following transitions: ν(𝒜)=γ(sec𝒜) and γ(𝒜)=ν(sec𝒜), for all 𝒜∈𝒫2(X).Definition 4.13.
An approach merotopic space(X,ν) is called separated if, for all 𝒜∈𝒫2(X),(4.3)ν({A⊆X:ν({A}∪𝒜)=0})≤ν(𝒜)∨γ(𝒜).Proposition 4.14.
Every regular approach merotopic space is separated.Proof.
Let(X,ν) be a regular approach merotopic space. Then since {A⊆X:ν({A}∪𝒜)=0}⊆{B⊆X: for all A∈𝒜,ν({A,B})=0}, therefore
(4.4)ν({A⊆X:ν({A}∪𝒜)=0})≤ν({B⊆X:∀A∈𝒜,ν({A,B})=0})=γ(𝒜)≤ν(𝒜)∨γ(𝒜).A morphismf in TOP is a retraction if and only if there exists a topological retraction r and a homeomorphism h such that f=h∘r. In other words, the retractions in TOP are (up to homeomorphism) exactly the topological retractions. Therefore a subspace Y of a space X is called a retract in TOP if there exists a continuous map f:X→Y with f(y)=y for all y∈Y so that if e:Y→X is an inclusion, then f∘e=idY (see [22]).Proposition 4.15.
Let(X,ν) be a regular approach nearness space. Then the map g:X+→X* that is defined as follows: for ℬ∈X+,g(ℬ)={A⊆X:ν({A}∪ℬ)=0} is a retraction.Proof.
Letℬ∈X+. Then secℬ⊆ℬ as ℬ is a grill on X yielding γ(ℬ)=0 which in turn gives that ν(g(ℬ))=0 applying the separability of ν. Thus g(ℬ) is a ν-cluster. Since ℬ⊆g(ℬ), therefore ℬ=g(ℬ), whenever ℬ∈X*. To prove that g is continuous, let ω⊆X+ and ℬ∈clX+(ω). Then we only need to show that ⋂g(ω)⊆g(ℬ). On the contrary if there is an A∈⋂g(ω) such that A∉g(ℬ), then ν({A}∪ℬ)≠0. But ν(ℬ∪{A})=ν({D⊆X:ν({A,X-D})>0 or there is B∈ℬ satisfying ν({B,X-D})>0}). Taking 𝒟={D⊆X:ν({A,X-D})>0 or there is B∈ℬ satisfying  ν({B,X-D})>0}, we get, by regularity of  ν,0≠ν(𝒟)≰ν(ℬ)=0 and so 𝒟⊀ℬ. Since ⋂ω⊆ℬ, therefore D∉⋂ω. Consequently there exists ℋ∈ω such that D∉ℋ. Also A∈g(ℋ) and ν({A,X-D})>0 imply that ν({A}∪ℋ)=0. But ℋ is a grill implying that X-D∈ℋ and therefore ν({A,X-D})=0, a contradiction. So, A∈g(ℬ) and hence ⋂g(ω)⊆g(ℬ). To show that g is a contraction, let Ω⊆𝒫(X+),𝒜=⋃{⋂τ:τ∈g(Ω)}, and ℰ={E⊆X: there is A∈𝒜 satisfying ν({A,X-E})>0}. Then ℰ⊆⋃{⋂ω:ω∈Ω}: for if E∈ℰ, then there exists A∈𝒜 such that ν({A,X-E})>0, for some E⊆X. Suppose that E∉⋂ω, for all ω∈Ω. Then for ω∈Ω, there exists ℬ∈ω such that E∉ℬ which yields that X-E∈ℬ because ℬ is a grill. Thus {A,X-E}≺{A}∪ℬ, where A∈𝒜. So ν({A,X-E})≤ν({A}∪ℬ)=0 which gives that ν({A,X-E})=0 for all A∈𝒜, a contradiction. Thus ν*(g(Ω))=ν(𝒜)=ν(ℰ)≤ν+(Ω). Hence g is a contraction.


## 5. Concluding Remark and Future Applications

The present paper is a unified study of the categoriesMET, TOP, ANEAR, AMER, AP, and AMT. Such type of unified study is relevant as can be seen in [16]. Various examples that are provided support the existence of approach merotopological spaces. Given an approach merotopological space, we can obtain a new approach merotopological space in general, employing the method given in Example 3.2(i) and (iv). Since ANEAR⊆AMT, therefore in this paper we have obtained a bunch completion (X+,ν+) of an approach nearness space (X,ν). In [21], we have obtained a cluster completion (X*,ν*) of the same structure. In fact, X* is a retract of X+, for a regular approach nearness space.In 1988, Smyth [26] suggested that nearness-like concepts provide a suitable vehicle for the study of problems in Theoretical Computer Science. For a comprehensive account with references, see Section 11 (Applications to Theoretical Science) in [27]. Using tools from nearness-like structures, Vakarelov [28] established a topological representation theorem for a connection-based class of systems. More clearly, all digital images used in computer vision and computer graphics can be viewed as nontrivial semi-proximity spaces, which can be seen in [29]. In 2002, work on a perceptual basis for near sets begun, which was motivated by image analysis. This work was inspired by a study of perception of the nearness of familiar physical objects in a philosophical manner in a poem entitled “How Near” written in 2002 and published in 2007 [30]. Since an approach merotopology measures how near a collection of sets is and AMT is a supercategory of almost all of the nearness-like structures, therefore it is our prediction that applying tools of approach merotopological spaces would help in obtaining better outputs in such studies of computer science.



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