Approach Merotopological Spaces and their Completion

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.

1. Introduction

Some of the applications of nearness-like structures within topology are unification, extensions, homology, and connectedness. The categories of 𝑅₀-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 Section 2. 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

Let 𝑋 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}; 𝒜corefines ℬ (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]). A merotopological 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 function 𝜈∶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 cl_𝜈(𝐴)={𝑥∈𝑋∶𝜈({{𝑥},𝐴})=0}, for all 𝐴⊆𝑋. Then cl_𝜈 is a ech closure operator on 𝑋.

An approach merotopy 𝜈 on 𝑋 is called an approach nearness on 𝑋 [21] if the following condition is satisfied:(AN5)𝜈({cl_𝜈(𝐴)∶𝐴∈𝒜})≥𝜈(𝒜).

In this case, cl_𝜈 is a Kuratowski closure operator on 𝑋. Denote cl(𝒜)={cl(𝐴)∶𝐴∈𝒜}.

Definition 2.3 (see [21]). Let 𝒜∈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,𝜈_𝑋) and (𝑌,𝜈_𝑌), a map 𝑓∶𝑋→𝑌 is called a contraction if 𝜈_𝑌(𝑓(𝒜))≤𝜈_𝑋(𝒜), for all 𝒜∈𝒫²(𝑋), or equivalently 𝜈_𝑋(𝑓⁻¹(ℬ))≥𝜈_𝑌(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 1_𝑋∶(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 on 𝑋 is also discussed.

Definition 3.1. An approach 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 on 𝑋.

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 𝜈_𝑔∶𝒫²(𝑋)→[0,∞] as follows: 𝜈_𝑔(𝒜)=inf{𝜈(𝒢)∶𝒜≺𝒢,𝒢∈𝔊(𝑋)},∀𝒜∈𝒫²(𝑋). (3.1) Then 𝜈_𝑔 is an approach merotopology with respect to cl on 𝑋 (see [19, 24]).
(ii) Let (𝑋,cl) be a topological space. Then (𝑋,𝜈_𝑔,cl) is an approach merotopological space, where 𝜈_𝑔 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)=sup_𝐺∈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)=inf_{𝐺∈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 cl_𝜈=cl in all of the above cases (note that a 𝑇₁-space is already a symmetrical topological space).
(iii) Let 𝑓∶(𝑋,cl_𝑋)→(𝑌,cl_𝑌) be a closed and continuous map. Define 𝜈∶𝔊(𝑋)→[0,∞] as follows: for 𝒢∈𝔊(𝑋),𝜈(𝒢)=0, if ⋂{cl_𝑌(𝑓(𝐺))∶𝐺∈𝒢})≠∅; and 𝜈(𝒢)=inf_{𝐺∈seccl(𝒢)}|𝐺|, otherwise. Then (𝑋,𝜈_𝑔,cl_𝑋) is an approach merotopological space. Further if 𝑦∈cl_𝑌(𝑓(𝑥))⇒𝑓(𝑥)∈cl_𝑌({𝑦}) for all 𝑦∈𝑌 and 𝑥∈𝑋, then cl_𝜈=cl_𝑋.
(iv) Let (𝑋,𝜈,cl) be an approach merotopological space. Define 𝜈_𝑐∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝜈_𝑐(𝒜)=sup{𝜈(ℬ)∶ℬ⊆𝒜,|ℬ|<ℵ₀}. (3.2) Then 𝜈_𝑐 is an approach merotopology with respect to the closure cl on 𝑋.
(v) Let (𝑋,cl) be a topological space and 𝑟∈(0,∞]. Define 𝜈^𝑟∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝜈^𝑟(𝒜)=0, if ⋂cl(𝒜)≠∅;𝜈^𝑟(𝒜)=∞, if ∅∈𝒜; and 𝜈^𝑟(𝒜)=𝑟, otherwise. Then 𝜈^𝑟 is an approach merotopology with respect to the closure cl on 𝑋.
(vi) The function 𝜈_𝑑∶𝒫²(𝑋)→[0,∞] this is defined as follows: for 𝒜∈𝒫²(𝑋),𝜈_𝑑(𝒜)=0, if ⋂𝒜≠∅, and 𝜈_𝑑(𝒜)=∞, otherwise, is an approach merotopology with respect to the discrete closure operator 𝑐𝑙_𝑑 on 𝑋 and cl_𝜈=cl_𝑑. We call (𝑋,𝜈_𝑑,cl_𝑑) the discrete approach merotopological space.
(vii) The function 𝜈_𝑖∶𝒫²(𝑋)→[0,∞] this is defined as follows: for 𝒜∈𝒫²(𝑋),𝜈_𝑖(𝒜)=0, if ∅∉𝒜, and 𝜈_𝑖(𝒜)=∞, otherwise, is an approach merotopology with respect to the indiscrete closure operator cl_𝑖 on 𝑋 and cl_𝜈=cl_𝑖. We call (𝑋,𝜈_𝑖,cl_𝑖) the indiscrete approach merotopological space.

Having established the existence of an adequate number of approach merotopologies on 𝑋 (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)⊆cl_𝑣(𝐴) for all 𝐴⊆𝑋, where cl_𝑣 denotes the closure operator induced by the approach merotopy 𝜈 on 𝑋.

Proof. Let 𝐴⊆𝑋 and 𝑥∈cl(𝐴). Then 𝜈({cl({𝑥}),cl(𝐴)})=0 which yields 𝜈({{𝑥},𝐴})=0 and hence 𝑥∈cl_𝜈(𝐴).

Theorem 3.4. The category AMT is a topological construct.

Proof. Let ((𝑌_𝑗,𝜈_𝑗,cl_𝑗))_𝑗∈𝐽 be a family of approach merotopological spaces indexed by 𝐽, and let (𝑓_𝑗∶𝑇(𝑋)→𝑇(𝑌_𝑗))_𝑗∈𝐽 be a source in TOP. Define 𝜈∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝜈(𝒜)=sup{in^𝑛f_𝑖=1sup_𝑗∈𝐽𝜈_𝑗(𝑓_𝑗(𝒜_𝑖))∶(𝒜)^𝑛_𝑖=1∈ℭ(𝒜)}, (3.3) where ℭ(𝒜) is the collection of all finite families (𝒜_𝑖)^𝑛_𝑖=1⊆𝒫²(𝑋) such that 𝒜₁∨𝒜₂∨⋯∨𝒜_𝑛≺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 (𝒜_𝑖)^𝑛_𝑖=1∈ℭ(𝒜). Then (cl(𝒜))^𝑛_𝑖=1∈ℭ(cl(𝒜)). Also since 𝑓_𝑗 is continuous for all 𝑗∈𝐽, therefore 𝜈_𝑗(𝑓_𝑗(𝒜_𝑖))≤𝜈_𝑗(𝑓_𝑗(cl(𝒜_𝑖))), for all (𝒜_𝑖)^𝑛_𝑖=1∈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,𝜈_𝑍,cl_𝑍) be an approach merotopological space and 𝑔∶𝑇(𝑍)→𝑇(𝑋) be a continuous map such that, for every 𝑗∈𝐽, the map 𝑓_𝑗∘𝑔∶𝑍→𝑌_𝑗 is an AMT-morphism. Then 𝑔∶𝑀(𝑍)→𝑀(𝑋) is a contraction since (𝑓_𝑗∶𝑀(𝑋)→𝑀(𝑌_𝑗))_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 ((𝑌_𝑗,𝜈_𝑗,cl_𝑗))_𝑗∈𝐽 be a family of approach merotopological spaces. Then (𝑓_𝑗∶𝑋→𝑌_𝑗)_𝑗∈𝐽 is an initial source in AMT if and only if (𝑓_𝑗∶𝑇(𝑋)→𝑇(𝑌_𝑗))_𝑗∈𝐽 is initial in TOP and (𝑓_𝑗∶𝑀(𝑋)→𝑀(𝑌_𝑗))_𝑗∈𝐽 is initial in AMER.

Proposition 3.6. Let ((𝑋_𝑗,𝜈_𝑗,cl_𝑗))_𝑗∈𝐽 be a family of approach merotopological spaces and (𝑓_𝑗∶𝑋_𝑗→𝑋)_𝑗∈𝐽 be a sink in AMT. Then (𝑓_𝑗∶𝑋_𝑗→𝑋)_𝑗∈𝐽 is a final sink in AMT if and only if both of the following conditions hold:(i)(𝑓_𝑗∶𝑇(𝑋_𝑗)→𝑇(𝑋))_𝑗∈𝐽   is final in TOP, (ii)the approach merotopology with respect to the closure operator cl induced by the final topology on 𝑋 is defined as follows: for 𝒜∈𝒫²(𝑋),𝜈(𝒜)=0,ifcl(𝒜)≠∅,inf_𝑗∈𝐽𝜈_𝑗(𝑓_𝑗⁻¹(cl(𝒜))),otherwise. (3.4)

Proof. To prove that 𝜈 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 (𝑓_𝑗∶𝑋_𝑗→𝑋)_𝑗∈𝐽 is final, let (𝑍,𝜈_𝑍,cl_𝑍) be an approach merotopological space and let 𝑔∶𝑋→𝑍 be a map such that, for every 𝑗∈𝐽,𝑔∘𝑓_𝑗∶𝑋_𝑗→𝑍 is an AMT-morphism. Then 𝑔∶𝑋→𝑍 is continuous since (𝑓_𝑗∶𝑇(𝑋_𝑗)→𝑇(𝑋))_𝑗∈𝐽 is final in TOP. Also for 𝒜⊆𝒫(𝑋) with ⋂cl(𝒜)=∅,𝜈(𝒜)=inf_𝑗∈𝐽𝜈_𝑗(𝑓_𝑗⁻¹(cl(𝒜)))≥inf_𝑗∈𝐽𝜈_𝑍(𝑔(𝑓_𝑗(𝑓_𝑗⁻¹(cl(𝒜)))))≥𝜈_𝑍(𝑔(cl(𝒜)))≥𝜈_𝑍(𝑔(𝒜)). (3.5) On the other hand, if ⋂cl(𝒜)≠∅, then there exists 𝑥∈𝑋 such that 𝑔(𝑥)∈𝑔(cl(𝐴))⊆cl_𝑍(𝑔(𝐴)), for all 𝐴∈𝒜. Hence 𝜈_𝑍(𝑔(𝒜))=0. Thus, 𝑔∶(𝑋,𝜈)→(𝑍,𝜈_𝑍) is a contraction implying that (𝑋,𝜈,cl) is the final approach merotopological space.

Let 𝜌 be a metric on 𝑋. Define 𝜈_𝜌∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝜈_𝜌(𝒜)=inf_𝑥∈𝑋sup_𝐴∈𝒜inf_𝑦∈𝐴𝜌(𝑥,𝑦). (3.6) Then 𝜈_𝜌 is an approach merotopology with respect to the closure cl induced by 𝜌 on 𝑋 and cl_𝜈𝜌=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,𝜈_𝜌,cl_𝜈𝜌)→(𝑌,𝜈_𝜌′,cl_𝜈𝜌′) 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,𝜈_𝜌,𝑐𝑙_𝜈𝜌) and 𝐹(𝑓)=𝑓.

Proposition 3.7. The category TOP 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 𝜈^𝑟 is defined as in Example 3.2 (v). Then 𝜈^𝑟∧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 1_𝑋∶(𝑋,𝜈,cl)→(𝑋,𝜈^𝑟∧𝜈,cl) is the TOP-bireflection of (𝑋,𝜈,cl).

Remark 3.8. (i) There is a full embedding of TOP into AMT that is defined by associating to a topological space 𝑋, the approach merotopy 𝜈^𝑟 as defined in Example 3.2 (v). Thus a topological space can be viewed as an approach merotopological space having the approach merotopy 𝜈^𝑟. 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 𝜈^𝑟.

(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 cl_𝜈 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=cl_𝜈. 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) Let 𝛿 be an approach space. Then the function 𝜈_𝛿∶𝒫²(𝑋)→[0,∞] that is defined as follows: for 𝒜∈𝒫²(𝑋),𝜈_𝛿(𝒜)=inf_𝑥∈𝑋sup_𝐴∈𝒜𝛿(𝑥,𝐴) (3.7) 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,𝜈_𝛿)→(𝑌,𝜈_𝛿′) is a contraction. So we get the functor 𝐹∶𝐀𝐏→𝐀𝐌𝐄𝐑 defined as follows: 𝐹((𝑋,𝛿))=(𝑋,𝜈_𝛿) 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 Remark 3.8, we have established that a topological space can be regarded as an approach merotopological space by associating the approach merotopy 𝜈^𝑟 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 cl_𝜈. 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 𝜈_𝑖 on ℝ be indiscrete. Then (ℝ,𝜈_𝑖,cl) is an approach merotopological space. Also observe that (ℝ,𝜈_𝑖) is an approach nearness space but cl_𝜈𝑖≠cl.

Remark 3.9. Let MERTOP 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,𝜈_𝜉,cl) is an approach merotopological space (where 𝜈_𝜉 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,𝜈_𝜉)→(𝑌,𝜈_𝜉′) 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,𝜈_𝜉,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,𝜉_𝜈,cl) and (𝑋,𝜉^𝜈,cl), where𝜉_𝜈={𝒜∈𝒫²𝜉(𝑋)∶𝜈(𝒜)=0},_𝜈={𝒜∈𝒫²(𝑋)∶𝜈(𝒜)<∞}, (3.8) are merotopological spaces; and if 𝑓∶(𝑋,𝜈)→(𝑌,𝜈^′) is a contraction, then 𝑓∶(𝑋,𝜉^𝜈)→(𝑌,𝜉^𝜈′) and 𝑓∶(𝑋,𝜉_𝜈)→(𝑌,𝜉_𝜈′) are merotopic maps. So this defines the functors 𝐺∶𝐀𝐌𝐓→𝐌𝐄𝐑𝐓𝐎𝐏 by 𝐺((𝑋,𝜈,cl))=(𝑋,𝜉_𝜈,cl) and 𝐺(𝑓)=𝑓 and 𝐺^′∶𝐀𝐌𝐓→𝐌𝐄𝐑𝐓𝐎𝐏 by 𝐺^′((𝑋,𝜈,cl))=(𝑋,𝜉^𝜈,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 1_𝑋∶(𝑋,𝜈_𝜉𝜈,cl)→(𝑋,𝜈,cl) and 1_𝑋∶(𝑋,𝜈,cl)→(𝑋,𝜈_𝜉^𝜈,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 on 𝑋 are constructed.

Definition 3.10. Let 𝜈 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 on 𝑋 forms a completely distributive complete lattice with respect to the partial order “≤”. The zero of this lattice is the indiscrete approach merotopology 𝜈_𝑖 with respect to the closure cl_𝑖 induced by the indiscrete topology (called indiscrete closure) on 𝑋 and the unit is the discrete approach merotopology 𝜈_𝑑 with respect to the discrete closure cl_𝑑 induced by the discrete topology (called discrete closure) on 𝑋.

Proof. Let {(𝑋_𝑗,𝜈_𝑗,cl_𝑗)∶𝑗∈𝐽} (where 𝐽 is an arbitrary index set) be a family of approach merotopological spaces. If 𝜏_𝑗 denotes the topology induced by cl_𝑗 for all 𝑗∈𝐽, then sup_𝑗∈𝐽𝜏_𝑗=⋂{𝑉_𝛼∶𝑉_𝛼 is a topology on 𝑋 and 𝜏_𝑗⊆𝑉_𝛼, for all 𝑗∈𝐽}. Let cl denote the closure operator induced by sup_𝑗∈𝐽𝜏_𝑗. Then for 𝐴⊆𝑋,cl(𝐴)=⋂{𝐵⊆𝑋∶𝐴⊆𝐵 and 𝐵 is sup_𝑗∈𝐽𝜏_𝑗-closed}. Define 𝜈_sup∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝜈_sup(𝒜)=sup{in^𝑛f_𝑖=1sup_𝑗∈𝐽𝜈_𝑗(𝒜_𝑖)∶(𝒜_𝑖)^𝑛_𝑖=1∈ℭ(𝒜)}, (3.9) where ℭ(𝒜) is the collection of all finite families (𝒜_𝑖)^𝑛_𝑖=1⊆𝒫²(𝑋) such that 𝒜₁∨𝒜₂∨⋯∨𝒜_𝑛≺𝒜. Then 𝜈_sup with respect to the closure operator cl is an approach merotopology on 𝑋 and is the supremum of the family of merotopologies {𝜈_𝑗∶𝑗∈𝐽} with respect to the closures {cl_𝑗∶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 𝜈_inf∶𝒫²(𝑋)→[0,∞] as follows: for 𝒜∈𝒫²(𝑋),𝑣_inf(𝐴)=inf_𝑗∈𝐽𝑣_𝑗(cl(𝐴)), (3.10) where cl∶𝒫(𝑋)→𝒫(𝑋) is defined as follows: for 𝐴⊆𝑋,cl(𝐴)=⋂{𝐵⊆𝑋∶𝐴⊆𝐵 and 𝑐𝑙_𝑗(𝐵)=𝐵, for all 𝑗∈𝐽}. Then 𝜈_inf is an approach merotopology with respect to the closure cl on 𝑋 which is also the infimum of the merotopologies {𝜈_𝑗∶𝑗∈𝐽} with respect to the closures {𝑐𝑙_𝑗∶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 𝜉_𝜈={𝒜∈𝒫(𝐿^𝑋)∶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⁺,cl_𝑋⁺) of an approach merotopological space (𝑋,𝜈,cl). Observe that 𝜈⁺ takes several values in [0,∞] if 𝜈 does so, and the completion of 𝜉_𝜈, that is, 𝜉⁺_𝜈, 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 a 𝜈-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. Let 𝒞 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 cl_𝑋⁺ denote the operator defined by cl_𝑋⁺(𝜔)={ℬ∈𝑋⁺∶⋂𝜔⊆ℬ}, for all 𝜔⊆𝑋⁺. Then cl_𝑋⁺ is a Kuratowski closure operator on 𝑋⁺.

Proof. Clearly cl_𝑋⁺(∅)=∅ and 𝜔⊆cl_𝑋⁺(𝜔), where 𝜔⊆𝑋⁺. Let ℬ∈cl_𝑋⁺(𝜔₁∪𝜔₂) but ℬ∉cl_𝑋⁺(𝜔₁)∪cl_𝑋⁺(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∈cl_𝑋⁺(𝜔₁)∪cl_𝑋⁺(𝜔₂). Hence cl_𝑋⁺(𝜔₁∪𝜔₂)⊆cl_𝑋⁺(𝜔₁)∪cl_𝑋⁺(𝜔₂). For the reverse inclusion, let ℬ∉cl_𝑋⁺(𝜔₁∪𝜔₂). Then ⋂𝜔₁⊈ℬ and ⋂𝜔₂⊈ℬ, giving that ℬ∉cl_𝑋⁺(𝜔₁)∪cl_𝑋⁺(𝜔₂). Finally let ℬ∈cl_𝑋⁺(cl_𝑋⁺(𝜔)) but ℬ∉cl_𝑋⁺(𝜔). Then ⋂cl_𝑋⁺(𝜔)⊆ℬ but ⋂𝜔⊈ℬ. Therefore there exists 𝐴∈⋂𝜔 such that 𝐴∉ℬ which implies that 𝐴∉⋂cl_𝑋⁺(𝜔) and thus ⋂𝜔⊈𝒩, for some 𝒩∈𝜔. Consequently 𝒩∉𝜔, which is not possible. Hence cl_𝑋⁺(cl_𝑋⁺(𝜔))=cl_𝑋⁺(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 𝔢_𝑋∶𝑋→𝑋⁺ as follows: for 𝑥∈𝑋,𝔢_𝑋(𝑥)={𝐴⊆𝑋∶𝑥∈cl(𝐴)}.

Theorem 4.4. The map 𝐹∶𝐀𝐌𝐓→𝐀𝐌𝐓 defined by 𝐹((𝑋,𝜈,cl))=(𝑋⁺,𝜈⁺,cl_𝑋⁺) 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⁺,cl_𝑋⁺) 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 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). (4.1) For (AM5) we have to show that ⋃{⋂𝜔∶𝜔∈Ω}⊆⋃{⋂cl_𝑋⁺(𝜔)∶cl_𝑋⁺(𝜔)∈cl_𝑋⁺(Ω)}, where cl_𝑋⁺(Ω)={cl_𝑋⁺(𝜔)∶𝜔∈Ω}. Let 𝐴∈⋃{⋂𝜔∶𝜔∈Ω}. Then 𝐴∈⋂𝜔, for some 𝜔∈Ω. So, 𝐴∈𝒜 for every 𝒜∈cl_𝑋⁺(𝜔) as ⋂𝜔⊆𝒜, for each 𝒜∈cl_𝑋⁺(𝜔). Thus 𝐴∈⋃{⋂cl_𝑋⁺(𝜔)∶cl_𝑋⁺(𝜔)∈cl_𝑋⁺(Ω)}. Hence (𝑋⁺,𝜈⁺,cl_𝑋⁺) is an approach merotopological space.
Next let 𝑓∶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∈cl_𝑋⁺(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 𝜈⁺_𝑌(𝑓⁺(Ω))≤𝜈⁺_𝑋(Ω) and 𝑓⁺∶(𝑋⁺,𝜈⁺_𝑋,cl_𝑋⁺)→(𝑌⁺,𝜈⁺_𝑌,cl_Y⁺) 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(𝑔∘𝑓)⁺(ℬ)={𝐺⊆𝑍∶thereisa𝐵∈ℬsuchthat(𝑔∘𝑓)(ℬ)⊆cl_𝑍(𝐺)}={𝐺⊆𝑍∶(𝑔∘𝑓)⁻¹(cl_𝑍(𝐺))∈ℬ}=𝑓⁺({𝐺⊆𝑍∶𝑔⁻¹(cl_𝑍(𝐺))∈ℬ})=(𝑔⁺∘𝑓⁺)(ℬ). (4.2)
Now to prove that 𝑒∶1→𝐹 is a natural transformation, we will first show that 𝔢_𝑋∶𝑋→𝑋⁺ is an AMT-morphism. For this, let 𝐴⊆𝑋 and 𝒜∈𝔢_𝑋(cl(𝐴)). Then ⋂𝔢_𝑋(cl(𝐴))⊆𝒜 and thus 𝐴∈𝒜 because 𝒜∈𝑋⁺. So, 𝒜∈cl_𝑋⁺𝔢_𝑋(𝐴). Therefore 𝔢_𝑋(cl(𝐴))⊆cl_𝑋⁺(𝔢_𝑋(𝐴)) and hence 𝔢_𝑋∶𝑋→𝑋⁺ is continuous. Also since ⋂𝔢_𝑋(𝐴)⊆𝒜 for all 𝐴∈𝒜∈𝑋⁺, therefore ⋃{⋂𝔢_𝑋(𝐴)∶𝐴∈𝒜}⊆𝒜 which in turn yields that 𝜈⁺(𝔢_𝑋(𝒜))≤𝜈(𝒜). Hence 𝔢_𝑋∶𝑋→𝑋⁺ is an AMT-morphism. Finally let 𝑥∈𝑋 and 𝐵∈𝔢_𝑌(𝑓(𝑥)). Then 𝑓(𝑥)∈cl_𝑌(𝐵) and therefore 𝑓⁻¹(cl_𝑌(𝐵))∈𝔢_𝑋(𝑥) which gives 𝐵∈(𝑓⁺∘𝔢_𝑋)(𝑥). On the other hand if 𝐵∈𝑓⁺(𝔢_𝑋(𝑥)), then there exists 𝐶∈𝔢_𝑋(𝑥) such that 𝑓(𝐶)⊆𝑐𝑙_𝑌(𝐵). Consequently 𝑓(𝑥)∈𝑐𝑙_𝑌(𝐵) and 𝐵∈𝔢_𝑌(𝑓(𝑥)). Hence 𝑓⁺∘𝔢_𝑋=𝔢_𝑌∘𝑓.

Definition 4.5. (i) Let (𝑋,𝜈_𝑋,cl_𝑋) and (𝑌,𝜈_𝑌,cl_𝑌) be approach merotopological spaces. A mapping 𝑓∶(𝑋,𝜈_𝑋,cl_𝑋)→(𝑌,𝜈_𝑌,cl_𝑌) is said to be strict in AMT if {cl_𝑌(𝑓(𝐴))∶𝐴⊆𝑋} is a base for closed subsets of 𝑌 (i.e., strict in TOP (see [25])), and 𝜈_𝑌(ℬ)≤sup{𝜈_𝑋(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(cl_𝑌(𝑓(𝐴)))⊆cl_𝑋(𝐴) for all 𝐴⊆𝑋 (called initial in TOP), and 𝜈_𝑋(𝒜)≤𝜈_𝑌(𝑓(𝒜)) for all 𝒜∈𝒫²(𝑋) (called initial in AMER).

Observe that an initial map 𝑓∶𝑋→𝑌 is strict in AMER if and only if the approach merotopy 𝜈_𝑌 on 𝑌 is defined as: 𝜈_𝑌(ℬ)=sup{𝜈_𝑋(𝒜)∶cl_𝑌(𝑓(𝒜))≺ℬ}, for all 𝐵∈𝒫²(𝑌). Also for any initial map 𝑓∶(𝑋,𝜈_𝑋,cl_𝑋)→(𝑌,𝜈_𝑌,cl_𝑌) in AMER, there is a strict approach merotopology 𝜇_𝑌 with respect to the closure operator cl_𝑌 on 𝑌 defined as: 𝜇_𝑌(ℬ)=sup{𝜈_𝑋(𝒜)∶cl_𝑌(𝑓(𝒜))≺ℬ}, for all ℬ∈𝒫²(𝑌).

Proposition 4.6. Let (𝑋,𝜈,cl) be an approach merotopological space. Then (i)the map 𝔢_𝑋∶𝑋→𝑋⁺ is initial and strict;(ii)cl_𝑋⁺(𝔢_𝑋(𝑋))=𝑋⁺;(iii)𝔢⁺_𝑋∶𝑋⁺→𝑋⁺⁺  is an injective map.Thus 𝑋⁺ can be regarded as a dense subspace of 𝑋⁺⁺.

Proof. (i) Let 𝐴⊆𝑋 and 𝑥∈𝔢_𝑋⁻¹(cl_𝑋⁺(𝔢_𝑋(𝐴))). Then we have 𝔢_𝑋(𝑥)∈cl_𝑋⁺(𝔢_𝑋(𝐴)), that is ⋂𝔢_𝑋(𝐴)⊆𝔢_𝑋(𝑥) which yields 𝑥∈cl(𝐴). Therefore 𝔢_𝑋∶𝑋→𝑋⁺ is initial in TOP. That 𝔢_𝑋 is initial in AMER, follows immediately by the relation 𝒜⊆⋃{⋂𝜔∶𝜔∈𝔢_𝑋(𝒜)}, for all 𝒜∈𝒫²(𝑋). Thus 𝔢_𝑋∶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 cl_𝑋⁺(𝔢_𝑋(𝒜))≺Ω. Thus 𝜈⁺(Ω)=sup{𝜈(𝒜)∶cl_𝑋⁺(𝔢_𝑋(𝒜))≺Ω} as 𝔢_𝑋 is initial in AMER.
(ii) Clearly cl_𝑋⁺(𝔢_𝑋(𝑋))⊆𝑋⁺. Let 𝐴∈⋂𝔢_𝑋(𝑋). Then 𝑋=cl(𝐴) implying that 𝐴∈ℬ, for all ℬ∈𝑋⁺. Hence cl_𝑋⁺(𝔢_𝑋(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⁺,cl_𝑋⁺) is a bunch complete approach merotopological space and (𝑋⁺,cl_𝑋⁺) is a 𝑇₀-space.

Proof. That (𝑋⁺,𝜈⁺,cl_𝑋⁺) 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⊆cl_𝑋⁺(𝔢_𝑋(𝐴∪𝐵))=cl_𝑋⁺(𝔢_𝑋(𝐴))∪cl_𝑋⁺(𝔢_𝑋(𝐵)). Since Ω is a grill on 𝑋⁺, therefore cl_𝑋⁺(𝔢_𝑋(𝐴))∪cl_𝑋⁺(𝔢_𝑋(𝐵))∈Ω which in turn gives that cl_𝑋⁺(𝔢_𝑋(𝐴))∈Ω or cl_𝑋⁺(𝔢_𝑋(𝐵))∈Ω. Also 𝐴∈⋂cl_𝑋⁺(𝔢_𝑋(𝐴))∈Ω and 𝐵∈⋂cl_𝑋⁺(𝔢_𝑋(𝐵))∈Ω yield that 𝐴∈ℬ or 𝐵∈ℬ. Now we will show that ⋂cl_𝑋⁺(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∈cl_𝑋⁺({ℬ₂}) and ℬ₂∈cl_𝑋⁺({ℬ₁}). Then ℬ₁⊆ℬ₂ and ℬ₂⊆ℬ₁. Hence cl_𝑋⁺ 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,𝜈^𝑟,cl), following the convention established previously, therefore each topological space (𝑋,cl) is bunch complete.

The following theorem shows that the bunch completion 𝑋⁺ of 𝑋 possesses a universal mapping property that shows it to be very large.

Theorem 4.11. Let (𝑋,𝜈_𝑋,cl_𝑋) be an approach merotopological space and let 𝔢_𝑋∶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,𝜈_𝑊,cl_𝑊) such that cl_𝑊(𝑓(𝑋))=𝑊, then the map 𝑣∶𝑊→𝑋⁺ that is defined as follows: for 𝑤∈𝑊,𝑣(𝑤)={𝐴⊆𝑋∶𝑤∈cl_𝑊(𝑓(𝐴))}, is such that 𝑣∘𝑓=𝔢_𝑋. Moreover,(i)𝑣∶𝑊→𝑋⁺ is an AMT-morphism(ii)if 𝑓∶𝑋→𝑊 is strict, then 𝑣∶𝑊→𝑋⁺ is initial, strict and cl_𝑋⁺(𝑣(𝑤))=𝑋⁺.

Proof. That 𝑣∶𝑊→𝑋⁺ is a map follows from the assumption that 𝑓∶𝑋→𝑊 is initial. Let 𝐴∈𝔢_𝑋(𝑥). Then 𝑥∈cl_𝑋(𝐴) and therefore 𝑓(𝑥)∈cl_𝑊(𝑓(𝐴)), that is, 𝐴∈𝑣(𝑓(𝑥)). Hence 𝑣∘𝑓=𝔢_𝑋.(i)For the continuity of 𝑣, let 𝐵⊆𝑊 and 𝑤∈cl_𝑊(𝐵) but 𝑣(𝑤)∉cl_𝑋⁺(𝑣(𝐵)). Then ⋂𝑣(𝐵)⊈𝑣(𝑤), that is, there exists 𝐴∈⋂𝑣(𝐵) such that 𝐴∉𝑣(𝑤). Consequently 𝐵⊈cl_𝑊(𝑓(𝐴)), and thus there is 𝑏∈𝐵 such that 𝐴∉𝑣(𝑏), a contradiction. Therefore 𝑣(𝑤)∈cl_𝑋⁺(𝑣(𝐵)) and 𝑣(cl_𝑊(𝐵))⊆cl_𝑋⁺(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 cl_𝑊(𝑓(ℐ))≺𝒞. By the initiality of 𝑓,𝜈⁺(𝑣(𝒞))≤𝜈_𝑊(𝒞). Hence 𝑣 is a contraction.(ii)It can be verified that if cl_𝑊(𝑓(𝒜))≺ℬ, then cl_𝑋⁺(𝔢_𝑋(𝒜))≺𝑣(ℬ), for all 𝒜∈𝒫²(𝑋) and ℬ∈𝒫²(𝑊). So, 𝜈_𝑋(𝒜)≤𝜈⁺(𝑣(ℬ)), for all 𝒜∈𝒫²(𝑋) such that cl_𝑊(𝑓(𝒜))≺ℬ, since 𝔢_𝑋∶𝑋→𝑋⁺ is initial. Thus sup{𝜈_𝑋(𝒜)∶cl_𝑊(𝑓(𝒜))≺ℬ}≤𝜈⁺(𝑣(ℬ)) implying that 𝜈_𝑊(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 that 𝑣 is strict, first we will show that cl_𝑋⁺(𝔢_𝑋(ℐ))≺Ω, where ℐ=⋃{⋂𝜔∶𝜔∈Ω}. On the contrary, suppose that cl_𝑋⁺(𝔢_𝑋(ℐ))⊀Ω. Then there exists 𝐵∈ℐ such that 𝜔⊈cl_𝑋⁺(𝔢_𝑋(𝐵)), for all 𝜔∈Ω, that is, for every 𝜔∈Ω, there is ℰ∈𝜔 such that ℰ∉cl_𝑋⁺(𝔢_𝑋(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 cl_𝑋⁺(𝔢_𝑋(ℐ))≺Ω. Since 𝑣∘𝑓=𝔢_𝑋, therefore 𝜈_𝑊(𝑓(ℐ))≤sup{𝜈_𝑊(𝒞)∶cl_𝑋⁺(𝑣(𝒞))≺Ω}. But by the initiality of 𝑓,𝜈⁺(Ω)≤sup{𝜈_𝑊(𝒞)∶cl_𝑋⁺(𝑣(𝒞))≺Ω} giving that 𝑣 is initial.
Finally for denseness of 𝑣, we know that cl_𝑊(𝑓(𝑋))=𝑊, which gives 𝑣(cl_𝑊(𝑓(𝑋)))=𝑣(𝑊). But 𝑣(cl_𝑊(𝑓(𝑋)))=cl_𝑋⁺((𝑣∘𝑓)(𝑋))=cl_𝑋⁺(𝔢_𝑋(𝑋))=𝑋⁺. Hence 𝑣(𝑊)=𝑋⁺ and therefore cl_𝑋⁺(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 𝔢_𝑋∶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 exists 𝐴∈𝒜 such that |𝐴|≤1⇒𝛾(𝒜)=0,(AM3^′)𝛾(∅)=∞,(AM4^′)𝛾(𝒜∪ℬ)≥𝛾(𝒜)∧𝛾(ℬ).

The function 𝛾 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↦𝜈_𝛾 and 𝜈↦𝛾_𝜈, where, for any 𝒜∈𝒫²(𝑋),𝜈_𝛾(𝒜)=𝛾(sec𝒜) and 𝛾_𝜈(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 that sec{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),𝜈({𝐴⊆𝑋∶𝜈({𝐴}∪𝒜)=0})≤𝜈(𝒜)∨𝛾(𝒜). (4.3)

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 𝜈({𝐴⊆𝑋∶𝜈({𝐴}∪𝒜)=0})≤𝜈({𝐵⊆𝑋∶∀𝐴∈𝒜,𝜈({𝐴,𝐵})=0})=𝛾(𝒜)≤𝜈(𝒜)∨𝛾(𝒜). (4.4)

A morphism 𝑓 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=id_𝑌 (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. Let ℬ∈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∈cl_𝑋⁺(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≺ℬ̸. 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 categories MET, 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.