Approach Merotopological Spaces and Their Completion

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 409804, 16 pages doi:10.1155/2010/409804

Research Article Approach Merotopological Spaces and their Completion

Mona Khare and Surabhi Tiwari

Department of Mathematics, University of Allahabad, Allahabad 211002, India

Correspondence should be addressed to Mona Khare, [email protected]

Received 24 July 2009; Accepted 13 April 2010

Academic Editor: Richard Wilson

Copyright q 2010 M. Khare and S. Tiwari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.In14, 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 2 International Journal of Mathematics and Mathematical Sciences 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.InSection 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 PX and the family of P P2 ℵ all subsets of X is denoted by X . We denote by 0 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 PX, A∨B≡{A ∪ B : A ∈A,B ∈B}; A corefines B written as A≺B if and only if for all A ∈Athere exists B ∈Bsuch that B ⊆ A. For A⊆PX, stackA{A ⊆ X : B ⊆ A for some B ∈A}and sec A {B ⊆ X : A ∩ B / ∅, for all A ∈A} {B ⊆ X : X − B /∈ stackA}. Observe that sec2A stack A, for all A∈P2X. A grill on X is a subset G of PX satisfying ∅ /∈G; if A ∈Gand A ⊆ B, then B ∈G;andifA ∪ B ∈G, then A ∈Gor 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 {clA : A ∈A}∈ξ ⇒A∈ξ, for all A∈P2X.

Deﬁnition 2.2 see 19, 21.Afunctionν : P2X → 0, ∞ is called an approach merotopy on X if for any A, B∈P2X the following conditions are satisﬁed:

AM1 A≺B⇒νA ≤ νB, AM2 A / ∅⇒νA0,

AM3 ∅∈A⇒νA∞,

AM4 νA∨B ≥ νA ∧ νB.

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 Cechˇ closure operator on X. International Journal of Mathematics and Mathematical Sciences 3

An approach merotopy ν on X is called an approach nearness on X 21 if the following condition is satisﬁed: { ∈A} ≥ A AN5 ν clν A : A ν . A{ ∈A} In this case, clν is a Kuratowski closure operator on X. Denote cl cl A : A .

Definition 2.3 see 21.LetA∈P2X and ν be an approach merotopy on X. Then we say that A is i near in ν if νA0, 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 A ≤ A A∈P2 −1B ≥ B contraction if νY f νX , for all X , or equivalently νX f νY , for B∈P2 ≤ all 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 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 νclA ≥ νA, for all A∈P2X. 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 GX be the family P2 → ∞ of all grills on X. Deﬁne νg : X 0, as follows:

A G A≺G G∈G ∀A ∈ P2 νg inf{ν : , X}, X. 3.1

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 G → ∞ space, where νg is defined as in the above example in which ν : X 0, can be defined in the following ways for G∈GX. G G ∅ G {| | ∈G | | ℵ } a ν 0, if cl / ;andν sup cl G : G and cl G < 0 , otherwise. G G ∅ G | | b ν 0, if cl / ;andν supG∈G stack cl G , otherwise. 4 International Journal of Mathematics and Mathematical Sciences c νG0, if clG / ∅;andνG| clG|, otherwise. G G ∅ G | | d ν 0, if cl / ;andν infG∈sec clG G , otherwise. e νG0, if clB / ∅ for every finite subset B of G;andνGinf{|G| : G ∈ sec clG}, otherwise. f νG0, if clG / ∅ or each element of G is infinite; and νGsup{|G| : G ∈G | | ℵ } and G < 0 , 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. → G → iii Let f : X, clX Y, clY be a closed and continuous map. Deﬁne ν : X ∞ G∈G G { ∈G} ∅ G 0, as follows: for X ,ν 0, if clY f G : G / ;andν | | infG∈sec clG 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. P2 → ∞ iv Let X, ν, cl be an approach merotopological space. Deﬁne νc : X 0, as follows: for A∈P2X,

A B B⊆A B ℵ νc sup{ν : , | | < 0}. 3.2

Then νc is an approach merotopology with respect to the closure cl on X. ∈ ∞ r P2 → ∞ v Let X, cl be a topological space and r 0, . Define ν : X 0, as follows: for A∈P2X,νr A0, if clA / ∅; νr A∞, if ∅∈A;andνr Ar, otherwise. Then νr is an approach merotopology with respect to the closure cl on X. P2 → ∞ A∈ vi The function νd : X 0, this is defined as follows: for P2 A A ∅ A∞ X ,νd 0, if / , and νd , 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. P2 → ∞ A∈ vii The function νi : X 0, this is defined as follows: for P2 A ∅ ∈A A∞ X ,νi 0, if / , and νi , 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 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 clv A for all ⊆ A X, where clv denotes the closure operator induced by the approach merotopy ν on X.

Proof. Let A ⊆ X and x ∈ clA. Then ν{cl{x}, clA}0 which yields ν{{x},A}0 ∈ and hence x clν A . International Journal of Mathematics and Mathematical Sciences 5

Theorem 3.4. The category AMT is a topological construct. Proof. Let Yj ,νj , clj j∈J be a family of approach merotopological spaces indexed by J, and → TOP P2 → ∞ let fj : T X T Yj j∈J be a source in . Deﬁne ν : X 0, as follows: for A∈P2X,

n n νA sup inf sup ν f A : A ∈ CA , j j i i 1 3.3 i 1 j∈J

CA A n ⊆P2 A ∨A ∨···∨A ≺ where is the collection of all ﬁnite families i i1 X such that 1 2 n A. Then ν is an approach merotopy on X see 19, Theorem 3.8. Let cl denote the closure A n ∈ CA An ∈ C A induced by the initial topology on X and i i1 . Then cl i1 cl . Also ∈ A ≤ A A n ∈ CA since fj is continuous for all j J, therefore νj fj i νj fj cl i , for all i i1 which in turn yields νA ≤ νclA. 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.ThusX, 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 AMT → TOP Yj j∈J is an initial source in if and only if fj : T X T Yj j∈J is initial in and → AMER fj : M X M Yj j∈J is initial in . Proposition 3.6. → Let Xj ,νj , clj j∈J be a family of approach merotopological spaces and fj : Xj AMT → AMT X j∈J beasinkin .Then fj : Xj X j∈J is a ﬁnal sink in if and only if both of the following conditions hold:

→ TOP i fj : T Xj T X j∈J is ﬁnal in ,

ii the approach merotopology with respect to the closure operator cl induced by the ﬁnal topology on X is deﬁned as follows: for A∈P2X,

⎧ ⎪ ⎨0, if clA / ∅, νA 3.4 ⎪ −1 A ⎩inf νj f cl , otherwise. j∈J j

Proof. To prove thatν is an approach merotopy with respect to cl on X, we will prove only AM5. For this, let clA / ∅. Then there exists x ∈ X such that A⊆{A ⊆ X : x ∈ clA}. A A → Hence ν cl 0 ν . 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 : → AMT → → Xj Z is an -morphism. Then g : X Z is continuous since fj : T Xj T X j∈J 6 International Journal of Mathematics and Mathematical Sciences is final in TOP.AlsoforA⊆PX with clA∅, A −1 A ν inf νj fj cl j∈J ≥ −1 A inf νZ g fj fj cl j∈J 3.5 ≥ A νZ gcl ≥ A νZ g . On the other hand, if clA / ∅, then there exists x ∈ X such that gx ∈ gclA ⊆ ∈A A → clZ g A , for all A . Hence νZ g 0. Thus, g : X, ν Z, νZ is a contraction implying that X, ν, cl is the final approach merotopological space.

P2 → ∞ A∈P2 Let ρ be a metric on X. Deﬁne νρ : X 0, as follows: for X , ν A inf sup inf ρ x, y . ρ ∈ ∈ 3.6 x X A∈A y A

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

→ AMT map if and only if f : X, νρ, clνρ Y, νρ , clνρ is an -morphism. Thus the category MET of metric spaces and nonexpansive maps is embedded as a full subcategory into AMT MET → AMT by the functor F : deﬁned as follows: F X, ρ X, νρ,clνρ 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{νA : A∈ P2X,νA / 0and∅ /∈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 of TOP into AMT that is deﬁned by associating to a topological space X, the approach merotopy νr as deﬁned in Example 3.2 v.Thusa 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 International Journal of Mathematics and Mathematical Sciences 7 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. P2 → ∞ iii Let δ be an approach space. Then the function νδ : X 0, that is deﬁned as follows: for A∈P2X,

ν A inf sup δx, A δ ∈ 3.7 x X A∈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 AP → AMER AP functor F : defined as follows: F X, δ X, νδ and F f f. Therefore 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 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 R of all closed sets. Then , cl is a non-symmetric and T0-space, where cl is the closure operator R associated with the topology on defined as above. Further let the approach merotopy νi on R R R be indiscrete. Then ,νi, cl is an approach merotopological space. Also observe that ,νi is an approach nearness space but clνi / 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, νξ, cl is an approach merotopological space where νξ is the induced approach merotopy on X deﬁned as follows: for A∈P2X,νA0, if A∈ξ;and νA∞, otherwise. Also for any merotopic spaces X, ξ and Y, ξ ,f : X, ξ → Y, ξ → MERTOP is a merotopic map if and only if f : X, νξ Y, νξ is a contraction. Thus 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 νP2X ⊆{0, ∞}. Also for any approach merotopological space ν X, ν, cl , the triples X, ξν, cl and X, ξ , cl , where A∈P2 A ξν X : ν 0 , 3.8 ξν A∈P2X : νA < ∞ ,

are merotopological spaces; and if f : X, ν → Y, ν is a contraction, then f : X, ξν → ν → Y, ξ and f : X, ξν Y, ξν are merotopic maps. So this defines the functors G : AMT → MERTOP AMT → by G X, ν, cl X, ξν, cl and G f f and G : MERTOP by G X, ν, cl X, ξν, cl and G ff. Hence we can conclude that the 8 International Journal of Mathematics and Mathematical Sciences 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 → MERTOP MERTOP 1X : X, ν, cl X, νξν , cl are the -bicoreflection and -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 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∈J τj { ⊆ ∈ } Vα : Vα is a topology on X and τj Vα, for all j J . Let cl denote the closure operator ⊆ { ⊆ ⊆ } induced by supj∈J τj . Then for A X, cl A B X : A B and B is supj∈J τj -closed . P2 → ∞ A∈P2 Deﬁne νsup : X 0, as follows: for X ,

n n ν A sup inf sup ν A : A ∈ CA , sup j i i i 1 3.9 i 1 j∈J

CA A n ⊆P2 A ∨A ∨···∨A ≺ where is the collection of all finite families i i1 X such that 1 2 n 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 . P2 → ∞ Now we construct the infimum of the given family. Define νinf : X 0, as follows: for A∈P2X,

v A inf v clA, 3.10 inf ∈ j J j

where cl : PX →PX is deﬁned as follows: for A ⊆ X, clA {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 . International Journal of Mathematics and Mathematical Sciences 9

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 {A ∈ P X A } associated merotopology ξν L : ν 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 νG0andclG ∈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 clG ∈C. Then ν{clC, clG}0, for all C ∈Cwhich by AM5 yields that ν{C, G}0, for all C ∈C. Consequently G ∈Cas 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 Gm{A ⊆ X : | clA|≥m} where m is an inﬁnite cardinal number. Consider the approach merotopological space X, ν, cl of the last case of Example 3.2 iif. Then for m >cwhere c is the cardinality of the set R of all real numbers, Gm is a ν-bunch which is not a ν-cluster Gm Gℵ Gℵ because is a proper subset of 0 and ν 0 0.

Theorem 4.3. Let X, ν, cl be an approach merotopological space and let clX denote the operator {B ∈ ⊆B} ⊆ deﬁned by clX ω X : ω , for all ω X . Then clX is a Kuratowski closure operator on X .

∅∅ ⊆ ⊆ B∈ ∪ Proof. Clearly clX and ω clX ω , where ω X . Let clX ω1 ω2 but B ∈ ∪ ∩ ⊆B ⊆B ⊆B / clX ω1 clX ω2 .Then ω1 ω2 but ω1 / and ω2 / . Therefore there ∈ ∈ ∈B ∈B ∪ ∈B ∈A exist A ω1 and B ω2 such that A / and B / , which gives A B / . Also A and ∈B A∈ B∈ ∪ ∈ ∪ B , for every ω1 and ω2, and consequently A B ω1 ω2 , which contradicts ∪ ⊆B B∈ ∪ ∪ ⊆ ∪ that ω1 ω2 . Thus clX ω1 clX ω2 . Hence clX ω1 ω2 clX ω1 clX ω2 . B ∈ ∪ ⊆B ⊆B For the reverse inclusion, let / clX ω1 ω2 . Then ω1 / and ω2/ , giving that B ∈ ∪ B∈ B ∈ ⊆B / clX ω1 clX ω2 . Finally let clX clX ω but / clX ω . Then clX ω but ⊆B ∈ ∈B ∈ ω / . Therefore there exists A ω such that A / which implies that A / clX ω 10 International Journal of Mathematics and Mathematical Sciences 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 { ⊆ ∈ } e → family A X : x cl A is a ν-bunch. Therefore we can deﬁne the map X : X X as ∈ e { ⊆ ∈ } follows: for x X, X x A X : x cl A .

Theorem 4.4. AMT → AMT The map F : defined by F X, ν, cl X ,ν , clX and F f f , where ν : P2X → 0, ∞ is defined as follows: for Ω ∈P2X,νΩ ν { ω : ω ∈ Ω}, and f : X → Y (when f : X → Y is defined as follows: for B∈X ,f B{A ⊆ Y : fB ⊆ clA, for some B ∈B}, is a functor. Moreover, e :1 → F is a natural transformation from the identity functor.

Ω I ∈ Proof. First we will show that X ,ν , clX is an approach merotopological space. Let , P2 Ω ≺ I { ∈ Ω}≺ { ∈ I} Ω ≤ I X and . Then ω : ω λ : λ . Thus ν ν . If Ω ∅ A∈ Ω ⊆A ∈ Ω / , then there exists and therefore ω , for every ω . Consequently { ∈ Ω}⊆A Ω ω : ω which yields ν 0. Condition AM3 follows by the convention ∅∈ ∅. Now

ν Ω ∨ I ν ω : ω ∈ Ω ∨ I

≥ ν ω : ω ∈ Ω ∨ ω : ω ∈ I 4.1 ≥ ν ω : ω ∈ Ω ∧ ν ω : ω ∈ I

ν Ω ∧ ν I.

{ ∈ Ω}⊆ { ∈ Ω} For AM5 we have to show that ω :ω clX ω :clX ω clX , where Ω { ∈ Ω} ∈ { ∈ Ω} ∈ ∈ Ω clX clX ω : ω . Let A ω : ω . Then A ω, for some ω . So, ∈A A∈ ⊆A A∈ ∈ { A for every clX ω as ω , for each clX ω . Thus A clX ω : ∈ Ω} clX ω clX . Hence X ,ν , clX is an approach merotopological space. Next let f : X → Y be an AMT-morphism. Then we will show that f : X → Y AMT B ≺ B is an -morphism. That f is a map follows from the fact that cl f f , for all B∈P2 → ⊆ B∈ ∈ X . For the continuity of f : X Y , let ω X and clX ω and A f ω . ∈ C C∈ ⊆ ∈B Then A f , for every ω. Consequently f B cl A , for some B and hence ⊆ B → f ω f which in turn gives that f : X Y is continuous. Now to show that → Ω ∈P2 ∈ { ∈ Ω} f : X Y is a contraction, let X and A f ω : f ω f . Then ∈ ∈ Ω B∈ ∈B there is an A f ω0 , for some ω0 . Thus for every ω0, there is a B such that fB ⊆ clA. Since fB ⊆ fB, therefore cl { f ω : f ω ∈ f Ω} ≺ f { ω : ∈ Ω} Ω ≤ Ω → AMT ω . Thus νY f νX and f : X ,νX, clX Y ,νY , clY is an - 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 International Journal of Mathematics and Mathematical Sciences 11

B∈X . Then ◦ B ⊆ ∈B ◦ B ⊆ g f G Z : there is a B such that g f clZG ⊆ ◦ −1 ∈B G Z : g f clZ G 4.2 ⊆ −1 ∈B ◦ B f G Z : g clZG g f .

→ e Now to prove that e :1 F is a natural transformation, we will ﬁrst show that X : → AMT ⊆ A∈e e ⊆A X X is an -morphism. For this, let A X and X cl A . Then X cl A ∈A A∈ A∈ e e ⊆ e and thus A because X . So, clX X A . Therefore X cl A clX X A and e → e ⊆A ∈A∈ hence X : X X is continuous. Also since X A for all A X , therefore { e ∈A}⊆A e A ≤ A e → X A : A which in turn yields that ν X ν . Hence X : X X AMT ∈ ∈ e ∈ is an -morphism. Finally let x X and B Y f x . Then f x clY B and therefore −1 ∈ e ∈ ◦ e ∈ e f clY B X x which gives B f X x . On the other hand if B f X x , then ∈ e ⊆ ∈ ∈ e there exists C X x such that f C clY B . Consequently f x clY B and B Y f x . ◦ e e ◦ Hence f X Y f. Deﬁnition 4.5. i Let X, νX, clX and Y, νY , clY be approach merotopological spaces. A → AMT { ⊆ } mapping f : X, νX, clX Y, νY , clY is said to be strict in if clY f A : A X TOP B ≤ { A is a base for closed subsets of Y i.e., strict in see 25 ,andνY sup νX : clfA ≺B}, for all B∈P2Xi.e., strict in AMER. AMT −1 ⊆ ⊆ ii Further, f said to be initial in if f clY f A clX A for all A X called TOP A ≤ A A∈P2 AMER initial in ,andνX νY f for all X called initial in .

Observe that an initial map f : X → Y is strict in AMER if and only if the B { A A ≺B} approach merotopy νY on Y is deﬁned as: νY sup νX :clY f , for all ∈P2 → AMER B Y . Also for any initial map f : X, νX, clX Y, νY , clY in , there is a strict approach merotopology μY with respect to the closure operator clY on Y deﬁned as: B { A A ≺B} B∈P2 μY sup νX :clY f , for all Y .

Proposition 4.6. Let X, ν, cl be an approach merotopological space. Then e → i the map X : X X is initial and strict; e ii clX X X X ; e → iii X : X X is an injective map. Thus X can be regarded as a dense subspace of X .

− ⊆ ∈ e 1 e e ∈ e Proof. i Let A X and x X clX X A . Then we have X x clX X A , that is e ⊆ e ∈ e → TOP e X A X x which yields x cl A . Therefore X : X X is initial in .That X AMER A⊆ { ∈ e A} is initial in , follows immediately by the relation ω : ω X , for all A∈P2 e → AMT TOP X . Thus X : X X is initial in . For strictness in , we refer to 20 . For AMER Ω ⊆P A { ∈ Ω} e A ≺ Ω strictness in ,let X and ω : ω . Then clX X . Thus Ω { A e A ≺ Ω} e AMER ν sup ν :clX X as X is initial in . e ⊆ ∈ e ∈B ii Clearly clX X X X . Let A X X . Then X cl A implying that A , B∈ e for all X . Hence clX X X X . iii See 20, Proposition 22. 12 International Journal of Mathematics and Mathematical Sciences

Deﬁnition 4.7. An approach merotopological space X, ν, cl is said to be bunch complete if clB / ∅, 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 νG0, has clG / ∅. G G Proof. Let X, ν, cl be bunch complete and let be a grill on X such that ν 0. Then G⊆{G ⊆ X :clG ∈G}. Thus clG / ∅. 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 B { ∈ Ω} B ∪ ∈B be a ν -bunch and ω : ω . Then is a ν-bunch: let A B . Then there exists ∈ Ω ∪ ∈ ⊆ e ∪ e ∪ e ω0 such that A B ω0. Consequently ω0 clX X A B clX X A clX X B . Ω e ∪ e ∈ Ω Since is a grill on X , therefore clX X A clX X B which in turn gives that e ∈ Ω e ∈ Ω ∈ e ∈ Ω ∈ e ∈ Ω clX X A or clX X B . Also A clX X A and B clX X B ∈B ∈B Ω ∅ ﬃ yield that A or B . Now we will show that clX / , for which it is su cient ⊆B ∈ Ω B B ∈ to note that ω , for all ω . Thus ν is bunch complete. Now let 1, 2 X and B ∈ {B } B ∈ {B } B ⊆B B ⊆B 1 clX 2 and 2 clX 1 . Then 1 2 and 2 1. 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 completion X of X possesses a universal mapping property that shows it to be very large. Theorem 4.11. e → Let X, νX, clX be an approach merotopological space and let 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, νW , clW such that clW f X W, then the map v : W X ∈ { ⊆ ∈ } ◦ e that is deﬁned as follows: for w W, v w A X : w clW f A , is such that v f 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 clX v w X .

Proof. That v : W → X is a map follows from the assumption that f : X → W is initial. ∈ e ∈ ∈ ∈ Let A X x . Then x clX A and therefore f x clW f A , that is, A v f x . Hence ◦ e v f X.

⊆ ∈ ∈ i For the continuity of v, let B W and w clW B but v w / clX v B . Then vB /⊆ vw, that is, there exists A ∈ vB such that A /∈ vw. 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 C∈P2 I is continuous. Now we will show that v is a contraction. Let W and { ∈C} I ≺C C ≤ C v C : C . Then clW f . By the initiality of f, ν v νW . Hence v is a contraction. International Journal of Mathematics and Mathematical Sciences 13

A ≺B e A ≺ B A∈P2 ii It can be veriﬁed that if clW f , then clX X v , for all X B∈P2 A ≤ B A∈P2 A ≺ and W . So, νX ν v , for all X such that clW f B e → { A A ≺B}≤ B , since X : X X is initial. Thus sup νX :clW f ν v B ≤ B AMER → implying that νW ν v , since f is strict in . Hence v : M W MX is initial in AMER. To show that v : TX → TX is initial in TOP,we refer to 20.

e I ≺ Ω I Now to show that v is strict, ﬁrst we will show that clX X , where { ∈ Ω} e I ≺ Ω ∈I ω : ω . On the contrary, suppose that clX X / . Then there exists B ⊆ e ∈ Ω ∈ Ω E∈ such that ω / clX X B , for all ω , that is, for every ω , there is ω such E ∈ e ∈ Ω E∈ ∈E that / clX X B , that is, for every ω , there is ω such that B / . Consequently, ∈ ∈ Ω ∈I e I ≺ Ω B / ω for all ω , and thus B / , a contradiction. Hence clX X . Since ◦ e I ≤ { C C ≺ Ω} v f X, therefore νW f sup νW :clX v . But by the initiality of Ω ≤ { C C ≺ Ω} f, ν sup νW :clX v giving that v is initial. Finally for denseness of v, we know that clW f X W, which gives v clW f X ◦ e v W . But v clW f X clX v f X clX X 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, νwhere X is the family of all ∗ e → ∗ ν-clusters by defining ν in the same way as ν and the map 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 : P2X → 0, ∞ such that the following conditions are satisfied for any A, B∈P2X:

AM1 A≺B⇒γA ≥ γB, AM2 there exists A ∈Asuch that |A|≤1 ⇒ γA0, AM3 γ∅∞, AM4 γA∪B ≥ γA ∧ γB.

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 νγ and ν γν, A∈P2 A A A A where, for any X ,νγ γ sec and γν ν sec .

Deﬁnition 4.12. An approach merotopic space X, ν is called regular if one of the following equivalent conditions holds:

i νAν{B ⊆ X : there is A ∈Asatisfying ν{A, X −B} > 0}, for all A∈P2X; ii γAν{B ⊆ X : for all A ∈A,ν{A, B}0}, for all A∈P2X; iii νAγ{B ⊆ X : for all A ∈A,ν{A, B}0}, for all A∈P2X; iv γAγ{B ⊆ X : there is A ∈Asatisfying ν{A, X −B} > 0}, for all A∈P2X. 14 International Journal of Mathematics and Mathematical Sciences

The above equivalences hold by noting that sec {B ⊆ X : there is A ∈Asatisfying ν{A, X − B} > 0} {B ⊆ X : for all A ∈A,ν{A, B}0} and by the following transitions: νAγsec A and γAνsec A, for all A∈P2X.

Deﬁnition 4.13. An approach merotopic space X, ν is called separated if, for all A∈P2X,

ν{A ⊆ X : ν{A} ∪A 0} ≤ νA ∨ γA. 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} 4.4 γA ≤ νA ∨ γA.

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 fyy 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 ,gB{A ⊆ X : ν{A}∪B0} is a retraction.

Proof. Let B∈X . Then sec B⊆Bas B is a grill on X yielding γB0 which in turn gives that νgB 0 applying the separability of ν. Thus gB is a ν-cluster. Since B⊆gB, therefore ∗ B B B∈ ⊆ B∈ g , whenever X .To prove that g is continuous, let ω X and clX ω . Then we only need to show that gω ⊆ gB. On the contrary if there is an A ∈ gω such that A /∈ gB, then ν{A}∪B / 0. But νB∪{A}ν{D ⊆ X : ν{A, X − D} > 0 or there is B ∈Bsatisfying ν{B, X − D} > 0}. Taking D {D ⊆ X : ν{A, X − D} > 0 or there ∈B { − } } D≤ B is B satisfying ν B, X D > 0 , we get, by regularity of ν, 0 / ν / ν 0andso D/≺B. Since ω ⊆B, therefore D /∈ ω. Consequently there exists H∈ω such that D /∈H. Also A ∈ gH and ν{A, X − D} > 0 imply that ν{A}∪H0. But H is a grill implying − ∈H { − } ∈ B that X D and therefore ν A, X D 0, a contradiction. So, A g and hence ⊆ B Ω ⊆P A { ∈ Ω} g ω g . To show that g is a contraction, let X , τ : τ g , and E {E ⊆ X : there is A ∈Asatisfying ν{A, X − E} > 0}. Then E⊆ { ω : ω ∈ Ω} :for ∈E ∈A { − } ⊆ if E , then there exists 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 /∈Bwhich yields that X − E ∈Bbecause B is a grill. Thus {A, X − E}≺{A}∪B, where A ∈A. So ν{A, X − E} ≤ ν{A}∪B0 which gives that ν{A, X − E}0 for all A ∈A, a contradiction. Thus ∗ ν gΩ νAνE ≤ 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 International Journal of Mathematics and Mathematical Sciences 15 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 3.2i 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.

References

1 H. Herrlich, “Topological structures,” in Topological Structures (Proc. Sympos. in Honour of Johannes de Groot (1914–1972), Amsterdam, 1973),vol.52ofMathematical Centre Tracts, pp. 59–122, Mathematica Centrum, Amsterdam, The Netherlands, 1974. 2 S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge Tracts in Mathematics and Mathematical Physics, no. 59, Cambridge University Press, London, UK, 1970. 3 Z. Yang, “A new proof on embedding the category of proximity spaces into the category of nearness spaces,” Fundamenta Informaticae, vol. 88, no. 1-2, pp. 207–223, 2008. 4 A.´ Csasz´ ar,´ “m-contiguities. I,” Acta Mathematica Hungarica, vol. 71, no. 3, pp. 241–262, 1996. 5 V. M. Ivanova and A. A. Ivanov, “Contiguity spaces and bicompact extensions of topological spaces,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 23, pp. 613–634, 1959. 6 M. Khare and R. Singh, “Complete ξ-grills and L, n-merotopies,” Fuzzy Sets and Systems, vol. 159, no. 5, pp. 620–628, 2008. 7 M. Khare and R. Singh, “L-contiguities and their order structure,” Fuzzy Sets and Systems, vol. 158, no. 4, pp. 399–408, 2007. 8 M. Khare and R. Singh, “L-guilds and binary L-merotopies,” Novi Sad Journal of Mathematics, vol. 36, no. 2, pp. 57–64, 2006. 9 Y. C. Kim and K. C. Min, “L-fuzzy preproximities and L-fuzzy topologies,” Information Sciences, vol. 173, no. 1–3, pp. 93–113, 2005. 10 P. Srivastava and M. Khare, “A note on classical and fuzzy f-proximities,” Fuzzy Sets and Systems, vol. 68, no. 2, pp. 239–241, 1994. 11 P. Srivastava and M. Khare, “Fuzzy f-proximities,” Fuzzy Sets and Systems, vol. 59, no. 2, pp. 197–203, 1993. 12 P. Srivastava and M. Khare, “Fuzzy grills, fuzzy ultrafilters and counterexamples,” Simon Stevin, vol. 67, pp. 65–75, 1993. 13 P. Srivastava and M. Khare, “On lattices of fuzzy basic proximities,” Indian Journal of Mathematics,vol. 34, no. 3, pp. 233–245, 1992. 14 R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1997. 15 R. Lowen and K. Robeys, “Compactifications of products of metric spaces and their relations to Cech-ˇ Stone and Smirnov compactifications,” Topology and Its Applications, vol. 55, no. 2, pp. 163–183, 1994. 16 International Journal of Mathematics and Mathematical Sciences

16 R. Lowen and M. Sioen, “A uniﬁed functional look at completion in MET, UNIF and AP,” Applied Categorical Structures, vol. 8, no. 3, pp. 447–461, 2000. 17 R. Lowen, M. Sioen, and D. Vaughan, “Completing quasi-metric spaces—an alternative approach,” Houston Journal of Mathematics, vol. 29, no. 1, pp. 113–136, 2003. 18 R. Lowen and D. Vaughan, “A non-quasi-metric completion for quasi-metric spaces,” Rendiconti dell’Istituto di Matematica dell’Universita` di Trieste, vol. 30, pp. 145–163, 1999. 19 R. Lowen and Y. J. Lee, “Approach theory in merotopic, Cauchy and convergence spaces. I,” Acta Mathematica Hungarica, vol. 83, no. 3, pp. 189–207, 1999. 20 H. L. Bentley and H. Herrlich, “Merotopological spaces,” Applied Categorical Structures, vol. 12, no. 2, pp. 155–180, 2004. 21 M. Khare and S. Tiwari, “Completion of approach nearness spaces,” submitted. 22 J. Adamek,´ H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, Pure and Applied Mathematics, Wiley-InterScience, New York, NY, USA, 1990. 23 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott, A Compendium of Continuous Lattices, Springer, Berlin, Germany, 1980. 24 M. Khare and S. Tiwari, “Grill determined L-approach merotopological spaces,” Fundamenta Informaticae, vol. 99, no. 1, pp. 1–12, 2010. 25 B. Banaschewski, “Extensions of topological spaces,” Canadian Mathematical Bulletin, vol. 7, pp. 1–28, 1964. 26 M. B. Smyth, “Quasi-uniformities: reconciling domains with metric spaces,” in Mathematical Foundations of Programming Language Semantics (New Orleans, LA, 1987), vol. 298 of Lecture Notes in Computer Science, pp. 236–253, Springer, Berlin, Germany, 1988. 27 H.-P.A.Kunzi,¨ “Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology,” in Handbook of the History of General Topology,C.E.Hulland R. Lowen, Eds., vol. 3, pp. 853–968, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. 28 D. Vakarelov, “Proximity approach to connection-based mereology: a representation theorem,” in Proceedings of the Seminar of the Group of Logics (MIM UW ’01), 2001. 29 L. Latecki and F. Prokop, “Semi-proximity continuous functions in digital images,” Pattern Recognition Letters, vol. 16, no. 11, pp. 1175–1187, 1995. 30 Z. Pawlak and J. Peters, Jak blisko (how near), Systemy Wspomagania Decyzji I, 2007.