Mathematica Aeterna, Vol. 3, 2013, no. 7, 565 - 577

A New Approach to the Study of Extended Metric Spaces

Surabhi Tiwari

Department of Mathematics Motilal Nehru National Institute of Technology Allahabad, U.P., India. Email: [email protected], [email protected]

James F. Peters

Computational Intelligence Laboratory Department of Electrical & Computer Engineering University of Manitoba Winnipeg, Manitoba R3T 51-V6, Canada. Email: [email protected]

Abstract

This paper studies the category theoretic aspects of ε-approach near- ness spaces and ε-approach merotopic spaces, respectively, ε ∈ (0, ∞], which are useful in measuring the degree of nearness (resemblance) of objects. Such structures measure the almost nearness of two collections of subsets of a nonempty set. The categories εANear and εAMer are shown to be full supercategories of various well-known categories, including the category sT op of symmetric topological spaces and con- tinuous maps, and the category Met∞ of extended metric spaces and nonexpansive maps. The results in this paper have important practical implications in the study of patterns in similar pictures.

Mathematics Subject Classification: 54E05, 54E17, 54C25.

Keywords: Approach merotopy, Nearness space, Topological space.

1 Introduction

In [13], the two-argument ε-approach nearness was axiomatized for the pur- pose of presenting an alternative approach for completing extended metric spaces. This completion of extended metric spaces was motivated by the recent work on approach merotopic spaces in [7, 6, 4, 5, 16], (see also [2, 3]). In [15], 566 Surabhi Tiwari and James F. Peters

ε-approach nearness was used to prove Niemytzki-Tychonoff theorem for sym- metric topological spaces. The notion of distance in approach spaces is closely related to the notion of nearness. The question “how near” the objects may be, was unsolved untill, in [5], a generalization of proximity, called approach merotopic structures were introduced which measure the degree of nearness of a collection of sets (see also [7, 4, 6]). An approach merotopic structure is a function ν : P2(X) −→ [0, ∞], and has only one argument. For comparision, we require atleast two objects, so this mainly motivates the introduction of a two-argument ε-approach merotopic structure ν : P2(X) × P2(X) −→ [0, ∞], to measure the degree of nearness between two digital images which can be considered as cllections of subimages. Further, using the notion defined in [8], two images I1 and I2 are either similar (ν(I1, I2) = 0) or dissimilar (ν(I1, I2) =6 0). Most of the time, when we are comparing two images, they are not exactly similar (near), instead they are almost near, for example they can have different shades of same color. Therefore, a function measuring almost nearness of digital images was required. This problem is solved by the positive real number ε associated with an ε-approach merotopy. The images I1 and I2 are said to be ε-near (or sufficiently near) if ν(I1, I2) < ε, where the choice of ε is in our hand. This new structure facilitates the study of the degree of near- ness between nonempty disjoint collections of sets (see also [10, 12, 14, 17]). From a practical point of view, it is helpful to consider the degree of nearnes of disjoint collections of subsets such as those found in regions-of-interest in pairs of digital images (see [11]). The focus of this paper is on the properties of the categories εAMer and εANear whose objects are ε-approach merotopic spaces and ε-approach -nearness spaces, respectively. The notation A ֒→ B reads category A is embed ded in category B. The categories εAMer and εANear are supercategories for a variety of familiar categories shown in Fig. 1. Let εANear denote the category of all ε-approach nearness spaces and contractions, and let εAMer denote the category of all ε-approach merotopic spaces and contractions. Among these familiar categories is sT op, the symmetric form of T op, the category with objects that are topological spaces and morphisms that are continuous maps between them [1, 9]. ∞ Again, for example, Met εAMer with objects that are extended metric ∞ spaces is a subcategory of εAP (hav- pMet ing objects ε-approach spaces and εANear gAP contractions) (see also [12, 16]). Met∞ εAP sAP The map f : (X, ρX ) −→ (Y, ρY ) is a contraction if and only if f : sT op (X, ν ) −→ (Y, ν ) is a contrac- Figure 1: Categorical Hierarchy DρX DρY tion. Thus εAP is embedded as a A New Approach to the Study of Extended Metric Spaces 567 full subcategory in εANear by the functor F : εAP −→ εANear de-

fined by F ((X, ρ)) = (X, νDρ ) and F (f) = f. Then f : (X, ρX ) −→ (Y, ρY ) is a contraction if and only if f : (X, ν ) −→ (Y, ν ) is a contraction. DρX DρY Thus εAP is embedded as a full subcategory in εANear by the functor

F : εAP −→ εANear defined by F ((X, ρ)) = (X, νDρ ) and F (f) = f. Since the category Met∞ of extended metric spaces and nonexpansive maps is a full subcategory of εAP , therefore, εANear is also a full supercategory of Met∞. Fig. 1 is the pictorial representation of the hierarchy of categories which is a consequence of the study done in this paper.

2 Preliminaries

In this section, generalized approach spaces are briefly discussed. A function δ : P(X) × P(X) → [0, ∞] is a distance on X, provided, for all nonempty A, B, C ∈ P(X), (D.1) δ(A, A) = 0, (D.2) δ(A, ∅) = ∞, (D.3) δ(A, B ∪ C) = min {δ(A, B), δ(A, C)}, (D.4) δ(A, B) ≤ δ(A, B(α)) + α, for α ∈ [0, ∞], where B(α) + {x ∈ X : δ({x}, B) ≤ α}. The distance δ first appeared in [13], an extension of the distance in [8]. The pair (X, δ) is called a generalized approach space. This leads to a new form of distance called an ε-approach merotopy. The following function called gap functional was introduced by C˘ech in his 1936–1939 seminar on topology [18]:

Example 2.1 For nonempty subsets A, B ∈ P(X), the distance function Dρ : P(X) × P(X) −→ [0, ∞] is defined by

inf {ρ(a, b) : a ∈ A, b ∈ B}, if A and B are not empty, Dρ(A, B) = (∞, if A or B is empty.

Observe that (X, Dρ) is a generalized approach space, where ρ is an extended psuedo-metric on X.

3 The Categories εANear and εAMer

In this section, we study the categories εANear and εAMer having ob- jects ε-approach nearness spaces and ε-approach merotopic spaces, respec- tively. Various known topological categories are shown to be full subcategories 568 Surabhi Tiwari and James F. Peters of these categories. Let

A ∨ B + {A ∪ B : A ∈ A, B ∈ B}, A ≺ B ⇔ ∀A ∈ A, ∃B ∈ B : B ⊆ A i.e., A corefines B.

Definition 3.1 Let ε ∈ (0, ∞]. Then a function ν : P2(X) × P2(X) −→ [0, ∞] is an ε-approach merotopy on X if and only if for any collections A, B, C ∈ P2(X), the properties (AN.1)-(AN.5) are satisfied:

(AN.1) A ≺ B =⇒ ν(C, A) ≤ ν(C, B),

(AN.2) A 6= ∅, B 6= ∅ and ( A) ∩ ( B) =6 ∅ =⇒ ν(A, B) < ε,

(AN.3) ν(A, B) = ν(B, A) aTnd ν(A,TA) = 0,

(AN.4) A 6= ∅ =⇒ ν(∅, A) = ∞,

(AN.5) ν(C, A ∨ B) ≥ ν(C, A) ∧ ν(C, B).

The pair (X, ν) is termed as an ε-approach merotopic space.

For an ε-approach merotopic space (X, ν), we define: clν(A) + {x ∈ X : ν({{x}}, {A}) < ε}, for all A ⊆ X. Then clν is a Cˇech closure operator on X.

Let clν(A) + {clν(A) : A ∈ A}. Then an ε-approach merotopy ν on X is called an ε-approach nearness on X, if the following condition is satisfied:

(AN.6) ν(clν(A), clν(B)) ≥ ν(A, B).

In this case, clν is a Kuratowski closure operator on X. For an ε-approach nearness ν that satisfies (AN.6), (X, ν) is an ε-approach nearness space. For a source of examples of ε-approach nearness on a nonempty set X, consider the following example:

2 Example 3.2 Let Dρ be a gap functional. Then the function νDρ : P (X)× P2(X) −→ [0, ∞] defined as

νDρ (A, B) + sup Dρ(A, B); νDρ (A, A) + sup Dρ(A, A) = 0, A∈A,B∈B A∈A is an ε-approach merotopy on X. Define clρ(A) = {x ∈ X : ρ({x}, A) < ε}, A ⊆ X. Then clρ is a Cˇech closure operator on X. Further, if ρ(clρ(A), clρ(B)) ≥ ρ(A, B), for all A, B ⊆ X, then clρ is a Kuratowski closure operator on X, and we call ρ as an ε-approach function on X; and (X, ρ) is an ε-approach space.

In this case, νDρ is an ε-approach nearness on X. A New Approach to the Study of Extended Metric Spaces 569

So, there are many instances of ε-approach nearness on X just as there are many instances of ε-approach spaces [8] and metric spaces on X.

2 2 Example 3.3 Let ε ∈ (0, ∞]. Then the function νd : P (X) × P (X) −→ 2 [0, ∞] defined as: for A, B ∈ P (X), νd(A, B) = 0, if (A and B are nonempty collections and ( A)∩( B) =6 ∅) or A = B, and νd(A, B) = ∞, otherwise, is an ε-approach nearness on X and clνd (A) = A, for all A ⊆ X. We call (X, νd) T T 2 a discrete ε-approach nearness space. Further, the function νi : P (X) × 2 2 P (X) −→ [0, ∞] defined as: for A, B ∈ P (X), νi(A, B) = 0, if (A 6= ∅ and B 6= ∅) or A = B, and νi(A, B) = ∞, otherwise, is an ε-approach nearness on X and clνi (A) = X, for all nonempty subsets A of X. We call (X, νi) an indiscrete ε-approach nearness space.

In the following set of examples, we develop methods of obtaining a new ε-approach nearness from a given ε-approach nearness on X.

Example 3.4 Let (X, ν) be an ε-approach nearness on X, ε < r < ∞ and ε′ < ε. Then

2 2 1. ν1 : P (X) × P (X) −→ [0, ∞] defined by

∞, if A = ∅ or B = ∅, ν1(A, B) = (ν(A, B) ∧ r, otherwise, is an ε-approach nearness on X.

2 2 2. ν2 : P (X) × P (X) −→ [0, ∞] defined by

ν(A, B) ∧ ε′, if ν(A, B) < ε, ν2(A, B) = (ν(A, B) ∨ r, otherwise, is an ε-approach nearness on X.

2 2 3. ν3 : P (X) × P (X) −→ [0, ∞] defined by

ν3(A, B) = sup{ν(C, D) : C ⊆ A and D ⊆ B,

such that 0 < |C| < ℵ0 and 0 < |D| < ℵ0}

and

ν3(A, A) = sup{ν(C, C) : C ⊆ A such that |C| < ℵ0} = 0,

is an ε-approach nearness on X. Here ℵ0 is the first infinite cardinal number. 570 Surabhi Tiwari and James F. Peters

Having established an adequate number of examples of ε-approach near- ness spaces, it is now relevant to study the category εANear. For this, we require the following definition:

Definition 3.5 For any ε-approach nearness spaces (X, ν) and (Y, ν′), a map f : X −→ Y is called a contraction if ν′(f(A), f(B)) ≤ ν(A, B), for all A, B ∈ P2(X).

Lemma 3.6 Let ε ∈ (0, ∞], and let (X, ν) and (Y, ν′) be ε-approach nearness spaces. Then f : (X, ν) −→ (Y, ν′) is a contraction if and only if ν(f −1(A), f −1(B)) ≥ ν′(A, B), for all A, B ∈ P2(Y ).

Remark 3.7 Let εANear denote the category of ε-approach nearness spaces and contractions, and εAP denote the category of ε-approach spaces and con- tractions. Suppose that (X, ρX ) and (Y, ρY ) are ε-approach spaces. Then f : (X, ρ ) −→ (Y, ρ ) is a contraction if and only if f : (X, ν ) −→ (Y, ν ) X Y DρX DρY is a contraction. Thus εAP is embedded as a full subcategory in εANear by the functor F : εAP −→ εANear defined as: F ((X, ρ)) = (X, νDρ ) and F (f) = f. Since the category Met∞ of extended metric spaces and nonex- pansive maps is a full subcategory of εAP , therefore, εANear is also a full supercategory of Met∞.

Theorem 3.8 Let ε ∈ (0, ∞]. Then the category εANear is a topological construct.

Proof. Clearly, the category εANear is concrete. Let ((Xj, νj))j∈J be a family of ε-approach nearness spaces and let (fj : X −→ Xj)j∈J be a source in εANear. Define ν : P2(X) × P2(X) −→ [0, ∞] by

m,n m C n C ν(A, B) + sup{ inf sup νj(fj(Ai), fj(Bk)) : (Ai)i=1 ∈ (A) and (Bk)k=1 ∈ (B)}, i,k=1 j∈J

2 C m for every A, B ∈ P (X), where (A) = {(Ai)i=1 : A1 ∨ A2 ∨ · · · ∨ Am ≺ N C n N N A, m ∈ } and (B) = {(Bi)i=1 : B1 ∨ B2 ∨ · · · ∨ Bn ≺ B, n ∈ }; here denotes the set of natural numbers. Then we will show that ν is the initial ε-approach nearness on X. For this, we will first show that ν is an ε-approach nearness on X. (AN.1) is obvious. Let A and B be nonempty collections and ( A) ∩ ( B) =6 ∅. Then there exists x ∈ A such that x ∈ B, for some m n x ∈ X. If (Ai) ∈ C(A) and (Bk) ∈ C(B), then T T i=1 k=1 T T νj(A1 ∨ A2 ∨ · · · ∨ Am, B1 ∨ B2 ∨ · · · ∨ Bn) < ε, for all j ∈ J, i.e., for all j ∈ J, we have

νj(A1, B1∨B2∨· · ·∨Bn)∧νj(A2, B1∨B2∨· · ·∨Bn)∧· · ·∧νj(Am, B1∨B2∨· · ·∨Bn) < ε A New Approach to the Study of Extended Metric Spaces 571

=⇒ νj(A1, B1)∧νj(A1, B2)∧· · ·∧νj(A1, Bn)∧· · ·∧νj(Am, Bn) < ε. Thus, for each j ∈ J, there exists i ∈ {1, 2, . . . , m} and k ∈ {1, 2, . . . , n} such that νj(Ai, Bk) < ε. Since each νj is a contraction, therefore νj(fj(Ai), fj(Bk)) < ε, for some i ∈ {1, 2, . . . , m} and k ∈ {1, 2, . . . , n} and j ∈ J. Consequently, (AN.2) follows. Clearly, ν(A, B) = ν(B, A). Let A1 ∨ A2 ∨ · · · ∨ An ≺ A and B1∨B2∨· · ·∨Bm ≺ A. Then νj(A1∨A2∨· · ·∨An, B1∨B2∨· · ·∨Bm) ≤ νj(A, A) = 0, for all j ∈ J. That is, for all j ∈ J, there exists i ∈ {1, 2, . . . , n} and k ∈ {1, 2, . . . , m} such that νj(Ai, Bk) = 0 which gives that νj(fj(Ai), fj(Bk)) = 0. n C Thus ν(A, A) = 0. (AN.4) is obvious. For (AN.5), let (Ai)i=1 ∈ (B1) and m C (Di)i=1 ∈ (B2). Then A1 ∨ A2 ∨ · · · ∨ An ∨ D1 ∨ D2 ∨ · · · ∨ Dm ≺ B1 ∨ B2. n m C C C C Thus (Ai)i=1 ∪ (Di)i=1 ∈ (B1 ∨ B2). As a result, (B1 ∨ B2) = (B1) ∨ (B2). Hence ν(A, B1 ∨ B2) = ν(A, B1) ∧ ν(A, B2). Finally,

ν(clν(A), clν(B))

m,n m C n C = sup{ inf sup νj(fj(Ai), fj(Bk)) : (Ai)i=1 ∈ (clν(A)) and (Bk)k=1 ∈ (clν(B))} i,k=1 j∈J m,n m C n C ≥ sup{ inf sup νj(fj(clν(Ai)), fj(clν(Bk))) : (Ai)i=1 ∈ (clν(A)) and (Bk)k=1 ∈ (clν(B))} i,k=1 j∈J m,n m C n C ≥ sup{ inf sup νj(clνj (fj(Ai)), clνj (fj(Bk))) : (Ai)i=1 ∈ (clν(A)) and (Bk)k=1 ∈ (clν(B))} i,k=1 j∈J

m,n m C n C = sup{ inf sup νj(fj(Ai), fj(Bk)) : (Ai)i=1 ∈ (clν(A)) and (Bk)k=1 ∈ (clν(B))} i,k=1 j∈J m,n m C n C ≥ sup{ inf sup νj(fj(Ai), fj(Bk)) : (Ai)i=1 ∈ (A) and (Bk)k=1 ∈ (B)} i,k=1 j∈J = ν(A, B). To show that ν is the initial ε-approach nearness on X, let (Y, ν′) be an ε- approach nearness space and g : Y −→ X be a map such that fj ◦g : Y −→ Xj is a contraction, for each j ∈ J. Then we will show that g is a contraction. Let A, B ∈ P2(Y ). Suppose that ν(g(A), g(B)) > ν′(A, B). Then, for some j ∈ J, we have

m,n ′ m C n C ν (A, B) < inf νj(fj(Ai), fj(Bk)), for some (Ai)i=1 ∈ (g(A)), (Bk)k=1 ∈ (g(B)) i,k=1

≤ νj(fj(A1) ∨ fj(A2) ∨ · · · ∨ fj(Am), fj(B1) ∨ fj(B2) ∨ · · · ∨ fj(Bn))

< νj(fj(A1 ∨ A2 ∨ · · · ∨ Am), fj(B1 ∨ B2 ∨ · · · ∨ Bn))

≤ νj(fj(g(A)), fj(g(B))) ′ ≤ ν (A, B), since fj ◦ g : Y −→ Xj is a contraction, 572 Surabhi Tiwari and James F. Peters i.e., ν′(A, B) < ν′(A, B), which is absurd. Therefore, ν′(A, B) ≥ ν(g(A), g(B)), for all A, B ∈ P2(X). Consequently, g is a contraction and ν is the initial ε- approach nearness on X. As a result, εANear is a topological construct. Let ε ∈ (0, ∞]. Suppose that εAMer denotes the category of all ε- approach merotopic spaces and contractions. Then we have the following re- sult.

Corollary 3.9 Let ε ∈ (0, ∞]. Then the category εAMer is a topological construct.

Proof. The proof is similar to the proof of Theorem 3.8, with the same initial structure. The above results confirm the existence of final structures of the cate- gories εANEAR and εAMer, ε ∈ (0, ∞]. In the following series of results, we explicitly construct the final structures of these categories.

Proposition 3.10 Let ε ∈ (0, ∞]. For any family ((Xj, νj))j∈J of εANear- 2 2 objects and a sink (fj : Xj −→ X)j∈J , the function ν : P (X) × P (X) −→ [0, ∞] defined by: A, B ∈ P2(X),

0, if ( cl(A)) ∩ ( cl(B)) =6 ∅, ν(A, B) + −1 −1 inf νj(fj (cl(A)), fj (cl(B))), otherwise, (j∈J T T

+ where cl(A) {B ⊆ X : A ⊆ B and clνj (B) = B, ∀j ∈ J}, is the final ε-approach nearness on X. T Proof. First we will show that ν is an ε-approach nearness on X. Clearly, ν satisfies (AN.1), (AN.2), (AN.3) and (AN.4). For (AN.5), let A, B1, B2 ∈ 2 P (X) such that ( cl(A)) ∩ ( cl(B1)) = ∅ and cl(A)) ∩ ( cl(B2)) = ∅. Then T T T T −1 −1 ν(A, B1 ∨ B2) = inf νj(fj (cl(A)), fj (cl(B1 ∨ B2))) j∈J −1 −1 −1 = inf νj(fj (cl(A)), fj (cl(B1)) ∨ fj (cl(B2))) j∈J −1 −1 −1 −1 = inf νj(fj (cl(A)), fj (cl(B1))) ∧ inf νj(fj (cl(A)), fj (cl(B2))) j∈J j∈J

= ν(A, B1) ∧ ν(A, B2).

For (AN.6), we will only show that clν = cl. Let x ∈/ clν(A), where A ⊆ X. Then ν({{x}}, {A}) ≥ ε which yields that cl({x}) ∩ cl(A) = ∅. Consequently, x ∈/ cl(A). Thus cl(A) ⊆ clν(A). For the reverse inclusion, let x ∈ clν(A) and cl({x}) ∩ cl(A) =6 ∅. Then there exists y ∈ X such that y ∈ cl({x}) and y ∈ A New Approach to the Study of Extended Metric Spaces 573 cl(A). Since y ∈ cl({x}) =⇒ x ∈ cl({y}), therefore x ∈ cl(A). Next suppose that cl({x})∩cl(A) = ∅ and x ∈/ cl(A). Then there exists B ⊆ X such that A ⊆

B and clνj (B) = B, for all j ∈ J but x ∈/ B. Therefore x ∈/ clνj (A) which yields that νj({{x}}, {A}) ≥ ε, for all j ∈ J. As a resultant, νj({cl({x})}, {cl(A)}) ≥ −1 −1 ε which in turn gives that νj({fj (cl({x}))}, {fj (cl(A))}) ≥ ε, for all j ∈ J, that is ν({{x}}, {A}) ≥ ε implying that x ∈/ clν(A). Thus clν(A) = cl(A), for all A ⊆ X. To show that ν is the final ε-approach nearness on X, let (Y, ν′) be an ε- approach nearness space and g : X −→ Y be a map such that g ◦ fj : X −→ Y −1 is a contraction for each j ∈ J. Let A ⊆ X. Then fj (cl(A)) ∈ Xj. Since each − 1 ′ contraction is a continuous map, therefore (g ◦ fj)(fj (cl(A))) ⊆ clν (g ◦ fj ◦ − 1 ′ ′ ′ ′ fj (cl(A))) ⊆ clν (g(cl(A))) ⊆ clν clν (g(A)) = clν (g(A)) (the last inclusion follows because cl is the final closure operator on X). Thus clν′ (g(A)) ≺ g◦fj ◦ −1 2 fj (cl(A))). Consequently, for A, B ∈ P (X) and ( cl(A)) ∩ ( cl(A)) = ∅, we have T T ′ ′ − − ′ ′ 1 1 ν (clν (g(A)), clν (g(B))) ≤ ν (g ◦ fj ◦ fj (cl(A))), g ◦ fj ◦ fj (cl(B)))) −1 −1 ≤ νj(fj (cl(A))), fj (cl(B)))), for all j ∈ J.

Therefore

′ − − ′ ′ 1 1 ν (clν (g(A)), clν (g(B))) ≤ inf νj(fj (cl(A))), fj (cl(B)))). j∈J

Hence

−1 −1 ν(A, B) = inf νj(fj (cl(A))), fj (cl(B)))) j∈J ′ ≥ ν (clν′ (g(A)), clν′ (g(B))) = ν′(g(A), g(B)).

Consequently, g : (X, ν) −→ (Y, ν′) is a contraction and ν is the final ε- approach nearness on X.

Corollary 3.11 Let ε ∈ (0, ∞]. Suppose that ((Xj, νj))j∈J be a family of ε- approach merotopic spaces and (fj : Xj −→ X)j∈J be a sink in εAMer. Then the final ε-approach merotopy on X is the function ν : P2(X) × P2(X) −→ [0, ∞] defined by: A, B ∈ P2(X),

0, if ( A) ∩ ( B) =6 ∅, ν(A, B) + −1 −1 inf νj(fj (A), fj (B)), otherwise. (j∈J T T 574 Surabhi Tiwari and James F. Peters

Remark 3.12 Let (X, δ) be an approach space (as defined by Lowen [8]). Let δ also satisfies inf δ(a, B) = inf δ(b, A). Then (X, δ) is a symmetric ap- a∈A b∈B proach space. Define ρδ : P(X) × P(X) −→ [0, ∞] by: for A, B ⊆ X,

ρδ(A, B) + inf δ(a, B). a∈A

Then ρδ is a generalized distance function on X. Let (X, δ1) and (Y, δ2) be approach spaces. Then f : (X, δ1) −→ (Y, δ2) is a contraction if and only if f : (X, ρδ1 ) −→ (Y, ρδ2 ) is a contraction. Thus the category sAP of symmetric approach spaces and contractions is embedded as a full subcategory into the category gAP of generalized approach spaces and contractions by the functor F : sAP −→ gAP defined as: F ((X, δ)) = (X, ρδ) and F (f) = f.

Further if (X, ρ) is a generalized approach space, then the function νρ : P2(X) × P2(X) −→ [0, ∞] defined by: for A, B ∈ P2(X),

νρ(A, B) + sup ρ(A, B) and νρ(A, A) + sup ρ(A, A) = 0, A∈A,B∈B A∈A is an ε-merotopy on X. Also for generalized approach spaces (X, ρ1) and (Y, ρ2), f : (X, ρ1) −→ (Y, ρ2) is a contraction if and only if f : (X, νρ1 ) −→ (Y, νρ2 ) is a contraction. Thus the category gAP is embedded as a full subcategory into the category εAMer by the functor F : gAP −→ εAMer defined as: F ((X, ρ)) = (X, νρ) and F (f) = f.

Example 3.13 Let (X, cl) be a symmetrical topological space. Define δ : X × P(X) −→ [0, ∞] by: for x ∈ X and A ⊆ X,

0, if x ∈ cl({a}) for some a ∈ cl(A), δ(x, A) = (∞, otherwise.

Then (X, δ) is a symmetric approach space [8].

Remark 3.14 Let (X, cl) be a symmetric topological space. Define δcl : X × P(X) −→ [0, ∞] by: for x ∈ X and A ⊆ X,

0, if x ∈ cl(A), δcl(x, A) + (∞, otherwise.

Then δcl is a distance function as shown by Lowen [8]. Also inf δ(a, B) = a∈cl(A)

inf δ(b, A). Further define ρδcl : P(X) × P(X) −→ [0, ∞] by: for A, B ⊆ X, b∈cl(B)

ρδcl (A, B) = inf δcl(a, B). a∈cl(A) A New Approach to the Study of Extended Metric Spaces 575

Then (X, ρδ ) is an ε-approach space and clρ = cl = clδ . Further, for cl δcl cl symmetric topological spaces (X, cl1) and (Y, cl2), f : (X, cl1) −→ (Y, cl2) is continuous if and only if f : (X, ρδcl1 ) −→ (Y, ρδcl2 ) is a contraction. Thus, the category sT op of symmetric topological spaces and continuous maps is embedded as a full subcategory into the category εAP of ε-approach spaces and contractions by the functor sT op −→ εAP defined as: F ((X, cl)) = (X, ρδcl ) and F (f) = f.

Let us now concentrate upon the lattice theoretic properties of ε-approach merotopies and ε-approach nearness on a nonempty set X. We, hereby, con- struct the exact join and meet of these lattices.

Definition 3.15 Let ε ∈ (0, ∞]. Suppose that ν and ν′ be ε-approach mero- topies on X. Then ν′ ≤ ν (ν is finer than ν′ or ν′ is coarser than ν) if and ′ only if 1X : (X, ν) −→ (X, ν ) is a contraction.

Theorem 3.16 Let ε ∈ (0, ∞]. Then the family of all ε-approach nearness on X forms a completely distributive complete lattice with respect to the partial order ‘≤’. The zero of this lattice is the indiscrete ε-approach nearness νi on X and the unit is the discrete ε-approach nearness νd on X.

Proof. Let {νj : j ∈ J} be a family of ε-approach nearness on X. Define 2 2 2 νsup : P (X) × P (X) −→ [0, ∞] as follows: for A, B ∈ P (X), n n C m C νsup(A, B) = sup{inf sup νj(Ai, Bk) : (Ai)i=1 ∈ (A) and (Bk)k=1 ∈ (B)}, i=1 j∈J

C n 2 where (A) is the collection of all finite families (Ai)i=1 ⊆ P (X) such that A1∨A2∨· · ·∨An ≺ A, and similarly C(B) is defined. Then νsup is an ε-approach nearness on X and is the supremum of the family of nearness {νj : j ∈ J} (techniques of the proof are similar to that of Theorem 3.8). Now we construct 2 2 the infimum of the given family. Define νinf : P (X) × P (X) −→ [0, ∞] as follows: for A, B ∈ P2(X),

νinf(A, B) = inf νj(cl(A), cl(B)), j∈J where cl : P(X) −→ P(X) is defined as: for A ⊆ X, cl(A) = {B ⊆ X : A ⊆ B and clν (B) = B, for all j ∈ J}. Then νinf is an ε-approach nearness on X j T and clνinf = cl. This νinf is also the infimum of the given family {νj : j ∈ J} (proof follows similarly as in Proposition 3.10).

Corollary 3.17 Let ε ∈ (0, ∞]. Then the family of all ε-approach mero- topies on X forms a completely distributive complete lattice with respect to the partial order ‘≤’. The zero of this lattice is the indiscrete ε-approach merotopy νi on X and the unit is the discrete ε-approach merotopy νd on X. 576 Surabhi Tiwari and James F. Peters

Proof. If {νj : j ∈ J} is a family of ε-approach merotopies on X, then its supremum is defined in a similar manner as in the above theorem. The 2 2 infimum of the given family νinf : P (X) × P (X) −→ [0, ∞] is defined as follows: for A, B ∈ P2(X),

νinf(A, B) = inf νj(A, B). j∈J

Concluding Remark. The present paper establishes a category theo- retic foundation of the category εANear having objects ε-approach nearness spaces that are useful in the study of degree of nearness of nonempty sets.

ACKNOWLEDGEMENTS. This research has been supported by Natural Science & Engineering Research Council of Canada grant 185986.

References

[1] J. Ad´amek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories. The Joy of Cats, Wiley-Interscience Publ., New York, 1990.

[2] B. Banaschewski, S. S. Hong, A. Pultr, On the completion of nearness frames, Quaest. Math. 21 (1-2) (1998) 19–37.

[3] H. L. Bentley, H. Herrlich, On the strict completion of a nearness space, Quaest. Math. 24 (1) (2001) 39–50.

[4] M. Khare, S. Tiwari, Approach merotopological spaces and their com- pletion, Internat. J. Math. Math. Sci. 2010 (2010) Article ID 409804, 16 Pages.

[5] M. Khare, S. Tiwari, Grill determined L-approach merotopological spaces, Fund. Inform. 48 (2010) 1–12.

[6] M. Khare, S. Tiwari, L-approach merotopies and their categorical per- spective, Demonstratio Math. 45 (3) (2012) 699–716.

[7] M. Khare, S. Tiwari, Completion in a common supercategory of Met, UAP, wsAP and Near, Demonstratio Math. 46 (1) (2013) 209–227.

[8] R. Lowen, Approach Spaces: The Missing Link in the Topology- Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford Uni- versity Press, Oxford, UK, 1997, viii + 253pp. A New Approach to the Study of Extended Metric Spaces 577

[9] S. MacLane, Categories for the Working Mathematician, Springer, Berlin, 1971, v + 262pp.

[10] J. Peters, S. Naimpally, Approach spaces and near families, Gen. Math. Notes 2 (1) (2011) 1–14.

[11] J. Peters, S. Naimpally, Applications of near sets, Amer. Math. Soc. No- tices 59 (4) (2012) 536–542.

[12] J. Peters, S. Tiwari, Approach merotopies and near filters. theory and application, Gen. Math. Notes 3 (1) (2011) 32–45.

[13] J. F. Peters, S. Tiwari, Completing extended metric spaces: an alternative approach, Appl. Math. Letters 25 (2012) 1544–1547.

[14] S. Tiwari, α∗-uniformities and their order structure, Afr. Mat. (2011) doi: 10.1007/s13370–012–0065–y.

[15] S. Tiwari, Ultrafilter completeness in ε-approach nearness spaces, Math. Comput. Sci. 7 (1) (2013) 107–111, doi: 10.1007/s11786–013–0148–7.

[16] S. Tiwari, Some aspects of general topology and applications. Approach merotopic structures and applications, supervisor: M. khare, Ph.D. thesis, Department of Mathematics, Allahabad (U.P.), India (Jan. 2010).

[17] S. Tiwari, J. F. Peters, Sufficiently near l-fuzzy sets, Creat. Math. Inform. 22 (1) (2013) to appear.

[18] E. C˘ech, Topological Spaces,, revised Ed. by Z. Frolik and M. Kat˘atov, John Wiley & Sons, NY, 1966.

Received: August, 2013