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A New Approach to the Study of Extended Metric Spaces 1 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.
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