Math.Comput.Sci. (2013) 7:107–111DOI 10.1007/s11786-013-0148-7 Mathematics in Computer Science

Ultrafilter Completeness in ε-approach Nearness Spaces

Surabhi Tiwari

Received: 18 August 2012 / Revised: 30 November 2012 / Accepted: 13 January 2013 / Published online: 20 February 2013© Springer Basel 2013

Abstract This paper presents a new approach to the proof of the Niemytzki–Tychonoff theorem for symmetrictopological spaces. The proof uses the concept of completeness in ε-approach nearness spaces which was introducedby Peters and Tiwari (Appl Math Lett 25:1544–1547, 2012), and of clusters that are a generalization of Cauchysequences.

Keywords Clusters · Completeness · Nearness spaces

Mathematics Subject Classification (2010) 54E05 · 54E17 · 54E50 · 30D40

1 Introduction

The concept of nearness had been of great importance in various branches of mathematics and theoretical computerscience. Topological spaces were the result of axiomatization of the concept of nearness between a set and a point(see, e.g. [2, §40], [3, p. 189–190n], [4, §§14–17]). Various structures involving this concept such as proximity[5], contiguity [6], uniformity [7,8], merotopy [9], near sets [10] and generalizations and variations of these near-ness-like concepts have always been fruitful in studying topological problems. In 1974, Herrlich [2] axiomatizednearness spaces which generalize proximity spaces, uniform spaces and (symmetric) topological spaces; and areconvenient tool for the study of extensions problems of topological spaces, such as completion, compactification,unification, homology, etc. The near set theory also provides a formal basis for observation, comparison and clas-sification of perceptual granules in the field of information sciences (see [11]). To measure the degree of nearnessbetween a set and a point, the notion of approach structure was introduced (see [12]). The notion of distance inapproach spaces is closely related to the notion of nearness. The question “how near” the objects may be, wasstill unsolved. To answer this question, in [13], a generalization of proximity, called approach merotopic structureswere introduced which measure the degree of nearness of a collection of sets (see also [14–16]). An approachmerotopic structure is a function ν :P2(X) −→ [0,∞], and has only one argument. For comparision, we requireatleast two objects, so this mainly motivates the introduction of a two-argument ε-approach merotopic structureν : P2(X) × P2(X) −→ [0,∞], to measure the degree of nearness between two digital images which can be

S. Tiwari (B)Department of Mathematics„ Motilal Nehru National Institute of Technology, Allahabad 211004 UP, Indiae-mail: au.surabhi@gmail.com

108 S. Tiwari

considered as collections of subimages. Further, using the notion defined in [12], two images I1 and I2 are eithersimilar (ν(I1, I2) = 0) or dissimilar (ν(I1, I2) �= 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 samecolor. Therefore, a function measuring almost nearness of digital images was required. This problem is solved bythe positive real number ε associated with an ε-approach merotopy. The images I1 and I2 are said to be ε-near ifν(I1, I2) < ε, where the choice of ε is in our hand.

The aim of this paper is to prove Niemytzki–Tychonoff theorem for ε-approach nearness spaces. In [1], a var-iation of approach nearness spaces [13–16] called ε-approach nearness spaces resulted from considering “almostnearness” of collections of sets, for the purpose of completing extended metric spaces. Various authors have definedcompleteness in nearness spaces via different generalizations of Cauchy sequences. For example, Herrlich [2]used clusters to complete nearness spaces; Carlson [17] used near ultrafilters to define complete nearness spaces;Herrlich and Bentley [18] used bunches for completion of merotopological spaces which are generalization ofnearness spaces. In this paper, we use ultrafilters to define complete ε-approach nearness spaces. A metric spaceis compact, if and only if, it is complete and totally bounded. Carlson [17] generalized this theorem and theNiemytzki–Tychonoff theorem for nearness spaces [2]. With the help of ultrafilter completeness, we have estab-lished these theorems in the framework of ε-approach nearness spaces.

2 Preliminaries

The usual requirement of convergence of Cauchy sequences for completion in a metric space was generalized byR. Lowen (see [12]) for approach spaces in terms of clusters with adherence points. 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 [19] and is a generalization of Lowen distance [12]. The pair (X, δ) is called ageneralized approach space. This leads to a new form of distance called an ε-approach merotopy. Let

A ∨ B � {A ∪ B : A ∈ A, B ∈ B},A ≺ B ⇐⇒ ∀A ∈ A, ∃B ∈ B : B ⊆ A i.e., A corefines B.

Assume ε ∈ (0,∞]. An ε-approach merotopy on X is a function ν :P2(X) × P2(X) −→ [0,∞] provided for anycollections A,B, C ∈ P2(X), we have

(AN.1) A ≺ B �⇒ ν(C,A) ≤ ν(C,B),(AN.2) A �= ∅,B �= ∅ and (

⋂ A) ∩ (⋂ B) �= ∅ �⇒ ν(A,B) < ε,

(AN.3) ν(A,B) = ν(B,A) and ν(A,A) = 0,

(AN.4) A �= ∅ �⇒ ν(∅,A) = ∞,(AN.5) ν(C,A ∨ B) ≥ ν(C,A) ∧ ν(C,B).

The pair (X, ν) is called an ε-approach merotopic space. There is continuing interest in the topological closureof a nonempty set (see, e.g., [16]). For an ε-approach merotopic space (X, ν), the function clν :P(X) −→ P(X)

defined by

clν(A) = {x ∈ X :ν({{x}}, {A}) < ε}, for all A ⊆ X,

satisfies the following properties for all A, B ∈ P(X):

(cl.1) clν(∅) = ∅,

(cl.2) A ⊂ clν(A),

Ultrafilter Completeness in ε-Approach Nearness Spaces 109

(cl.3) clν(A ∪ B) = clν(A) ∪ clν(B).

That is, clν is a Cech topological closure of the set X [20, Def. 15.A.1]. For A ∈ P2(X), let clν(A) = {clν(A) :A ∈ A}. An ε-approach merotopy ν on X is called an ε-approach nearness on X for A,B ∈ P2(X), provided

(AN.6) ν(clν(A), clν(B)) ≥ ν(A,B).

Then clν also satisfies

(cl.4) clν(clν(A)) = clν(A).

That is, clν is a Kuratowski closure operator on X [7,20]. Also, (X, clν) is a symmetrical topological space. By asymmetric topological space (X, cl), we mean that y ∈ cl({x}) �⇒ x ∈ cl({y}) (see [2]).

Remark 2.1 For an ε-approach nearness ν that satisfies (AN.6), (X, ν) is an ε-approach nearness space. For asource of examples of ε-approach nearness on a nonempty set X , consider the gap distance function introduced byCech [20] in his 1936–1939 seminar on topology.

Definition 2.2 For nonempty subsets A, B ∈ P(X), the distance function Dρ : P(X) × P(X) −→ [0,∞] isdefined by

Dρ(A, B) ={

inf {ρ(a, b) :a ∈ A, b ∈ B}, if A and B are not empty,

∞, if A or B is empty.

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

Example 2.3 The function νDρ :P2(X) × P2(X) −→ [0,∞] defined as

νDρ (A,B) � supA∈A,B∈B

Dρ(A, B), where νDρ (A,A) � supA∈A

Dρ(A, A) = 0

satisfies (AN.1)–(AN.5). Hence, νDρ is an ε-approach merotopy on X . Let ρ be an extended metric on X andclνDρ

(A) = {clνDρ(A) : A ∈ A}, where clνDρ

is defined by

clνDρ(A) = {x ∈ X :νDρ ({{x}}, {A}) < ε}, for all A ⊆ X.

Then νDρ satisfies (AN.6). Hence, (X, νDρ ) is an ε-approach nearness space.

Definition 2.4 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).

Remark 2.5 Let εANear denote the category of ε-approach nearness spaces and contractions, and let Met∞ denotethe category of extended metric spaces and nonexpansive maps. Suppose that (X, ρX ) and (Y, ρY ) are extendedmetric spaces. Then f :(X, ρX ) −→ (Y, ρY ) is a nonexpansive map, if and only if, f :(X, νDρX

) −→ (Y, νDρY) is

a contraction. Thus, Met∞ is embedded as a full subcategory in εANear by the functor F : Met∞ −→ εANeardefined by F((X, ρ)) = (X, νDρ ) and F( f ) = f.

3 The Main Results

In [1], completion of an extended metric space was done via an ε-approach nearness space. In this section, we haveemployed the definition of ultrafilter completeness (which is a generalization of the definition of completeness ofextended metric spaces) for proving the Niemytzki–Tychonoff theorem for ε-approach nearness space. Let us firstrecall the following definitions for ε-approach nearness spaces.

110 S. Tiwari

Definition 3.1 [1]. Let C ∈ P2(X). Suppose that ε ∈ (0,∞] and (X, ν) is an ε-approach nearness space. Then Cis a ν-cluster, provided the following conditions are satisfied:

(i) C, D ∈ C �⇒ ν({C}, {D}) < ε,(ii) ν({A}, {C}) < ε, for all C ∈ C �⇒ A ∈ C,

(iii) C ∪ D ∈ C �⇒ C ∈ C or D ∈ C.

Definition 3.2 [1]. An ε-approach nearness space (X, ν) is said to be complete if, and only if,⋂

clν(A) �= ∅, forall ν-clusters A ∈ P2(X).

Definition 3.3 Let ε ∈ (0,∞]. Then an ε-approach nearness space (X, ν) is said to be totally bounded, if and onlyif,F is a filter on X �⇒ there exists an ultrafilter U such that U is a ν-cluster and F ⊆ U .

Lemma 3.4 Let ε ∈ (0,∞] and 0 < r < ε. Suppose that (X, cl) be a symmetric topological space. Defineμ :G(X) × G(X) −→ [0,∞] by

μ(G1,G2) ={

r, (⋂

cl(G1)) ∩ (⋂

cl(G2)) �= ∅,

inf{|G| :G ∈ sec(cl(G1 ∪ G2))}, otherwise,

for all G1,G2 ∈ G(X). Next define νμ :P2(X) × P2(X) −→ [0,∞] as follows: for A,B ∈ P2(X),

νμ(A,B) = inf{μ(G1,G2) :A ≺ G1 and B ≺ G2}.Then νμ is a compatible (i.e., clμ = cl) totally bounded ε-approach nearness space.

Proof The proof follows from noting that μ(x, x) = r, where x = {A ⊆ X : x ∈ A} ∈ G(X);G1 ⊆ G2 �⇒μ(G1,G3) ≤ μ(G2,G3); and μ(G1,G2) = μ(G2,G1), for all G1,G2,G3 ∈ G(X). ��

A known characterization for a metric space is: “A metric space is compact, if and only if, it is complete and totallybounded”. Using the definition of completeness of ε-approach nearness space defined in [1], this characterizationis not valid in the framework of ε-approach nearness spaces. For this, consider the following example.

Example 3.5 Consider the closed unit interval [0, 1] with the usual topology and r ∈ (0,∞]. Then it is well knownthat [0, 1] is compact with the usual topology. Define ν : P2([0, 1]) × P2([0, 1]) −→ [0,∞] as follows: forA,B ∈ P2([0, 1]),

ν(A,B) ={

0, if⋂

cl(A) ∩ ⋂cl(B) �= ∅ or each element of A and B are infinite;

inf{sup{|G| :G ∈ G and |G| < ℵ0} :G ∈ G(X) and A ∪ B ≺ G}, otherwise,

where cl is the closure operature induced by the usual topology on [0,1]. Then ([0, 1], ν) is a compatible ε-approachnearness space which is totally bounded but not complete: for let A = { 1

n :n = 2, 3, 4, · · · }, {1 − 1n :n = 3, 4, · · · }

and M = {M ⊆ X : A ⊆ M or B ⊆ M}. Then M is a ν-cluster and⋂

cl(M) = ∅.

Therefore, let us define another generalization of completeness of metric spaces in terms of ultrafilters in theframework of ε-approach nearness spaces.

Definition 3.6 Let ε ∈ (0,∞]. An ε-approach nearness space (X, ν) is said to be ultrafilter complete if for anyultrafilter U such that ν(U) is a ν-cluster, we have

⋂clν(U) �= ∅, that is U is convergent. If (X, ν) is an ultrafilter

complete ε-approach nearness space, then we call ν an ultrafilter complete ε-approach nearness on X.

Observe that the definitions of total boundedness and ultrafilter completeness of ε-approach nearness spacescoincides with the usual definition of total boundedness and completeness in the case of metric spaces.

Theorem 3.7 Let ε ∈ (0,∞] and (X, ν) be an ε-approach nearness space. Then the underlying topology of (X, ν)

is compact, if and only if, (X, ν) is ultrafilter complete and totally bounded.

Ultrafilter Completeness in ε-Approach Nearness Spaces 111

Proof Let (X, clν) be compact and let U be an ultrafilter on X such that U is a ν-cluster. Then clν(U) is a set ofclosed sets such that any finite subset B of clν(U) satisfies ∩B �= ∅. Thus

⋂clν(U) �= ∅, which yields that (X, ν)

is ultrafilter complete. Next for total boundedness, let F be a filter on X. Then⋂

clν(F) �= ∅, that is, there existsx ∈ X satisfying x ∈ clν(F), for all F ∈ F . Thus, F ⊆ {A ⊆ X : x ∈ clν(A)}, which is an ultrafilter and aν-cluster. Conversely, let (X, ν) be ultrafilter complete and totally bounded and A be a collection of closed subsetsof X such that for all finite subset B of A,∩B �= ∅. Then there exists a filter F such that A ⊆ F . Since (X, ν)

is totally bounded, therefore there exists an ultrafilter U such that U is a ν-cluster and A ⊆ F ⊆ U . Further, theultrafilter completeness of ν yields that

⋂ A �= ∅. Hence (X, clν) is compact.

Theorem 3.8 A symmetric topological space is compact, if and only if, every compatible ε-approach nearness isultrafilter complete.

Proof If (X, τ ) is a R0-space and every compatible approach nearness is ultrafilter complete, then (X, νμ) definedin Lemma 3.4 is also ultrafilter complete. Therefore by Theorem 3.7, (X, νμ) is ultrafilter complete as (X, νμ) istotally bounded. The converse follows by Theorem 3.7.

Remark 3.9 Theorem 3.8 is the generalization of Niemytzki–Tychonoff theorem for symmetrical topological spaces,in the framework of ε-approach nearness spaces, where ε ∈ (0,∞].

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