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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}.
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