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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 409804, 16 pages
http://dx.doi.org/10.1155/2010/409804
Research Article

Approach Merotopological Spaces and their Completion

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

Received 24 July 2009; Accepted 13 April 2010

Academic Editor: Richard Wilson

Copyright © 2010 Mona Khare and Surabhi 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.

Abstract

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 -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 -fuzzy theory can be seen in [613]. In [14], 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 [1518]. 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 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. In Section 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 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

2. Preliminaries and Basic Results

Let be a nonempty ordinary set. The power set of is denoted by and the family of all subsets of is denoted by We denote by the first infinite cardinal number, by the cardinality of where and by an arbitrary index set. For , subsets of , ; corefines (written as ) if and only if for all there exists such that . For , stack for some and sec  for all stack Observe that for all A on is a subset of satisfying ; if and , then ; and if , then or . For basic definitions and results of merotopic spaces and nearness spaces, we refer to [1].

Definition 2.1 (see [20]). A merotopological space is the triple where is a merotopy and is a Kuratowski closure operator on such that for all

Definition 2.2 (see [19, 21]). A function is called an approach merotopy on if for any the following conditions are satisfied:(AM1),(AM2),(AM3),(AM4).

The pair is called an approach merotopic space. For an approach merotopic space , we define , for all . Then is a ech closure operator on .

An approach merotopy on is called an approach nearness on [21] if the following condition is satisfied:.

In this case, is a Kuratowski closure operator on . Denote

Definition 2.3 (see [21]). Let and be an approach merotopy on Then we say that is(i)near in if ,(ii)-clan if is a near grill,(iii)-closed if for all ,(iv)-cluster if is a -closed -clan.

For any approach merotopic spaces and a map is called a contraction if for all or equivalently for all For any approach merotopies and on ( is finer than or is coarser than ) if the identity mapping 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 is also discussed.

Definition 3.1. An approach merotopological space is a triple where is an approach merotopy on and is a Kuratowski closure operator on such that the following condition is satisfied:(AM5) for all We call to be an approach merotopology with respect to the closure operator cl on

Example 3.2. (i) Let be an approach merotopological space and let be the family of all grills on Define as follows: Then is an approach merotopology with respect to on (see [19, 24]).
(ii) Let be a topological space. Then is an approach merotopological space, where is defined as in the above example in which can be defined in the following ways for
(a) if and otherwise.(b) if and otherwise.(c) if and otherwise.(d) if and otherwise.(e) if for every finite subset of and otherwise.(f) if or each element of is infinite; and and otherwise. (In this case, is a -closure operator on )
Observe that if is a symmetrical topological space (i.e., for all ), then in all of the above cases (note that a -space is already a symmetrical topological space).
(iii) Let be a closed and continuous map. Define as follows: for if and otherwise. Then is an approach merotopological space. Further if for all and then
(iv) Let be an approach merotopological space. Define as follows: for Then is an approach merotopology with respect to the closure on
(v) Let be a topological space and Define as follows: for if if and otherwise. Then is an approach merotopology with respect to the closure on
(vi) The function this is defined as follows: for if and otherwise, is an approach merotopology with respect to the discrete closure operator on and We call the discrete approach merotopological space.
(vii) The function this is defined as follows: for if and otherwise, is an approach merotopology with respect to the indiscrete closure operator on and We call the indiscrete approach merotopological space.

Having established the existence of an adequate number of approach merotopologies on (taking several values in ), it is now relevant to study the category AMT having objects as approach merotopological spaces. A morphism in AMT is a map 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 and 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 be an approach merotopological space. Then for all where denotes the closure operator induced by the approach merotopy on

Proof. Let and Then which yields and hence

Theorem 3.4. The category AMT is a topological construct.

Proof. Let be a family of approach merotopological spaces indexed by and let be a source in TOP. Define as follows: for where is the collection of all finite families such that Then is an approach merotopy on (see [19, Theorem ]). Let denote the closure induced by the initial topology on and Then Also since is continuous for all therefore for all which in turn yields Hence is an approach merotopological space. To show that the source is initial, let be an approach merotopological space and be a continuous map such that, for every the map is an AMT-morphism. Then is a contraction since is initial in AMER (see [19, Theorem ]). Thus is the initial source in AMT and hence AMT is a topological construct.

Corollary 3.5. Let be a family of approach merotopological spaces. Then is an initial source in AMT if and only if is initial in TOP and is initial in AMER.

Proposition 3.6. Let be a family of approach merotopological spaces and be a sink in AMT. Then is a final sink in AMT if and only if both of the following conditions hold:(i)   is final in TOP, (ii)the approach merotopology with respect to the closure operator induced by the final topology on is defined as follows: for

Proof. To prove that is an approach merotopy with respect to on we will prove only (AM5). For this, let Then there exists such that Hence To show that is final, let be an approach merotopological space and let be a map such that, for every is an AMT-morphism. Then is continuous since is final in TOP. Also for with On the other hand, if then there exists such that for all Hence Thus, is a contraction implying that is the final approach merotopological space.

Let be a metric on Define as follows: for Then is an approach merotopology with respect to the closure induced by on and Also for any metric spaces and is a nonexpansive map if and only if is an AMT-morphism. Thus the category MET of metric spaces and nonexpansive maps is embedded as a full subcategory into AMT by the functor defined as follows: and

Proposition 3.7. The category TOP is a bireflective full subcategory of AMT.

Proof. Let be any approach merotopological space. Set and and is defined as in Example 3.2 (v). Then is an approach merotopology on with respect to the closure operator which is coarser than Therefore for any approach merotopological space the identity mapping is the TOP-bireflection of

Remark 3.8. (i) There is a full embedding of TOP into AMT that is defined by associating to a topological space the approach merotopy as defined in Example 3.2 (v). Thus a topological space can be viewed as an approach merotopological space having the approach merotopy Thus following this convention, the forgetful functor shall be viewed as keeping the topology but replacing the approach merotopy with that of

(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 the topology given by the closure operator induced by 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 can be regarded as a special approach merotopological space for which 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 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 if we associate the discrete closure operator on then we obtain an approach merotopological space Thus the category AMER can also be embedded into AMT.

(iii) Let be an approach space. Then the function that is defined as follows: for is an approach merotopy on Also for any approach spaces and is a contraction map if and only if is a contraction. So we get the functor defined as follows: and Therefore AP 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 with the topology on It should be clarified here that, in general for an approach merotopological space the topology on is not determined by its approach merotopy it happens so for a topological space only if is a symmetrical topological space. Thus, in general, for an approach merotopological space need not be equal to This fact can be supported by the following example: consider with as the collection of all closed sets. Then is a non-symmetric and -space, where is the closure operator associated with the topology on defined as above. Further let the approach merotopy on be indiscrete. Then is an approach merotopological space. Also observe that is an approach nearness space but

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 the pair is an approach merotopological space (where is the induced approach merotopy on defined as follows: for if and otherwise. Also for any merotopic spaces and is a merotopic map if and only if is a contraction. Thus MERTOP is embedded as a full subcategory in AMT by the functor such that and
An approach merotopological space is induced by a merotopological space if and only if Also for any approach merotopological space the triples and where are merotopological spaces; and if is a contraction, then and are merotopic maps. So this defines the functors by and and by and Hence we can conclude that the category MERTOP is a bicoreflective and bireflective subcategory of AMT: for any approach merotopological space the identity mappings and are the MERTOP-bicoreflection and MERTOP-bireflection of , 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 are constructed.

Definition 3.10. Let and be approach merotopologies with respect to the closures and on respectively. Define the ordering “” as follows: if and the topology induced by is weaker than the topology induced by on

Theorem 3.11. The family of all approach merotopologies on forms a completely distributive complete lattice with respect to the partial order “”. The zero of this lattice is the indiscrete approach merotopology with respect to the closure induced by the indiscrete topology called indiscrete closure on and the unit is the discrete approach merotopology with respect to the discrete closure induced by the discrete topology called discrete closure on

Proof. Let (where is an arbitrary index set) be a family of approach merotopological spaces. If denotes the topology induced by for all then is a topology on and for all Let denote the closure operator induced by Then for and is Define as follows: for where is the collection of all finite families such that Then with respect to the closure operator is an approach merotopology on and is the supremum of the family of merotopologies with respect to the closures on respectively (techniques of the proof are similar to those of Theorem 3.4). Now we construct the infimum of the given family. Define as follows: for where is defined as follows: for and , for all Then is an approach merotopology with respect to the closure on which is also the infimum of the merotopologies with respect to the closures on respectively (proof follows similarly as in Proposition 3.6).

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 associated merotopology 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 of an approach merotopological space Observe that takes several values in if does so, and the completion of that is, is not, in general, equal to A similar type of study can also be seen in [17], where Lowen et al. had constructed the completion of a -approach space using the clusters of which are precisely the clusters of the associated merotopic space.

Definition 4.1. Let be an approach merotopological space. A nonempty grill on is said to be a -bunch if and

By a -cluster in an approach merotopological space we mean that is a -cluster of the approach merotopic space

Proposition 4.2. Let be an approach merotopological space. Then every -cluster is a maximal near -bunch.

Proof. Let be a -cluster. Then clearly is a near element in Let Then for all which by (AM5) yields that for all Consequently as is -closed.

For any approach merotopological space let denote the family of all -bunches on It must be clarified here that a -bunch need not be a -cluster. For example, let where is an infinite cardinal number. Consider the approach merotopological space of the last case of Example 3.2 (ii) (f). Then for (where is the cardinality of the set of all real numbers), is a -bunch which is not a -cluster because is a proper subset of and

Theorem 4.3. Let be an approach merotopological space and let denote the operator defined by for all Then is a Kuratowski closure operator on

Proof. Clearly and where Let but Then but and Therefore there exist and such that and which gives Also and for every and and consequently which contradicts that Thus Hence For the reverse inclusion, let Then and giving that Finally let but Then but Therefore there exists such that which implies that and thus for some Consequently which is not possible. Hence

It can be easily verified that, for any approach merotopological space the family is a -bunch. Therefore we can define the map as follows: for

Theorem 4.4. The map defined by and where is defined as follows: for and (when is defined as follows: for for some is a functor. Moreover, is a natural transformation from the identity functor.

Proof. First we will show that is an approach merotopological space. Let and Then Thus If then there exists and therefore for every Consequently which yields Condition (AM3) follows by the convention Now For (AM5) we have to show that where Let Then for some So, for every as for each Thus Hence is an approach merotopological space.
Next let be an AMT-morphism. Then we will show that is an AMT-morphism. That is a map follows from the fact that for all For the continuity of let and and Then for every Consequently for some and hence which in turn gives that is continuous. Now to show that is a contraction, let and Then there is an for some Thus for every there is a such that Since therefore Thus and is an AMT-morphism. Hence is well defined. Clearly preserves identity. To show that is a functor, we finally show that, for AMT-morphisms and Let Then
Now to prove that is a natural transformation, we will first show that is an AMT-morphism. For this, let and Then and thus because So, Therefore and hence is continuous. Also since for all therefore which in turn yields that Hence is an AMT-morphism. Finally let and Then and therefore which gives On the other hand if then there exists such that Consequently and Hence

Definition 4.5. (i) Let and be approach merotopological spaces. A mapping is said to be strict in AMT if is a base for closed subsets of (i.e., strict in TOP (see [25])), and for all (i.e., strict in AMER).
(ii) Further, said to be initial in AMT if for all (called initial in TOP), and for all (called initial in AMER).

Observe that an initial map is strict in AMER if and only if the approach merotopy on is defined as: for all Also for any initial map in AMER, there is a strict approach merotopology with respect to the closure operator on defined as: for all

Proposition 4.6. Let be an approach merotopological space. Then (i)the map is initial and strict;(ii)(iii)  is an injective map.Thus can be regarded as a dense subspace of

Proof. (i) Let and Then we have that is which yields Therefore is initial in TOP. That is initial in AMER, follows immediately by the relation for all Thus is initial in AMT. For strictness in TOP, we refer to [20]. For strictness in AMER, let and Then Thus as is initial in AMER.
(ii) Clearly Let Then implying that for all Hence
(iii) See [20, Proposition ].

Definition 4.7. An approach merotopological space is said to be bunch complete if for all

Proposition 4.8. An approach merotopological space is bunch complete if and only if for every grill on one such that one has

Proof. Let be bunch complete and let be a grill on such that Then Thus Converse is obvious.

Proposition 4.9. Let be an approach merotopological space. Then is a bunch complete approach merotopological space and is a -space.

Proof. That is an approach merotopological space follows from Theorem 4.4. Let be a -bunch and Then is a -bunch: let Then there exists such that Consequently Since is a grill on therefore which in turn gives that or Also and yield that or Now we will show that for which it is sufficient to note that for all Thus is bunch complete. Now let and and Then and Hence is a -closure operator on

Remark 4.10. Since any topological space can be viewed as an approach merotopological space following the convention established previously, therefore each topological space is bunch complete.

The following theorem shows that the bunch completion of possesses a universal mapping property that shows it to be very large.

Theorem 4.11. Let be an approach merotopological space and let be the canonical mapping into the bunch completion. If is an initial AMT-morphism into the approach merotopological space such that then the map that is defined as follows: for is such that Moreover,(i) is an AMT-morphism(ii)if is strict, then is initial, strict and

Proof. That is a map follows from the assumption that is initial. Let Then and therefore that is, Hence (i)For the continuity of let and but Then that is, there exists such that Consequently and thus there is such that a contradiction. Therefore and for all Hence is continuous. Now we will show that is a contraction. Let and Then By the initiality of Hence is a contraction.(ii)It can be verified that if then for all and So, for all such that since is initial. Thus implying that since is strict in AMER. Hence is initial in AMER. To show that is initial in TOP, we refer to [20].
Now to show that is strict, first we will show that where On the contrary, suppose that Then there exists such that for all that is, for every there is such that that is, for every there is such that Consequently, for all and thus a contradiction. Hence Since therefore But by the initiality of giving that is initial.
Finally for denseness of we know that which gives But Hence and therefore

We now concentrate on approach nearness spaces. In [21], we have constructed the cluster completion of an approach nearness space (where is the family of all -clusters) by defining in the same way as and the map 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 by Proposition 4.2. We introduce the property of regularity in approach merotopic spaces and prove that is a retract of for a regular approach nearness space For this, we first consider the following.

Lowen and Lee [19] also gave an equivalent description of an approach merotopy on (by generalizing the concept of micromeric collections of a nonempty set) as a function such that the following conditions are satisfied for any :(AM),(AM)there exists such that ,(AM),(AM).

The function is called an approach merotopy on and is an approach merotopic space. The relation between an approach merotopy (generalizing micromeric collections) and an approach merotopy (generalizing near collections) on is given by and where, for any and

Definition 4.12. An approach merotopic space is called regular if one of the following equivalent conditions holds:(i) for all (ii) for all (iii) for all (iv) for all

The above equivalences hold by noting that there is satisfying for all and by the following transitions: and for all

Definition 4.13. An approach merotopic space is called separated if, for all

Proposition 4.14. Every regular approach merotopic space is separated.

Proof. Let be a regular approach merotopic space. Then since for all therefore

A morphism in TOP is a retraction if and only if there exists a topological retraction and a homeomorphism such that In other words, the retractions in TOP are (up to homeomorphism) exactly the topological retractions. Therefore a subspace of a space is called a retract in TOP if there exists a continuous map with for all so that if is an inclusion, then (see [22]).

Proposition 4.15. Let be a regular approach nearness space. Then the map that is defined as follows: for is a retraction.

Proof. Let Then as is a grill on yielding which in turn gives that applying the separability of Thus is a -cluster. Since therefore whenever To prove that is continuous, let and Then we only need to show that On the contrary if there is an such that then But or there is satisfying Taking or there is satisfying we get, by regularity of and so Since therefore Consequently there exists such that Also and imply that But is a grill implying that and therefore a contradiction. So, and hence To show that is a contraction, let and there is satisfying Then for if then there exists such that for some Suppose that for all Then for there exists such that which yields that because is a grill. Thus where So which gives that for all a contradiction. Thus Hence 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 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 3.2(i) and (iv). Since therefore in this paper we have obtained a bunch completion of an approach nearness space In [21], we have obtained a cluster completion of the same structure. In fact, is a retract of 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 (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.

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