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Journal of Mathematics
Volume 2018, Article ID 9758415, 6 pages
https://doi.org/10.1155/2018/9758415
Research Article

Some Topological Properties of Fuzzy Antinormed Linear Spaces

University of Niš, Faculty of Sciences and Mathematics, 18000 Niš, Serbia

Correspondence should be addressed to Ljubiša D. R. Kočinac; moc.liamg@canicokl

Received 17 January 2018; Accepted 9 May 2018; Published 8 August 2018

Academic Editor: Hee S. Kim

Copyright © 2018 Ljubiša D. R. Kočinac. 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

The definition of fuzzy antinorm is modified. Some topological properties of finite dimensional fuzzy antinormed linear space are studied. Fuzzy anticonvergence and statistical fuzzy anticonvergence are defined and their properties are studied. We also discuss some boundedness properties in fuzzy antinormed linear spaces.

1. Introduction

The concept of fuzzy set was introduced by Zadeh [1] in 1965. Thereafter, fuzzy set theory found applications in different areas of mathematics and its applications in other sciences.

The concept of fuzzy norm was introduced by Katsaras [2] in 1984. In 1992, by using fuzzy numbers, Felbin [3] introduced the fuzzy norm on a linear space. Cheng and Mordeson [4] introduced another idea of fuzzy norm on a linear space, and in 2003 Bag and Samanta [5] modified the definition of fuzzy norm of Cheng-Mordeson [4]. In [6] a comparative study of the fuzzy norms defined by Katsaras [2], Felbin [3], and Bag and Samanta [5] was given.

In the paper [6] the idea of fuzzy antinorm was introduced. On the basis of this idea, Jebril and Samanta [7] introduced the concept of fuzzy antinorm on a linear space based on the notion of continuous triangular conorm, first applied in investigation of probabilistic metric spaces [8]. Dinda, Samanta, and Jebril further modified this concept in [9] and also defined fuzzy -anticonvergence. In this paper we use this later approach to investigate statistical versions of anticonvergence. For more information on statistical convergence see [10]. Recall that statistical convergence is defined by using the asymptotic or natural density of a subset of natural numbers , defined by if this limit exists. is said to be statistically dense if .

The paper is organized in the following way. After this introductory section, in Section 2 we give basic definitions and preliminary results which we use in the sequel. Section 3 is devoted to fuzzy anticonvergence, and in Section 4 we consider some covering properties in fuzzy antinormed linear spaces.

2. Preliminaries

This section contains some basic definition and preliminary results which we need for further exposition.

Definition 1 (see [8]). A binary operation is said to be a continuous -conorm if it satisfies the following conditions: (i) is commutative and associative.(ii) is continuous.(iii),  .(iv) and imply for .

Classical examples of continuous -conorms are ,  ,  .

We often use idempotent -conorms (i.e., satisfying for each ).

Recall now the notion of fuzzy antinorm in a linear space with respect to a continuous -conorm following [9].

In what follows will denote a real linear space with the zero element and will be a continuous -conorm.

Definition 2 (see [9]). Let be a real linear space and a -conorm. A fuzzy subset of is called a fuzzy antinorm on with respect to the -conorm if, for all ,(FaN1) for each ,  ;(FaN2) for each ,   if and only if ;(FaN3) for each ,   if ;(FaN4) for all ,  ;(FaN5) .

Note that if is the antinorm in the definition above, then is nonincreasing with respect to for each .

The following are examples of fuzzy antinorms with respect to a corresponding -conorm and show how a fuzzy antinorm can be obtained from a norm.

Example 3. (1) ([9]) Let be a normed linear space and let the -norm be given by . Define by Then is a fuzzy antinorm on with respect to the -conorm .
(2) ([7]) Let be a normed linear space and let the -conorm be given by . Define by Then is a fuzzy antinorm on with respect to the -conorm .

Example 4 (see [9]). Let be a normed linear space and let the -conorm be given by . Define byThen is a fuzzy antinorm on with respect to the -conorm . This antinorm satisfies also the following:(FaN6) for each implies .

Example 5 (see [9]). Let be a normed linear space and consider the -conorm defined by . Define by Then is a fuzzy antinorm on with respect to the -norm . Note that this satisfies the condition (FaN6) and also the following:(FaN7) is a continuous function on and strictly decreasing on the subset of .

3. Statistical Fuzzy (Anti)Convergence

Definition 6. A sequence in a fuzzy antinormed linear space is said to be -convergent to a point if for each and each there is such that In this case we write .

Definition 7. A sequence in a fuzzy antinormed linear space is said to be statistically -convergent to a point if for each and each In this case we write .

The following theorem is a slight modification of a theorem from [9].

Theorem 8 (see [9]). Let be a fuzzy antinormed linear space with respect to an idempotent -conorm , and let satisfy   Then for each the function defined by is a norm on (called an -norm generated by ), and is an ascending family of norms on .

Convention. We use the notation for the family of normed linear spaces and call also a fuzzy antinormed linear space.

Lemma 9. In a fuzzy antinormed linear space with respect to an idempotent -conorm satisfying   and ,  a sequence is statistically -convergent if and only if it is statistically -convergent for each .

Proof. : Let be a sequence in such that , i.e., for each . Fix . So, . There exists a statistically dense set so that, for each , Since we have for all . As was arbitrary, for each , by (FaN6), we have which statistically converges to .
: Suppose now that for each , statistically converges to . This means that for each and each there is a statistically dense such that for each . Therefore, implies for each and each , which means that is,

The relations of fuzzy -anticonvergence, fuzzy -anticauchyness, and fuzzy -anticompactness with respect to their corresponding increasing family of norms are studied in the following part of this section.

Definition 10. Let be a fuzzy antinormed linear space and . A sequence in is said to be fuzzy statistically -anticonvergent in if there exist and with such that, for all , In this case we write and is called a fuzzy statistical -antilimit of .

Theorem 11. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm satisfying   Then statistical fuzzy -antilimit of a fuzzy statistically -anticonvergent sequence is unique.

Proof. Let be a fuzzy statistically -anticonvergent sequence converging to distinct points and in . This means that for each there are two subsets and of with such that we have for each and for each . The set is sequentially dense in , and by the assumption on for each we have Therefore, for each . By (FaN6) one obtains , i.e., .

Theorem 12. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm satisfying   Then: (1)If and , then ;(2)If and , then .

Proof. (1) Since for each there is with such that for each . Similarly, from it follows that for each there is a set with such that for each . Then is such that and for each and each we have which means .
(2) The fact implies that for each there is a statistically dense subset of such that for each . Then for each for each ; i.e., .

Theorem 13. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm . If is a fuzzy statistically -anticonvergent sequence in statistically converging to , then statistically converges to .

Proof. By assumption there is a set with such that for each and each we have . In other words, for each . Since was arbitrary we have that statistically converges to .

Definition 14. Let . A sequence in a fuzzy antinormed linear space (with respect to a -conorm ) is said to be fuzzy statistically -anti-Cauchy if for every there is a set such that and for all ,  .

Theorem 15. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm satisfying  and . Then every fuzzy statistically -anticonvergent sequence in is fuzzy statistically -anti-Cauchy.

Proof. Since is fuzzy statistically -anticonvergent to some , for each , there is a set with such that for each . Then for all we have which means that is fuzzy statistically -anti-quasi-Cauchy in .

Theorem 16. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm . Then every statistically Cauchy sequence in , , is fuzzy statistically -anti-quasi-Cauchy in .

Proof. Let be arbitrary and fixed. Since is a statistical Cauchy sequence in , for any there is a set with such that for all we have . It means that for each which implies the existence of such that . It follows that , and as was arbitrary, we conclude that for each and all . This means that is fuzzy statistically -anti-quasi-Cauchy sequence in . But, also was an arbitrary element in so that we have that is fuzzy statistically -anti-quasi-Cauchy in for each .

Definition 17. A fuzzy antinormed linear space (with respect to a -conorm ) is said to be fuzzy statistically -anticomplete, , if every fuzzy statistically -anti-Cauchy sequence in fuzzy statistically -anticonverges in .

Theorem 18. Let be a fuzzy antinormed linear space with respect to an idempotent -conorm . If is fuzzy statistically -anticomplete, then is statistically complete with respect to for each .

Proof. Let be fixed and let be a statistically Cauchy sequence in with respect to . By the previous theorem is fuzzy statistically -anti-Cauchy in . Therefore, there is and a subset of with such that, for each and each , . By Theorem 13, this means statistically converges to ; i.e., statistically converges to with respect to . Therefore, is statistically complete.

4. Some Covering Properties

Let be a fuzzy antinormed linear space where is idempotent. Given , , and , the set is called the open ball with center and radius with respect to .

For each point there is an open ball with center contained in . Let and set . We prove . Let . Then so that we have i.e., .

Therefore, the collection is a base of a topology on ; denote this topology by . Notice that the collection is also a base for . The topology is Hausdorff and first countable.

The following definitions are motivated by definitions of the classical Menger, Rothberger, and Hurewicz covering properties (for details see the papers [1114]).

Recall that a topological space has the Menger (Rothberger, Hurewicz) covering property if for each sequence of open covers of there is a sequence (resp., , ) such that, for each , is a finite subset of (resp., , is a finite subset of ) and (resp., , each belongs to for all but finitely many ).

Definition 19. A fuzzy antinormed linear space is said to be M: Menger-bounded (or M-bounded),R: Rothberger-bounded (or R-bounded),H: Hurewicz-bounded (or H-bounded) if for each sequence of elements of and each there is a sequence M: of finite subsets of such that ,R: of elements of such that ,H: of finite subsets of such that for each there is such that for all .

A fuzzy antinormed linear space is said to be precompact (respectively, pre-Lindelöf) if for every and every there is a finite (respectively, countable) set such that .

Evidently,

Example 20. Let be a normed linear space with the Menger (Rothberger, Hurewicz) property. Consider the fuzzy antinormed linear space , where and are as in Example 3(1). Then this fuzzy antinormed linear space is M-bounded (R-bounded, H-bounded).
Consider only the M-bounded case because the other two are shown quite similarly.
Let be a sequence in and let . As has the Menger covering property, there is a sequence of finite subsets of such that where .
Let . There is and a point satisfying . ThenIf , then so that . If and , then ; i.e., in this case also .
Therefore, ; i.e., is M-bounded.

Example 21. Let and , and let be defined as in Examples 3(1) and 20. Then the fuzzy antinormed space is M-bounded by the previous example and the fact that has the Menger property [12, 13]. On the other hand, is not R-bounded.
Indeed, if we take the sequence and , then cannot be covered by the open balls for any choice of elements ,  , from . Otherwise, we would have that for every the following holds: which means However, it is impossible.

We end with the following result on M-boundedness.

If is a fuzzy antinormed space and , then , where , is also a fuzzy antinormed space and it is called the fuzzy antinormed subspace of .

Theorem 22. Every fuzzy antinormed subspace of an M-bounded (H-bounded) fuzzy antinormed space is also M-bounded (H-bounded).

Proof. Consider the M-bounded case; the proof for H-boundedness is quite similar. Let be a sequence of elements of and . Continuity of implies that for each there is such that .
For each let Further, for each choose an element such that and set Let us show that the sequence of finite subsets of witnesses for and that is M-bounded.
Let be an arbitrary element. There are and such that . By the definition of it follows that so that there is such that ; hence . So, we have By (FaN4) we have which means . As was arbitrary and we conclude i.e., is M-bounded.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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