# Some Topological Properties of Fuzzy Antinormed Linear Spaces

Authors: Ljubiša D. R. Kočinac
Journal: Journal of Mathematics
 (2018)

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## 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.


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## Body

## 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 α-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 δ(A) of a subset A of natural numbers N, defined by δ(A)=limn→∞⁡|{k∈A:k≤n}|/n if this limit exists. A⊂N is said to bestatistically dense if δ(A)=1.The paper is organized in the following way. After this introductory section, in Section2 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⋄:[0,1]×[0,1]→[0,1] is said to be acontinuoust-conorm if it satisfies the following conditions: (i)
⋄ is commutative and associative.(ii)
⋄ is continuous.(iii)
a⋄0=a,  a∈[0,1].(iv)
a≤c and b≤d imply a⋄b≤c⋄d for a,b,c,d∈[0,1].Classical examples of continuoust-conorms are a⋄b=a+b-ab,  a⋄b=max⁡{a,b},  a⋄b=min⁡{a+b,1}.We often use idempotentt-conorms ⋄ (i.e., satisfying a⋄a=a for each a∈[0,1]).Recall now the notion of fuzzy antinorm in a linear space with respect to a continuoust-conorm following [9].In what followsE will denote a real linear space with the zero element θ and ⋄ will be a continuous t-conorm.Definition 2 (see [9]).
LetE be a real linear space and ⋄ a t-conorm. A fuzzy subset ν:E×R→R of E×R is called afuzzy antinorm on E with respect to the t-conorm ⋄ if, for all x,y∈E,(FaN1) for each t∈(-∞,0],  ν(x,t)=1;(FaN2) for each t∈(0,∞),  ν(x,t)=0 if and only if x=θ;(FaN3) for each t∈(0,∞),  ν(cx,t)=ν(x,t/|c|) if c≠0;(FaN4) for all s,t∈R,  ν(x+y,s+t)≤ν(x,s)⋄ν(y,t);(FaN5) limt→∞⁡ν(x,t)=0.Note that ifν is the antinorm ν in the definition above, then ν(x,t) is nonincreasing with respect to t for each x∈E.The following are examples of fuzzy antinorms with respect to a correspondingt-conorm and show how a fuzzy antinorm can be obtained from a norm.Example 3.
(1) ([9]) Let (E,·) be a normed linear space and let the t-norm ⋄ be given by a⋄b=a+b-ab. Define ν:E×R→[0,1] by(1)νx,t=0,ift>x;xt+x,if0<t≤x;1,ift≤0. Then ν is a fuzzy antinorm on E with respect to the t-conorm ⋄.
(2) ([7]) Let (E,·) be a normed linear space and let the t-conorm ⋄ be given by a⋄b=max⁡{a,b}. Define ν:E×R→[0,1] by(2)νx,t=xt+x,ift>0;1,ift≤0. Then ν is a fuzzy antinorm on E with respect to the t-conorm ⋄.Example 4 (see [9]).
Let(E,·) be a normed linear space and let the t-conorm ⋄ be given by a⋄b=a+b-ab. Define ν:E×R→[0,1] by(3)νx,t=0,ift>x;1,ift≤x.Then ν is a fuzzy antinorm on E with respect to the t-conorm ⋄. This antinorm ν satisfies also the following:(FaN6) ν(x,t)<1 for each t>0 implies x=θ.Example 5 (see [9]).
Let(E,·) be a normed linear space and consider the t-conorm ⋄ defined by a⋄b=min⁡{a+b,1}. Define ν:E×R→[0,1] by(4)νx,t=x2t-x,ift>x;1,ift≤x. Then ν is a fuzzy antinorm on E with respect to the t-norm ⋄. Note that this ν satisfies the condition (FaN6) and also the following:(FaN7) ν(x,·) is a continuous function on R and strictly decreasing on the subset {t:0<ν(x,t)<1} of R.


## 3. Statistical Fuzzy (Anti)Convergence

Definition 6.
A sequence(xn)n∈N in a fuzzy antinormed linear space (E,ν,⋄) is said to be ν-convergent to a point x∈E if for each ε>0 and each t>0 there is n0∈N such that(5)νxn-x,t<εfor  eachn≥n0. In this case we write (xn)→νx.Definition 7.
A sequence(xn)n∈N in a fuzzy antinormed linear space (E,ν,⋄) is said to bestatisticallyν-convergent to a point x∈E if for each ε>0 and each t>0(6)δn∈N:νxn-x,t<ε=0. In this case we write (xn)→st-νx.The following theorem is a slight modification of a theorem from [9].Theorem 8 (see [9]).
Let(E,ν,⋄) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄, and let ν satisfy (FaN6).  Then for each λ∈(0,1) the function xλ:X→[0,∞) defined by(7)xλ=⋀t>0:νx,t≤1-λ is a norm on E (called an λ-norm generated by ν), and Λ≔{·λ:λ∈(0,1)} is an ascending family of norms on E.Convention.We use the notation (E,Λ) for the family of normed linear spaces {(E,·λ):λ∈(0,1)} and call (E,Λ) also a fuzzy antinormed linear space.Lemma 9.
In a fuzzy antinormed linear space(E,Λ) with respect to an idempotent t-conorm ⋄ satisfying (FaN6)  and (FaN7),  a sequence is statistically ν-convergent if and only if it is statistically λ-convergent for each λ∈(0,1).Proof.
( ⇒ ): Let (xn)n∈N be a sequence in E such that (xn)n∈N→st-νx, i.e., for each t>0.(8)st-limn→∞⁡νxn-x,t=0. Fix λ∈(0,1). So, st-limn→∞ν(xn-x,t)=0<1-λ. There exists a statistically dense set M⊂N so that, for each n∈M,(9)νxn-x,t<1-λ. Since xn-xλ=⋀{t>0:ν(xn-x,t)≤1-λ} we have xn-xλ≤t for all n∈M. As t>0 was arbitrary, for each λ∈(0,1), by (FaN6), we have xn-xλ which statistically converges to 0.
( ⇐ ): Suppose now that for each λ∈(0,1), xn-xλ statistically converges to 0. This means that for each λ∈(0,1) and each ε>0 there is a statistically dense Mλ⊂N such that(10)xn-xλ<ε for each n∈Mλ. Therefore,(11)νxn-x,ε=⋀1-λ:xn-xλ≤ε implies(12)νxn-x,ε≤1-λ for each λ∈(0,1) and each n∈Mλ, which means(13)νxn-x,ε→st-ν0; that is,(14)xn→st-νx.The relations of fuzzyλ-anticonvergence, fuzzy λ-anticauchyness, and fuzzy λ-anticompactness with respect to their corresponding increasing family of norms are studied in the following part of this section.Definition 10.
Let(E,Λ) be a fuzzy antinormed linear space and λ∈(0,1). A sequence (xn)n∈N in E is said to befuzzy statisticallyλ-anticonvergent in E if there exist x∈E and M⊂N with δ(M)=1 such that, for all t>0,(15)νxn-x,t<1-λ. In this case we write (xn)→a-st-λx and x is called afuzzy statisticalλ-antilimit of (xn)n∈N.Theorem 11.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄ satisfying (FaN6).  Then statistical fuzzy λ-antilimit of a fuzzy statistically λ-anticonvergent sequence is unique.Proof.
Let(xn)n∈N be a fuzzy statistically λ-anticonvergent sequence converging to distinct points x and y in E. This means that for each t>0 there are two subsets Mx and My of N with δ(Mx)=1=δ(My) such that we have ν(xn-x,t/2)<1-λ for each n∈Mx and ν(xn-y,t/2)<1-λ for each n∈My. The set M=Mx∩My is sequentially dense in N, and by the assumption on ⋄ for each n∈M we have (16)νx-y,t=νx-xn+xn-y,t≤νxn-x,t2⋄νxn-y,t2<1-λ⋄1-λ=1-λTherefore, ν(x-y,t)<1 for each t>0. By (FaN6) one obtains x-y=θ, i.e., x=y.Theorem 12.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄ satisfying (FaN6).  Then: (1)
If(xn)→a-st-λx and (yn)→a-st-λy, then (xn+yn)→a-st-λx+y;(2)
If(xn)→a-st-λx and r∈R, then (r·xn)→a-st-λrx.Proof.
(1) Since(xn)→a-st-λx for each t>0 there is M1⊂N with δ(M1)=1 such that ν(xn-x,t/2)<1-λ for each n∈M1. Similarly, from (yn)→a-st-λz it follows that for each t>0 there is a set M2⊂N with δ(M2)=1 such that ν(yn-y,t/2)<1-λ for each n∈M2. Then M=M1∩M2 is such that δ(M)=1 and for each t>0 and each n∈M we have (17)νxn+yn-x-y,t≤νxn-x,t2⋄νyn-y,t2<1-λ⋄1-λ=1-λ, which means (xn+yn)→a-st-λx+y.
(2) The fact(xn)→a-st-λx implies that for each t>0 there is a statistically dense subset M of N such that ν(xn-y,t)<1-λ for each n∈M. Then for each t>0(18)νrxn-rx,t=νxn-x,tr⁡<1-λ for each n∈M; i.e., (r·xn)→a-st-λrx.Theorem 13.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄. If (xn)n∈N is a fuzzy statistically λ-anticonvergent sequence in (E,Λ) statistically converging to x∈E, then xn-xλ statistically converges to 0.Proof.
By assumption there is a setM⊂N with δ(M)=1 such that for each t>0 and each n∈M we have ν(xn-x,t)<1-λ. In other words, xn-xλ<t for each n∈M. Since t was arbitrary we have that xn-xλ statistically converges to 0.Definition 14.
Letλ∈(0,1). A sequence (xn)n∈N in a fuzzy antinormed linear space (E,Λ) (with respect to a t-conorm ⋄) is said to befuzzy statisticallyλ-anti-Cauchy if for every t>0 there is a set M⊂N such that δ(M)=1 and for all m,n∈M,  ν(xn-xm,t)<1-λ.Theorem 15.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄ satisfying (FaN6) and λ∈(0,1). Then every fuzzy statistically λ-anticonvergent sequence (xn)n∈N in (E,Λ,⋄) is fuzzy statistically λ-anti-Cauchy.Proof.
Since(xn)n∈N is fuzzy statistically λ-anticonvergent to some x∈E, for each t>0, there is a set M⊂N with δ(M)=1 such that ν(xn-x,t/2)<1-λ for each n∈M. Then for all m,n∈M we have (19)νxn-xm,t=νxn-x+x-xm,t≤νxn-x,t2⋄νxm-x,t2<1-λ⋄1-λ=1-λ which means that (xn)n∈N is fuzzy statistically λ-anti-quasi-Cauchy in (E,Λ).Theorem 16.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄. Then every statistically Cauchy sequence (xn)n∈N in (E,·λ), λ∈(0,1), is fuzzy statistically λ-anti-quasi-Cauchy in (E,Λ).Proof.
Letλ∈(0,1) be arbitrary and fixed. Since (xn)n∈N is a statistical Cauchy sequence in (E,·λ), for any ε>0 there is a set M⊂N with δ(M)=1 such that for all m,n∈M we have xn-xmλ<ε. It means that for each m,n∈M(20)⋀t>0:νxn-xm,t≤1-λ<ε which implies the existence of t0<ε such that ν(xn-xm,t0)<1-λ. It follows that ν(xn-xm,ε)<1-λ, and as ε was arbitrary, we conclude that ν(xn-xm,t)=1-λ for each t>0 and all m,n∈M. This means that (xn)n∈N is fuzzy statistically λ-anti-quasi-Cauchy sequence in (E,Λ). But, λ also was an arbitrary element in (0,1) so that we have that (xn)n∈N is fuzzy statistically λ-anti-quasi-Cauchy in (E,Λ) for each λ∈(0,1).Definition 17.
A fuzzy antinormed linear space(E,Λ) (with respect to a t-conorm ⋄) is said to befuzzy statisticallyλ-anticomplete, λ∈(0,1), if every fuzzy statistically λ-anti-Cauchy sequence in E fuzzy statistically λ-anticonverges in E.Theorem 18.
Let(E,Λ) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄. If (E,Λ) is fuzzy statistically λ-anticomplete, then E is statistically complete with respect to ·λ for each λ∈(0,1).Proof.
Letλ∈(0,1) be fixed and let (xn)n∈N be a statistically Cauchy sequence in E with respect to ·λ. By the previous theorem (xn)n∈N is fuzzy statistically λ-anti-Cauchy in (E,Λ). Therefore, there is x∈E and a subset M of N with δ(M)=1 such that, for each t>0 and each n∈M, ν(xn-x,t)<1-λ. By Theorem 13, this means xn-xλ statistically converges to 0; i.e., (xn)n∈N statistically converges to x with respect to ·λ. Therefore, (E,·λ) is statistically complete.


## 4. Some Covering Properties

Let(E,ν,⋄) be a fuzzy antinormed linear space where ⋄ is idempotent. Given x∈E, ε∈(0,1), and t>0, the set(21)Bνx,ε,t≔y∈E:νy-x,t<ε is called theopen ball with center x and radius ε with respect to t.For each pointy∈Bν(x,ε,t) there is an open ball with center y contained in Bν(x,ε,t). Let ν(y-x,t)=α<ε and set β=ε-α. We prove Bν(y,β,t/2)⊂Bν(x,ε,t). Let p∈Bν(y,β,t/2). Then ν(p-y,t/2)<β so that we have (22)νp-x,t=νp-y+y-x,t2+t2=νp-y,t2⋄νy-p,t2<ε-α⋄ε≤ε⋄ε=ε, i.e., p∈Bν(x,ε,t).Therefore, the collection(23)Bx,ε,t:x∈E,ε∈0,1,t>0 is a base of a topology on E; denote this topology by τν. Notice that the collection(24)Bx,1n,t:x∈E,n∈N,t>0 is also a base for τν. The topology τν 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 [11–14]).Recall that a topological space has the Menger (Rothberger, Hurewicz) covering property if for each sequence(Un:n∈N) of open covers of X there is a sequence (Vn:n∈N) (resp., (Un:n∈N), (Wn:n∈N)) such that, for each n∈N, Vn is a finite subset of Un (resp., Un∈Un, Wn is a finite subset of Un) and X=⋃n∈N⋃Vn (resp., X=⋃n∈NUn, each x∈X belongs to ∪Wn for all but finitely many n).Definition 19.
A fuzzy antinormed linear space(E,ν,⋄) is said to be M:Menger-bounded (orM-bounded),R:Rothberger-bounded (orR-bounded),H:Hurewicz-bounded (orH-bounded)
if for each sequence(εn:n∈N) of elements of (0,1) and each t>0 there is a sequence M: (An:n∈N) of finite subsets of E such that E=⋃n∈N⋃a∈AnB(a,εn,t),R: (xn:n∈N) of elements of E such that E=⋃n∈NB(xn,εn,t),H: (An:n∈N) of finite subsets of E such that for each x∈E there is n0∈N such that x∈⋃a∈AnB(a,εn,t) for all n≥n0.A fuzzy antinormed linear space(E,ν,⋄) is said to beprecompact (respectively,pre-Lindelöf) if for every ε∈(0,1) and every t>0 there is a finite (respectively, countable) set A⊂E such that E=⋃a∈ABν(a,ε,t).Evidently,(25)precompact⟹H-bounded⟹M-bounded⟹pre-Lindelöf,R-bounded⟹M-bounded.Example 20.
Let(E,·) be a normed linear space with the Menger (Rothberger, Hurewicz) property. Consider the fuzzy antinormed linear space (E,ν,⋄), where ν and ⋄ are as in Example 3(1). Then this fuzzy antinormed linear space isM-bounded (R-bounded,H-bounded).
Consider only theM-bounded case because the other two are shown quite similarly.
Let(εn:n∈N) be a sequence in (0,1) and let t>0. As (E,·) has the Menger covering property, there is a sequence (An:n∈N) of finite subsets of X such that(26)E=⋃n∈N⋃a∈AnKa,εn, where K(a,ε)={y∈X:a-y<ε}.
Letx∈X. There is n∈N and a point an∈An satisfying x-an<εn. Then(27)νx-an,t=0,ift>x-an;x-ant+x-an,ift≤x-an,t>0;If t>x-an, then ν(x-an,t)=0 so that x∈Bν(an,εn,t). If t>0 and t≤x-an, then ν(x-an,t)=x-an/t+x-anεn; i.e., in this case also x∈Bν(an,εn,t).
Therefore,E=⋃n∈N⋃a∈AnBν(a,εn,t); i.e., (E,ν,⋄) isM-bounded.Example 21.
LetE=R and ·=|·|, and let ⋄ be defined as in Examples 3(1) and 20. Then the fuzzy antinormed space (E,ν,⋄) isM-bounded by the previous example and the fact that (R,|·|) has the Menger property [12, 13]. On the other hand, (E,ν,⋄) is notR-bounded.
Indeed, if we take the sequence(2-n:n∈N)⊂(0,1) and t=2-1, then X cannot be covered by the open balls B(xn,2-n,2-1) for any choice of elements xn,  n∈N, from E. Otherwise, we would have that for every x∈R the following holds:(28)x-xn2-1+x-xn<2-n, which means(29)R=⋃n∈Nxn-2-n,xn+2-n. However, it is impossible.We end with the following result onM-boundedness.If(E,ν,♢) is a fuzzy antinormed space and S⊂E, then (S,νS,⋄), where νS=ν↾(S×R), is also a fuzzy antinormed space and it is called thefuzzy antinormed subspace of (X,ν,⋄).Theorem 22.
Every fuzzy antinormed subspace(S,νS,⋄) of anM-bounded (H-bounded) fuzzy antinormed space (E,ν,⋄) is alsoM-bounded (H-bounded).Proof.
Consider theM-bounded case; the proof forH-boundedness is quite similar. Let (εn:n∈N) be a sequence of elements of (0,1) and t>0. Continuity of ⋄ implies that for each n∈N there is δn∈(0,1) such that δn⋄δn<εn.
For eachn∈N let(30)Qn=q∈Pn:thereisy∈Swithy∈Bνq,δn,t2. Further, for each q∈Qn choose an element yq∈S such that yq∈Bν(q,δn,t/2) and set(31)An=yq:q∈Qn. Let us show that the sequence (An:n∈N) of finite subsets of S witnesses for (εn:n∈N) and t that (S,νS,⋄) isM-bounded.
Letz∈S be an arbitrary element. There are n∈N and p∈Pn such that z∈Bν(p,δn,t/2). By the definition of Qn it follows that p∈Qn so that there is zp∈An such that zp∈Bν(p,δn,t/2); hence p∈Bν(zp,δn,t/2). So, we have(32)νp-zp,t2<δn. By (FaN4) we have(33)νz-zp,t≤νz-p,t2⋄νp,zp,t2<δn⋄δn<εn, which means z∈Bν(zp,εn,t). As z∈S was arbitrary and zp∈An we conclude(34)S=⋃n∈N⋃a∈AnBνa,εn,t; i.e., S isM-bounded.



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