Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

1. Introduction

In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics, computer and engineering [1]. After the excellent work of Zadeh [2], a large number of research work have been done on fuzzy set theory and its applications as well as fuzzy analogues of the classical theories. It has a wide number of applications in various fields such as population dynamics [3], nonlinear dynamical system [4], chaos control [5], computer programming [6], etc. In 2006, Saadati and Park [7] introduced the concept of intuitionistic fuzzy normed spaces after that the concept of statistical convergence in intuitionistic fuzzy normed space was studied for single sequence in [8]. The study of intuitionistic fuzzy topological spaces [9], intuitionistic fuzzy 2-normed space [10] and intuitionistic fuzzy Zweier ideal convergent sequence spaces [11] are the latest developments in fuzzy topology.

First, let us recall some notions, basic definitions and concepts which are used in sequel.

Definition 1.1. (See Ref. [7]). The five-tuple Xμν∗⋄is said to be an intuitionistic fuzzy normed space (for short, IFNS) if Xis a vector space, ∗is a continuous t-norm, ⋄is a continuous t-conorm, and μand νare fuzzy sets on X×0∞satisfying the following conditions for every x,y∈Xand s,t>0:

(a) μxt+νxt≤1,

(b) μxt>0,

(c) μxt=1if and only if x=0,

(d) μαxt=μxt∣α∣for each α≠0,

(e) μxt∗μys≤μx+yt+s,

(f) μx.:0∞→01is continuous,

(g) limt→∞μxt=1and limt→0μxt=0,

(h) νxt<1,

(i) νxt=0if and only if x=0,

(j) ναxt=νxt∣α∣for each α≠0,

(k) νxt⋄νys≥νx+yt+s,

(l) νx.:0∞→01is continuous,

(m) limt→∞νxt=0and limt→0νxt=1.

In this case μνis called an intuitionistic fuzzy norm.

Example 1.1. Let X∥.∥be a normed space. Denote a∗b=aband a⋄b=mina+b1for all a,b∈01and let μ0and ν0be fuzzy sets on X×0∞defined as follows:

for all t∈ℝ+. Then Xμν∗⋄is an intuitionistic fuzzy normed space.

Definition 1.2. Let Xμν∗⋄be an IFNS. Then a sequence x=xkis said to be convergent to L∈Xwith respect to the intuitionistic fuzzy norm μνif, for every ε>0and t>0, there exists k0∈ℕsuch that μxk−Lt>1−εand νxk−Lt<εfor all k≥k0. In this case we write μν-limx=L.

In 1951, the concept of statistical convergence was introduced by Steinhaus [12] and Fast [13] in their papers “Sur la convergence ordinaire et la convergence asymptotique” and “Sur la convergence statistique,” respectively. Later on, in 1959, Schoenberg [14] reintroduced this concept. It is a very useful functional tool for studying the convergence of numerical problems through the concept of density. The concept of ideal convergence, which is a generalization of statistical convergence, was introduced by Kostyrko et al. [15] and it is based on the ideal Ias a subsets of the set of positive integers and further studied in [16, 17, 18, 19, 20].

Let Xbe a non-empty set then a family I⊂2Xis said to be an ideal in Xif ∅∈I, Iis additive, i.e., for all A,B∈I⇒A∪B∈Iand Iis hereditary, i.e., for all A∈I,B⊆A⇒B∈I. A non empty family of sets F⊂2Xis said to be a filter on Xif for all A,B∈Fimplies A∩B∈Fand for all A∈Fwith A⊆Bimplies B∈F. An ideal I⊂2Xis said to be nontrivial if I≠2X, this non trivial ideal is said to be admissible if I⊇x:x∈Xand is said to be maximal if there cannot exist any nontrivial ideal J≠Icontaining Ias a subset. For each ideal I, there is a filter FIcalled as filter associate with ideal I, that is (see [15]),

A sequence x=xk∈ωis said to be I-convergent [21, 22] to a number Lif for every ε>0, we have k∈ℕ:xk−L≥ε∈I.In this case, we write I−limxk=L.

2. IF-ideal convergent sequence spaces using compact operator

This section consists of some double sequence spaces with respect to intuitionistic fuzzy normed space and study the fuzzy topology on the said spaces. First we recall some basic definitions on compact operator.

Definition 2.1. (See [23]). Let Xand Ybe two normed linear spaces and T:DT→Ybe a linear operator, where D⊂X.Then, the operator Tis said to be bounded, if there exists a positive real ksuch that

The set of all bounded linear operators BXY[24] is a normed linear spaces normed by

and BXYis a Banach space if Yis a Banach space.

Definition 2.2. (See [23]). Let Xand Ybe two normed linear spaces. An operator T:X→Yis said to be a compact linear operator (or completely continuous linear operator), if

(i) Tis linear,

(ii) Tmaps every bounded sequence xkin Xon to a sequence Txkin Ywhich has a convergent subsequence.

The set of all compact linear operators CXYis a closed subspace of BXYand CXYis Banach space, if Yis a Banach space.

In 2015, Khan et al. [11] introduced the following sequence spaces:

Motivated by this, we introduce the following sequence spaces with the help of compact operator in intuitionistic fuzzy normed spaces:

Here, we also define an open ball with center xand radius rwith respect to tas follows:

Now, we are ready to state and prove our main results. This theorem is based on the linearity of new define sequence spaces which is stated as follows.

Theorem 2.1. The sequence spaces MμνITand M0μνITare linear spaces.

Proof. Let x=xk,y=yk∈MμνITand α,βbe scalars. Then for a given ε>0, we have the sets:

This implies

Now, we define the set P3=P1∪P2, so that P3∈I. It shows that P3cis a non-empty set in FI. We shall show that for each xk,yk∈MμνIT.

Let m∈P3c, in this case

and

Thus, we have

and

This implies that

Therefore, the sequence space MμνITis a linear space.

Similarly, we can proof for the other space. □

In the following theorems, we discussed the convergence problem in the said sequence spaces. For this, firstly we have to discuss about the topology of this space. Define

Then τμνITis a topology on MμνIT.

Theorem 2.2. Let MμνITis an IFNS and τμνITis a topology on MμνIT. Then a sequence xk∈MμνIT,xk→xif and only if μTxk−Txt→1and νTxk−Txt→0as k→∞.

Proof. Fix t0>0. Suppose xk→x. Then for r∈01, there exists n0∈ℕsuch that xk∈BxrtTfor all k≥n0. So, we have

such that Bxcrt0T∈FI. Then 1−μTxk−Txt0<rand νTxk−Txt0<r. Hence μTxk−Txt0→1and νTxk−Txt0→0as k→∞.

Conversely, if for each t>0,μTxk−Txt→1and νTxk−Txt→0as k→∞, then for r∈01,there exists n0∈ℕ, such that 1−μTxk−Txt<rand νTxk−Txt<r, for all k≥n0. It shows that μTxk−Txt>1−rand νTxk−Txt<rfor all k≥n0.Therefore xk∈BxcrtTfor all k≥n0and hence xk→x.

There are some facts that arise in connection with the convergence of sequences in these spaces. Let us proceed to the next theorem on Ideal convergence of sequences in these new define spaces.

Theorem 2.3. A sequence x=xk∈MμνITis I-convergent if and only if for every ε>0and t>0there exists a number N=Nxεtsuch that

Proof. Suppose that Iμν−limx=Land let t>0. For a given ε>0, choose s>0such that 1−ε∗1−ε>1−sand ε⋄ε<s.Then for each x∈MμνIT,

which implies that

Conversely, let us choose N∈Rc. Then

Now, we want to show that there exists a number N=Nxεtsuch that

For this, we define for each x∈MμνIT

So, we have to show that S⊂R. Let us suppose that S⊊R, then there exists n∈Sand n∉R. Therefore, we have

In particular μTxN−Lt2>1−ε.Therefore, we have

which is not possible. On the other hand

In particular νTxN−Lt2<ε.So, we have

which is not possible. Hence S⊂R. R∈Iwhich implies S∈I.□

3. IF-ideal convergent sequence spaces using Orlicz function

In this section, we have discussed the ideal convergence of sequences in Intuitionistic fuzzy I-convergent sequence spaces defined by compact operator and Orlicz function. We shall now define the concept of Orlicz function, which is basic definition in our work.

Definition 3.1. An Orlicz function is a function F:0∞→0∞,which is continuous, non-decreasing and convex with F0=0,Fx>0for x>0and Fx→∞as x→∞. If the convexity of Orlicz function Fis replaced by Fx+y≤Fx+Fy, then this function is called modulus function.

Remark 3.1. If Fis an Orlicz function, then Fλx≤λFxfor all λwith 0<λ<1.

In 2009, Mohiuddine and Lohani [18] introduced the concept of statistical convergence in intuitionistic fuzzy normed spaces in their paper published in Chaos, Solitons and Fractals. This motivated us to introduced some sequence spaces defined by compact operator and Orlicz function which are as follows:

We also define an open ball with center xand radius rwith respect to tas follows:

We shall now consider some theorems of these sequence spaces and invite the reader to verify the linearity of these sequence spaces.

Theorem 3.1. Every open ball BxrtTFis an open set in MμνITF.

Proof. Let BxrtTFbe an open ball with center xand radius rwith respect to t. That is

Let y∈BxcrtTF, then FμTxk−Tyktρ>1−rand

FνTxk−Tyktρ<r.Since FμTxk−Tyktρ>1−r,there exists t0∈0tsuch that FμTxk−Tykt0ρ>1−rand FνTxk−Tykt0ρ<r.

Putting r0=FμTxk−Tykt0ρ, so we have r0>1−r, there exists s∈01such that r0>1−s>1−r. For r0>1−s, we have r1,r2∈01such that r0∗r1>1−sand 1−r0⋄1−r0≤s.Putting r3=maxr1r2. Now we consider a ball Byc1−r3t−t0TF. And we prove that

Let z=zk∈Byc1−r3t−t0TF, then FμTyk−Tzkt−t0ρ>r3and FνTyk−Tzkt−t0ρ<1−r3. Therefore, we have

and

Thus z∈BxcrtTFand hence, we get

Remark 3.2. MμνITFis an IFNS.

Define

Then τμνITFis a topology on MμνITF.

In the above result we can easily verify that the open sets in these spaces are open ball in the same spaces. This theorem itself will have various applications in our future work.

Theorem 3.2. The topology τμνITFon M0μνITFis first countable.

Proof. Bx1n1nTF:n=1,2,3…is a local base at x, the topology τμνITFon M0μνITFis first countable.□

Theorem 3.3. MμνITFand M0μνITFare Hausdorff spaces.

Proof. Let x,y∈MμνITFsuch that x≠y. Then 0<FμTx−Tytρ<1and 0<FνTx−Tytρ<1.

Putting r1=FμTx−Tytρ, r2=FνTx−Tytρand r=maxr11−r2.For each r0∈r1there exists r3and r4such that r3∗r4≥r0and 1−r3⋄1−r4≤1−r0.

Putting r5=maxr31−r4and consider the open balls Bx1−r5t2and By1−r5t2. Then clearly Bxc1−r5t2∩Byc1−r5t2=ϕ. For if there exists z∈Bxc1−r5t2∩Byc1−r5t2, then

and

which is a contradiction. Hence, MμνITFis Hausdorff. Similarly the proof follows for M0μνITF.□

4. Conclusion

The concept of defining intuitionistic fuzzy ideal convergent sequence spaces as it generalized the fuzzy set theory and give quite useful and interesting applications in many areas of mathematics and engineering. This chapter give brief introduction to intuitionistic fuzzy normed spaces with some basic definitions of convergence applicable on it. We have also summarized different types of sequence spaces with the help of ideal, Orlicz function and compact operator. At the end of this chapter some theorems and remarks based on these new defined sequence spaces are discussed for proper understanding.

Conflict of interest

The authors declare that they have no competing interests.