Section 2 Fuzzy Mathematics

Chapter 2

Spaces

Abstract

Some Topological Properties of

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

In 1986, Atanassov introduced the concept of intuitionistic fuzzy set theory which is based on the extensions of definitions of fuzzy set theory given by Zadeh. This theory provides a variable model to elaborate uncertainty and vagueness involved in decision making problems. In this chapter, we concentrate our study on the ideal convergence of sequence spaces with respect to intuitionistic fuzzy norm

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

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

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

Intuitionistic Fuzzy Normed

and discussed their topological and algebraic properties.

compact operator, I-convergence

developments in fuzzy topology.

(a) μð Þþ x; t νð Þ x; t ≤1,

(d) μ αð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>μ</sup> <sup>x</sup>; <sup>t</sup>

(c) μð Þ¼ x; t 1 if and only if x ¼ 0,

∣α∣

for each <sup>α</sup> 6¼ <sup>0</sup>,

(b) μð Þ x; t > 0,

1. Introduction

sequel.

13

Keywords: ideal, intuitionistic fuzzy normed spaces, Orlicz function

## Chapter 2

## Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

### Abstract

In 1986, Atanassov introduced the concept of intuitionistic fuzzy set theory which is based on the extensions of definitions of fuzzy set theory given by Zadeh. This theory provides a variable model to elaborate uncertainty and vagueness involved in decision making problems. In this chapter, we concentrate our study on the ideal convergence of sequence spaces with respect to intuitionistic fuzzy norm and discussed their topological and algebraic properties.

Keywords: ideal, intuitionistic fuzzy normed spaces, Orlicz function compact operator, I-convergence

## 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 ð Þ X; μ; ν; ∗; ⋄ is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm, and μ and ν are fuzzy sets on X � ð Þ 0; ∞ satisfying the following conditions for every x, y ∈X and s, t > 0 :


(e) μð Þ x; t ∗ μð Þ y; s ≤μð Þ x þ y; t þ s , (f) μð Þ x; : : ð Þ! 0; ∞ ½ � 0; 1 is continuous, (g) limt!<sup>∞</sup> μð Þ¼ x; t 1 and limt!<sup>0</sup> μð Þ¼ x; t 0, (h) νð Þ x; t < 1, (i) νð Þ¼ x; t 0 if and only if x ¼ 0, (j) ν αð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>ν</sup> <sup>x</sup>; <sup>t</sup> ∣α∣ for each <sup>α</sup> 6¼ <sup>0</sup>, (k) νð Þ x; t ⋄ νð Þ y; s ≥νð Þ x þ y; t þ s , (l) νð Þ x; : : ð Þ! 0; ∞ ½ � 0; 1 is continuous, (m) limt!<sup>∞</sup> νð Þ¼ x; t 0 and limt!<sup>0</sup> νð Þ¼ x; t 1:

In this case ð Þ μ; ν is called an intuitionistic fuzzy norm.

Example 1.1. Let Xð Þ ; ∥:∥ be a normed space. Denote a ∗ b ¼ ab and a ⋄ b ¼ minð Þ a þ b; 1 for all a, b∈½ � 0; 1 and let μ<sup>0</sup> and ν<sup>0</sup> be fuzzy sets on X � ð Þ 0; ∞ defined as follows:

∥Tx∥≤k∥x∥, for all x∈Dð Þ T :

The set of all bounded linear operators Bð Þ X; Y [24] is a normed linear spaces

Definition 2.2. (See [23]). Let X and Y be two normed linear spaces. An operator T : X ! Y is said to be a compact linear operator (or completely continuous

(ii) T maps every bounded sequence ð Þ xk in X on to a sequence ð Þ T xð Þ<sup>k</sup> in Y

The set of all compact linear operators Cð Þ X; Y is a closed subspace of Bð Þ X; Y

� �≤<sup>1</sup> � <sup>ε</sup> or <sup>ν</sup> <sup>x</sup><sup>=</sup>

n o∈<sup>I</sup> n o,

n o∈<sup>I</sup> n o:

� �≤<sup>1</sup> � <sup>ε</sup> or <sup>ν</sup> <sup>x</sup><sup>=</sup>

<sup>k</sup> � L; t � �≥<sup>ε</sup>

k; t � �≥<sup>ε</sup>

or νð Þ T xð Þ� <sup>k</sup> L; t ≥ε∈I} (2)

or νð Þ T xð Þ<sup>k</sup> ; t ≥ε∈I}: (3)

� �; <sup>t</sup> � �≤<sup>1</sup> � <sup>ε</sup>

� �; t � �≥ε∈I}: (4)

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> and <sup>α</sup>, <sup>β</sup> be scalars. Then for a given <sup>ε</sup>>0,

� �≥<sup>ε</sup>

� � � <sup>L</sup>2; <sup>t</sup>

� � <sup>&</sup>lt; <sup>ε</sup>

� �≥<sup>ε</sup>

2∣α∣

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T are linear spaces.

2∣α∣

2∣β∣

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

<sup>k</sup> � L; t

k; t

Motivated by this, we introduce the following sequence spaces with the help of

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup> ð Þ xk <sup>∈</sup>ℓ<sup>∞</sup> : {<sup>k</sup> : <sup>μ</sup>ð Þ T xð Þ� <sup>k</sup> <sup>L</sup>; <sup>t</sup> <sup>≤</sup><sup>1</sup> � <sup>ε</sup>

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup> ð Þ xk <sup>∈</sup>ℓ<sup>∞</sup> : {<sup>k</sup> : <sup>μ</sup>ð Þ T xð Þ<sup>k</sup> ; <sup>t</sup> <sup>≤</sup><sup>1</sup> � <sup>ε</sup>

Here, we also define an open ball with center x and radius r with respect to t as

� �∈ℓ<sup>∞</sup> : {<sup>k</sup> : <sup>μ</sup> T xð Þ� <sup>k</sup> T yk

Now, we are ready to state and prove our main results. This theorem is based on

� �≤<sup>1</sup> � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>k</sup> <sup>L</sup>1; <sup>t</sup>

� �>1 � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>k</sup> <sup>L</sup>1; <sup>t</sup>

� �∈I;

� �∈I:

� �∈Fð Þ<sup>I</sup> ;

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> and <sup>M</sup><sup>I</sup>

or ν T xð Þ� <sup>k</sup> T yk

the linearity of new define sequence spaces which is stated as follows.

2∣α∣

2∣β∣ � �≤<sup>1</sup> � <sup>ε</sup> or <sup>ν</sup> T yk

2∣α∣

� �∈M<sup>I</sup>

� � � <sup>L</sup>2; <sup>t</sup>

∥Tx∥

<sup>∥</sup>T<sup>∥</sup> <sup>¼</sup> sup <sup>x</sup>∈X, <sup>∥</sup>x∥¼<sup>1</sup>

and Bð Þ X; Y is a Banach space if Y is a Banach space.

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

DOI: http://dx.doi.org/10.5772/intechopen.82528

normed by

linear operator), if (i) T is linear,

ZI

follows:

ZI

which has a convergent subsequence.

and Cð Þ X; Y is Banach space, if Y is a Banach space.

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> <sup>¼</sup> ð Þ xk <sup>∈</sup><sup>ω</sup> : <sup>k</sup> : <sup>μ</sup> <sup>x</sup><sup>=</sup>

compact operator in intuitionistic fuzzy normed spaces:

ð Þ <sup>μ</sup>;<sup>ν</sup> <sup>¼</sup> ð Þ xk <sup>∈</sup><sup>ω</sup> : <sup>k</sup> : <sup>μ</sup> <sup>x</sup><sup>=</sup>

M<sup>I</sup>

M<sup>I</sup>

Bxð Þ r; t ð Þ¼ T yk

Theorem 2.1. The sequence spaces M<sup>I</sup>

<sup>P</sup><sup>1</sup> <sup>¼</sup> <sup>k</sup> : <sup>μ</sup> T xð Þ� <sup>k</sup> <sup>L</sup>1; <sup>t</sup>

<sup>1</sup> <sup>¼</sup> <sup>k</sup> : <sup>μ</sup> T xð Þ� <sup>k</sup> <sup>L</sup>1; <sup>t</sup>

Proof. Let x ¼ ð Þ xk , y ¼ yk

P<sup>2</sup> ¼ k : μ T yk

we have the sets:

This implies

Pc

15

$$\mu\_0(\mathbf{x}, t) = \frac{t}{t + \|\mathbf{x}\|}, \quad \text{and} \quad \nu\_0(\mathbf{x}, t) = \frac{\|\mathbf{x}\|}{t + \|\mathbf{x}\|}$$

for all t∈ ℝþ. Then ð Þ X; μ; ν; ∗; ⋄ is an intuitionistic fuzzy normed space.

Definition 1.2. Let ð Þ X; μ; ν; ∗; ⋄ be an IFNS. Then a sequence x ¼ ð Þ xk is said to be convergent to L∈X with respect to the intuitionistic fuzzy norm ð Þ μ; ν if, for every ε > 0 and t > 0, there exists k<sup>0</sup> ∈ℕ such that μð Þ xk � L; t >1 � ε and νð Þ xk � L; t < ε for all k≥k0. In this case we write ð Þ μ; ν -limx ¼ 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 I as a subsets of the set of positive integers and further studied in [16–20].

Let X be a non-empty set then a family I ⊂ 2<sup>X</sup> is said to be an ideal in X if ∅∈I, I is additive, i.e., for all A, B∈I ) A∪B∈I and I is hereditary, i.e., for all <sup>A</sup>∈I, B⊆<sup>A</sup> ) <sup>B</sup>∈I. A non empty family of sets <sup>F</sup>⊂2<sup>X</sup> is said to be a filter on <sup>X</sup> if for all A, B∈F implies A∩B∈F and for all A∈F with A⊆B implies B∈F. An ideal I⊂2<sup>X</sup> is said to be nontrivial if <sup>I</sup> 6¼ <sup>2</sup><sup>X</sup>, this non trivial ideal is said to be admissible if I⊇f g f gx : x∈X and is said to be maximal if there cannot exist any nontrivial ideal J 6¼ I containing I as a subset. For each ideal I, there is a filter Fð ÞI called as filter associate with ideal I, that is (see [15]),

$$\mathcal{F}(I) = \{K \underline{\subset} X : K^c \boxplus I\}, \quad \text{where} \quad K^c = X \backslash K. \tag{1}$$

A sequence x ¼ ð Þ xk ∈ω is said to be I-convergent [21, 22] to a number L if for every ε>0, we have f g k∈ℕ : jxk � Lj≥ε ∈I: In this case, we write I � lim xk ¼ 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 X and Y be two normed linear spaces and T : Dð Þ! T Y be a linear operator, where D⊂X: Then, the operator T is said to be bounded, if there exists a positive real k such that

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces DOI: http://dx.doi.org/10.5772/intechopen.82528

$$\|Tx\| \le k \|x\|, \quad \text{for all } \ x \in \mathcal{D}(T).$$

The set of all bounded linear operators Bð Þ X; Y [24] is a normed linear spaces normed by

$$\|T\| = \sup\_{\mathbf{x} \in \mathcal{X}, \ \|\mathbf{x}\| = 1} \|T\mathbf{x}\|.$$

and Bð Þ X; Y is a Banach space if Y is a Banach space.

Definition 2.2. (See [23]). Let X and Y be two normed linear spaces. An operator T : X ! Y is said to be a compact linear operator (or completely continuous linear operator), if

(i) T is linear,

(e) μð Þ x; t ∗ μð Þ y; s ≤μð Þ x þ y; t þ s , (f) μð Þ x; : : ð Þ! 0; ∞ ½ � 0; 1 is continuous, (g) limt!<sup>∞</sup> μð Þ¼ x; t 1 and limt!<sup>0</sup> μð Þ¼ x; t 0,

(i) νð Þ¼ x; t 0 if and only if x ¼ 0,

(k) νð Þ x; t ⋄ νð Þ y; s ≥νð Þ x þ y; t þ s , (l) νð Þ x; : : ð Þ! 0; ∞ ½ � 0; 1 is continuous, (m) limt!<sup>∞</sup> νð Þ¼ x; t 0 and limt!<sup>0</sup> νð Þ¼ x; t 1:

associate with ideal I, that is (see [15]),

∣α∣ 

for each α 6¼ 0,

In this case ð Þ μ; ν is called an intuitionistic fuzzy norm.

μ0ð Þ¼ x; t

Example 1.1. Let Xð Þ ; ∥:∥ be a normed space. Denote a ∗ b ¼ ab and

t

every ε > 0 and t > 0, there exists k<sup>0</sup> ∈ℕ such that μð Þ xk � L; t >1 � ε and νð Þ xk � L; t < ε for all k≥k0. In this case we write ð Þ μ; ν -limx ¼ L.

subsets of the set of positive integers and further studied in [16–20].

is additive, i.e., for all A, B∈I ) A∪B∈I and I is hereditary, i.e., for all

a ⋄ b ¼ minð Þ a þ b; 1 for all a, b∈½ � 0; 1 and let μ<sup>0</sup> and ν<sup>0</sup> be fuzzy sets on X � ð Þ 0; ∞

for all t∈ ℝþ. Then ð Þ X; μ; ν; ∗; ⋄ is an intuitionistic fuzzy normed space.

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 I as a

Definition 1.2. Let ð Þ X; μ; ν; ∗; ⋄ be an IFNS. Then a sequence x ¼ ð Þ xk is said to be convergent to L∈X with respect to the intuitionistic fuzzy norm ð Þ μ; ν if, for

In 1951, the concept of statistical convergence was introduced by Steinhaus [12]

Let X be a non-empty set then a family I ⊂ 2<sup>X</sup> is said to be an ideal in X if ∅∈I, I

<sup>F</sup>ðÞ¼ <sup>I</sup> <sup>K</sup>⊆<sup>X</sup> : <sup>K</sup><sup>c</sup> f g <sup>∈</sup><sup>I</sup> , where <sup>K</sup><sup>c</sup> <sup>¼</sup> <sup>X</sup>\K: (1)

<sup>A</sup>∈I, B⊆<sup>A</sup> ) <sup>B</sup>∈I. A non empty family of sets <sup>F</sup>⊂2<sup>X</sup> is said to be a filter on <sup>X</sup> if for all A, B∈F implies A∩B∈F and for all A∈F with A⊆B implies B∈F. An ideal I⊂2<sup>X</sup> is

A sequence x ¼ ð Þ xk ∈ω is said to be I-convergent [21, 22] to a number L if for every ε>0, we have f g k∈ℕ : jxk � Lj≥ε ∈I: In this case, we write I � lim xk ¼ L:

said to be nontrivial if <sup>I</sup> 6¼ <sup>2</sup><sup>X</sup>, this non trivial ideal is said to be admissible if I⊇f g f gx : x∈X and is said to be maximal if there cannot exist any nontrivial ideal J 6¼ I containing I as a subset. For each ideal I, there is a filter Fð ÞI called as filter

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.

Definition 2.1. (See [23]). Let X and Y be two normed linear spaces and T : Dð Þ! T Y be a linear operator, where D⊂X: Then, the operator T is said to be

First we recall some basic definitions on compact operator.

bounded, if there exists a positive real k such that

14

<sup>t</sup> <sup>þ</sup> <sup>∥</sup>x<sup>∥</sup> , and <sup>ν</sup>0ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>∥</sup>x<sup>∥</sup>

t þ ∥x∥

(h) νð Þ x; t < 1,

Fuzzy Logic

defined as follows:

(j) ν αð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>ν</sup> <sup>x</sup>; <sup>t</sup>

(ii) T maps every bounded sequence ð Þ xk in X on to a sequence ð Þ T xð Þ<sup>k</sup> in Y which has a convergent subsequence.

The set of all compact linear operators Cð Þ X; Y is a closed subspace of Bð Þ X; Y and Cð Þ X; Y is Banach space, if Y is a Banach space.

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

$$\begin{split} \mathcal{Z}\_{(\mu,\iota)}^{I} &= \left\{ (\boldsymbol{\infty}\_{k}) \boldsymbol{\mathsf{E}} \boldsymbol{\alpha} : \left\{ \boldsymbol{k} : \boldsymbol{\mu} \Big(\boldsymbol{\mathsf{x}}\_{k}^{\ulcorner} - \boldsymbol{L}, t \Big) \boldsymbol{\leq} \boldsymbol{1} - \boldsymbol{\varepsilon} \quad \boldsymbol{\sigma} \,\, \boldsymbol{\nu} \left(\boldsymbol{\mathsf{x}}\_{k}^{\ulcorner} - \boldsymbol{L}, t \Big) \succeq \boldsymbol{\varepsilon} \right\} \boldsymbol{\in} I \right\}, \\ \mathcal{Z}\_{0(\boldsymbol{\mu},\iota)}^{I} &= \left\{ (\boldsymbol{\infty}\_{k}) \boldsymbol{\mathsf{E}} \boldsymbol{\alpha} : \left\{ \boldsymbol{k} : \boldsymbol{\mu} \Big(\boldsymbol{\mathsf{x}}\_{k}^{\ulcorner}, t \Big) \preceq \boldsymbol{1} - \boldsymbol{\varepsilon} \quad \boldsymbol{\sigma} \,\, \boldsymbol{\nu} \left(\boldsymbol{x}\_{k}^{\ulcorner}, t \Big) \succeq \boldsymbol{\varepsilon} \right\} \boldsymbol{\in} I \right\}. \end{split}$$

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

$$\mathcal{M}^{I}\_{(\mu,\nu)}(T) = (\mathfrak{x}\_{k}) \boxplus \ell\_{\infty} : \{k : \mu(T(\mathfrak{x}\_{k}) - L, t) \le 1 - \varepsilon\}$$

$$\text{or} \quad \nu(T(\mathfrak{x}\_{k}) - L, t) \ge \varepsilon \in I\} \tag{2}$$

$$\mathcal{M}^{I}\_{0(\mu,\nu)}(T) = (\mathfrak{x}\_{k}) \boxplus \ell\_{\infty} : \{k : \mu(T(\mathfrak{x}\_{k}), t) \le 1 - \varepsilon\}$$

$$\text{or} \quad \nu(T(\mathfrak{x}\_{k}), t) \ge \varepsilon \in I\}. \tag{3}$$

Here, we also define an open ball with center x and radius r with respect to t as follows:

$$\mathcal{B}\_{\mathbf{x}}(r,t)(T) = \begin{pmatrix} \boldsymbol{y}\_{k} \end{pmatrix} \in \ell\_{\infty} : \{ \boldsymbol{k} : \boldsymbol{\mu}(T(\mathbf{x}\_{k}) - T(\mathbf{y}\_{k}), t) \le \mathbf{1} - \boldsymbol{\varepsilon} \\ \quad \text{or} \quad \boldsymbol{\nu} \left( T(\mathbf{x}\_{k}) - T(\mathbf{y}\_{k}), t \right) \ge \boldsymbol{\varepsilon} \in I \}. \tag{4}$$

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 M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> and <sup>M</sup><sup>I</sup> <sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T are linear spaces. � �∈M<sup>I</sup>

Proof. Let x ¼ ð Þ xk , y ¼ yk ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> and <sup>α</sup>, <sup>β</sup> be scalars. Then for a given <sup>ε</sup>>0, we have the sets:

$$P\_1 = \left\{ k : \mu\left(T(\mathbf{x}\_k) - L\_1, \frac{t}{2|a|}\right) \le \mathbf{1} - \varepsilon \text{ or } \nu\left(T(\mathbf{x}\_k) - L\_1, \frac{t}{2|a|}\right) \ge \varepsilon \right\} \in \mathcal{I};$$

$$P\_2 = \left\{ k : \mu\left(T(\mathbf{y}\_k) - L\_2, \frac{t}{2|\beta|}\right) \le \mathbf{1} - \varepsilon \text{ or } \nu\left(T(\mathbf{y}\_k) - L\_2, \frac{t}{2|\beta|}\right) \ge \varepsilon \right\} \in \mathcal{I}.$$

This implies

$$P\_1^\varepsilon = \left\{ k : \mu\left( T(\mathbf{x}\_k) - L\_1, \frac{t}{2|a|} \right) \succ 1 - \varepsilon \quad \text{or} \quad \nu\left( T(\mathbf{x}\_k) - L\_1, \frac{t}{2|a|} \right) < \varepsilon \right\} \mathsf{E}\mathcal{F}(I);$$

$$P\_2^\varepsilon = \left\{ k : \mu\left( T\left( \_{jk} \right) - L\_2, \frac{t}{2|\beta|} \right) \succ 1 - \varepsilon \quad \text{or} \ \nu\left( \left( T\left( \_{jk} \right) - L\_2, \frac{t}{2|\beta|} \right) < \varepsilon \right) \in \mathcal{F}(I) . \right\}$$

Theorem 2.2. Let M<sup>I</sup>

νðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 0 as k ! ∞.

ð Þ xk ∈Bxð Þ r; t ð Þ T for all k≥n0. So, we have

DOI: http://dx.doi.org/10.5772/intechopen.82528

sequences in these new define spaces.

Theorem 2.3. A sequence <sup>x</sup> <sup>¼</sup> ð Þ xk <sup>∈</sup>M<sup>I</sup>

N : μ T xð Þ� <sup>N</sup> L;

R ¼ k : μ T xð Þ� <sup>k</sup> L;

Rc <sup>¼</sup> <sup>k</sup> : <sup>μ</sup> T xð Þ� <sup>k</sup> <sup>L</sup>;

μ T xð Þ� <sup>N</sup> L;

Conversely, let us choose N∈R<sup>c</sup>

For this, we define for each x∈M<sup>I</sup>

and n∉R. Therefore, we have

17

which implies that

<sup>ν</sup>ð Þ T xð Þ� <sup>k</sup> T xð Þ; <sup>t</sup> <sup>&</sup>lt; <sup>r</sup> for all <sup>k</sup>≥n0: Therefore ð Þ xk <sup>∈</sup>B<sup>c</sup>

Then a sequence ð Þ xk <sup>∈</sup>M<sup>I</sup>

such that B<sup>c</sup>

as k ! ∞.

xk ! x.

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is an IFNS and <sup>τ</sup><sup>I</sup>

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

Proof. Fix t0>0. Suppose xk ! x. Then for r∈ð Þ 0; 1 , there exists n0∈ℕ such that

Bxð Þ r; t<sup>0</sup> ð Þ¼ T f g k : μð Þ T xð Þ� <sup>k</sup> T xð Þ; t ≤1 � r or νð Þ T xð Þ� <sup>k</sup> T xð Þ; t<sup>0</sup> ≥r ∈I,

<sup>x</sup>ð Þ r; t<sup>0</sup> ð Þ T ∈Fð ÞI . Then 1 � μð Þ T xð Þ� <sup>k</sup> T xð Þ; t<sup>0</sup> < r and νð Þ T xð Þ� <sup>k</sup> T xð Þ; t<sup>0</sup> < r. Hence μðT xð Þ� <sup>k</sup> T xð Þ; t0Þ ! 1 and νðT xð Þ� <sup>k</sup> T xð Þ; t0Þ ! 0

Conversely, if for each t>0, μðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 1 and νðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 0 as k ! ∞, then for r∈ð Þ 0; 1 , there exists n0∈ℕ, such that 1 � μð Þ T xð Þ� <sup>k</sup> T xð Þ; t < r and νð Þ T xð Þ� <sup>k</sup> T xð Þ; t < r, for all k≥n0. It shows that μð Þ T xð Þ� <sup>k</sup> T xð Þ; t >1 � r and

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

� �>1 � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>N</sup> <sup>L</sup>;

n o∈Fð Þ<sup>I</sup> :

Proof. Suppose that Ið Þ <sup>μ</sup>;<sup>ν</sup> � lim x ¼ L and let t>0. For a given ε>0, choose s>0

� �≤<sup>1</sup> � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>k</sup> <sup>L</sup>;

� �>1 � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>k</sup> <sup>L</sup>;

. Then

� �>1 � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>N</sup> <sup>L</sup>;

Now, we want to show that there exists a number N ¼ N xð Þ ; ε; t such that

f g k : μð Þ T xð Þ� <sup>k</sup> T xð Þ <sup>N</sup> ; t ≤1 � s or νð Þ T xð Þ� <sup>k</sup> T xð Þ <sup>N</sup> ; t ≥s ∈I:

S ¼ f g k : μð Þ T xð Þ� <sup>k</sup> T xð Þ <sup>N</sup> ; t ≤1 � s or νð Þ T xð Þ� <sup>k</sup> T xð Þ <sup>N</sup> ; t ≥s ∈I: So, we have to show that S⊂R. Let us suppose that S⊊R, then there exists n∈S

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T

μð Þ T xð Þ� <sup>n</sup> T xð Þ <sup>N</sup> ; t ≤1 � s or μ T xð Þ� <sup>n</sup> L;

n o∈I,

n o∈Fð Þ<sup>I</sup> :

every ε>0 and t>0 there exists a number N ¼ N xð Þ ; ε; t such that

t 2

such that 1ð Þ � <sup>ε</sup> <sup>∗</sup>ð Þ <sup>1</sup> � <sup>ε</sup> >1 � <sup>s</sup> and <sup>ε</sup>⋄<sup>ε</sup> <sup>&</sup>lt; <sup>s</sup>: Then for each <sup>x</sup>∈M<sup>I</sup>

t 2

t 2

t 2 ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is a topology on <sup>M</sup><sup>I</sup>

<sup>x</sup>ð Þ r; t ð Þ T for all k≥n<sup>0</sup> and hence

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T is I-convergent if and only if for

t 2

t 2 � � <sup>&</sup>lt; <sup>ε</sup>

t 2 � � <sup>&</sup>lt; <sup>ε</sup>:

t 2 � �>1 � <sup>ε</sup>:

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T ,

t 2

� �≥<sup>ε</sup>

� � <sup>&</sup>lt; <sup>ε</sup>

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> , xk ! <sup>x</sup> if and only if <sup>μ</sup>ðT xð Þ� <sup>k</sup> T xð Þ; <sup>t</sup>Þ ! 1 and

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T .

Now, we define the set <sup>P</sup><sup>3</sup> <sup>¼</sup> <sup>P</sup><sup>1</sup> <sup>∪</sup> <sup>P</sup>2, so that <sup>P</sup>3∈I. It shows that Pc <sup>3</sup> is a nonempty set in Fð ÞI . We shall show that for each ð Þ xk , yk ∈M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T :

$$\begin{array}{c} P\_3^\varepsilon \subset \{ k : \mu\big( (aT(\varkappa\_k) + \beta T(\mathcal{y}\_k)) - (aL\_1 + \beta L\_2), t \big) \succ 1 - \varepsilon \\ \text{or} \quad \nu\big( (aT(\varkappa\_k) + \beta T(\mathcal{y}\_k)) - (aL\_1 + \beta L\_2), t \big) \prec \varepsilon \}. \end{array}$$

Let m∈Pc 3, in this case

$$\mu\left(T(\mathfrak{x}\_m) - L\_1, \frac{t}{2|a|}\right) \succ \mathbf{1} - \varepsilon \quad \text{or} \quad \nu\left(T(\mathfrak{x}\_m) - L\_1, \frac{t}{2|a|}\right) < \varepsilon$$

and

$$\mu\left(T(\mathbb{y}\_m) - L\_2, \frac{t}{2|\beta|}\right) > 1 - \varepsilon \quad \text{or} \quad \nu\left(T(\mathbb{y}\_m) - L\_2, \frac{t}{2|\beta|}\right) < \varepsilon.$$

Thus, we have

$$\begin{aligned} &\mu\left(\left(\alpha T(\mathfrak{x}\_m) + \beta T(\mathfrak{y}\_m)\right) - \left(\alpha L\_1 + \beta L\_2\right), t\right) \\ \geq &\mu\left(\alpha T(\mathfrak{x}\_m) - \alpha L\_1, \frac{t}{2}\right) \ast \mu\left(\beta T(\mathfrak{x}\_m) - \beta L\_2, \frac{t}{2}\right) \\ = &\mu\left(T(\mathfrak{x}\_m) - L\_1, \frac{t}{2|\alpha|}\right) \ast \mu\left(T(\mathfrak{x}\_m) - L\_2, \frac{t}{2|\beta|}\right) \\ > &(1 - \varepsilon) \ast (1 - \varepsilon) = 1 - \varepsilon. \end{aligned}$$

and

$$\begin{aligned} &\nu\left(\left(aT(\boldsymbol{\kappa}\_m) + \beta T(\boldsymbol{\jmath}\_m)\right) - (a\boldsymbol{L}\_1 + \beta \boldsymbol{L}\_2), t\right) \\ &\leq \nu\left(aT(\boldsymbol{\kappa}\_m) - a\boldsymbol{L}\_1, \frac{t}{2}\right) \diamond \nu\left(\beta T(\boldsymbol{\kappa}\_m) - \beta \boldsymbol{L}\_2, \frac{t}{2}\right) \\ &= \mu\left(T(\boldsymbol{\kappa}\_m) - \boldsymbol{L}\_1, \frac{t}{2|a|}\right) \diamond \mu\left(T(\boldsymbol{\kappa}\_m) - \boldsymbol{L}\_2, \frac{t}{2|\beta|}\right) \\ &\qquad \quad < \varepsilon \diamond \varepsilon = \varepsilon. \end{aligned}$$

This implies that

$$\begin{array}{c} P\_3^\varepsilon \subset \{ k : \mu\big( (aT(\boldsymbol{\kappa}\_k) + \beta T(\boldsymbol{\jmath}\_k)) - (aL\_1 + \beta L\_2), t \big) \succ 1 - \varepsilon \} \\ \text{or} \quad \nu\big( (aT(\boldsymbol{\kappa}\_k) + \beta T(\boldsymbol{\jmath}\_k)) - (aL\_1 + \beta L\_2), t \big) < \varepsilon. \end{array}$$

Therefore, the sequence space M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T is 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

$$
\pi^l\_{(\mu,\nu)}(T) = A \subset \mathcal{M}^l\_{(\mu,\nu)}(T) : \text{for each } \mu \in \mathcal{A} \text{ there exists } t > 0 \text{ and } \ r \in (0,1) \text{ such that } \mathcal{B}\_\mathbf{x}(r,t)(T) \subset \mathcal{A}.
$$

Then τ<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is a topology on <sup>M</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T .

Theorem 2.2. Let M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is an IFNS and <sup>τ</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is a topology on <sup>M</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T . Then a sequence ð Þ xk <sup>∈</sup>M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> , xk ! <sup>x</sup> if and only if <sup>μ</sup>ðT xð Þ� <sup>k</sup> T xð Þ; <sup>t</sup>Þ ! 1 and νðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 0 as k ! ∞.

Proof. Fix t0>0. Suppose xk ! x. Then for r∈ð Þ 0; 1 , there exists n0∈ℕ such that ð Þ xk ∈Bxð Þ r; t ð Þ T for all k≥n0. So, we have

$$\mathcal{B}\_{\mathbf{x}}(r, t\_0)(T) = \{k : \mu(T(\mathbf{x}\_k) - T(\mathbf{x}), t) \le 1 - r \text{ or } \nu(T(\mathbf{x}\_k) - T(\mathbf{x}), t\_0) \ge r\} \in I,$$

such that B<sup>c</sup> <sup>x</sup>ð Þ r; t<sup>0</sup> ð Þ T ∈Fð ÞI . Then 1 � μð Þ T xð Þ� <sup>k</sup> T xð Þ; t<sup>0</sup> < r and νð Þ T xð Þ� <sup>k</sup> T xð Þ; t<sup>0</sup> < r. Hence μðT xð Þ� <sup>k</sup> T xð Þ; t0Þ ! 1 and νðT xð Þ� <sup>k</sup> T xð Þ; t0Þ ! 0 as k ! ∞.

Conversely, if for each t>0, μðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 1 and νðT xð Þ� <sup>k</sup> T xð Þ; tÞ ! 0 as k ! ∞, then for r∈ð Þ 0; 1 , there exists n0∈ℕ, such that 1 � μð Þ T xð Þ� <sup>k</sup> T xð Þ; t < r and νð Þ T xð Þ� <sup>k</sup> T xð Þ; t < r, for all k≥n0. It shows that μð Þ T xð Þ� <sup>k</sup> T xð Þ; t >1 � r and <sup>ν</sup>ð Þ T xð Þ� <sup>k</sup> T xð Þ; <sup>t</sup> <sup>&</sup>lt; <sup>r</sup> for all <sup>k</sup>≥n0: Therefore ð Þ xk <sup>∈</sup>B<sup>c</sup> <sup>x</sup>ð Þ r; t ð Þ T for all k≥n<sup>0</sup> and 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 <sup>x</sup> <sup>¼</sup> ð Þ xk <sup>∈</sup>M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T is I-convergent if and only if for every ε>0 and t>0 there exists a number N ¼ N xð Þ ; ε; t such that

$$\left\{ N : \mu\left( T(\mathfrak{x}\_N) - L, \frac{t}{2} \right) \succ 1 - \varepsilon \quad \text{or} \quad \nu\left( T(\mathfrak{x}\_N) - L, \frac{t}{2} \right) < \varepsilon \right\} \in \mathcal{F}(I). \; \square$$

Proof. Suppose that Ið Þ <sup>μ</sup>;<sup>ν</sup> � lim x ¼ L and let t>0. For a given ε>0, choose s>0 such that 1ð Þ � <sup>ε</sup> <sup>∗</sup>ð Þ <sup>1</sup> � <sup>ε</sup> >1 � <sup>s</sup> and <sup>ε</sup>⋄<sup>ε</sup> <sup>&</sup>lt; <sup>s</sup>: Then for each <sup>x</sup>∈M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T ,

$$R = \left\{ k : \mu\left(T(\mathbf{x}\_k) - L, \frac{t}{2}\right) \le 1 - \varepsilon \quad \text{or} \quad \nu\left(T(\mathbf{x}\_k) - L, \frac{t}{2}\right) \ge \varepsilon \right\} \in I\_1$$

which implies that

Pc

Fuzzy Logic

Let m∈Pc

and

and

Define

16

Then τ<sup>I</sup>

Thus, we have

This implies that

τI

Pc

Therefore, the sequence space M<sup>I</sup>

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup> <sup>A</sup>⊂M<sup>I</sup>

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup> is a topology on <sup>M</sup><sup>I</sup>

<sup>2</sup> ¼ k : μ T yk

Pc

3, in this case

μ T ym

<sup>μ</sup> T xð Þ� <sup>m</sup> <sup>L</sup>1; <sup>t</sup>

� <sup>L</sup>2; <sup>t</sup>

� <sup>L</sup>2; <sup>t</sup>

empty set in Fð ÞI . We shall show that for each ð Þ xk , yk

<sup>3</sup>⊂{k : μ αT xð Þþ <sup>k</sup> βT yk

2∣α∣

2∣β∣

μ αT xð Þþ <sup>m</sup> βT ym

ν αT xð Þþ <sup>m</sup> βT ym

≤ν αT xð Þ� <sup>m</sup> αL1;

<sup>¼</sup> <sup>μ</sup> T xð Þ� <sup>m</sup> <sup>L</sup>1; <sup>t</sup>

<sup>3</sup>⊂{k : μ αT xð Þþ <sup>k</sup> βT yk

or ν αT xð Þþ <sup>k</sup> βT yk

≥μ αT xð Þ� <sup>m</sup> αL1;

<sup>¼</sup> <sup>μ</sup> T xð Þ� <sup>m</sup> <sup>L</sup>1; <sup>t</sup>

2∣β∣

Now, we define the set <sup>P</sup><sup>3</sup> <sup>¼</sup> <sup>P</sup><sup>1</sup> <sup>∪</sup> <sup>P</sup>2, so that <sup>P</sup>3∈I. It shows that Pc

or ν αT xð Þþ <sup>k</sup> βT yk

>1 � ε or ν T yk

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup> >1 � <sup>ε</sup>

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup> <sup>&</sup>lt; <sup>ε</sup>}:

>1 � <sup>ε</sup> or <sup>ν</sup> T xð Þ� <sup>m</sup> <sup>L</sup>1; <sup>t</sup>

∗μ βT xð Þ� <sup>m</sup> βL2;

<sup>∗</sup><sup>μ</sup> T xð Þ� <sup>m</sup> <sup>L</sup>2; <sup>t</sup>

⋄ν βT xð Þ� <sup>m</sup> βL2;

<sup>⋄</sup><sup>μ</sup> T xð Þ� <sup>m</sup> <sup>L</sup>2; <sup>t</sup>

>1 � ε or ν T ym

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup>

t 2

2∣α∣

> 1ð Þ � ε ∗ð Þ¼ 1 � ε 1 � ε:

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup>

< ε⋄ε ¼ ε:

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.

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T .

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup> >1 � <sup>ε</sup>

� ð Þ <sup>α</sup>L<sup>1</sup> <sup>þ</sup> <sup>β</sup>L<sup>2</sup> ; <sup>t</sup> <sup>&</sup>lt; <sup>ε</sup>:

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T is a linear space.

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T : for each x∈A there exists t>0

and r∈ð Þ 0; 1 such that Bxð Þ r; t ð Þ T ⊂A:

t 2

2∣α∣

� <sup>L</sup>2; <sup>t</sup>

∈M<sup>I</sup>

� <sup>L</sup>2; <sup>t</sup>

2∣β∣

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T :

2∣α∣

2∣β∣

t 2

2∣β∣

t 2

2∣β∣

< ε

< ε:

< ε

∈Fð ÞI :

<sup>3</sup> is a non-

$$\mathcal{R}^{\varepsilon} = \left\{ k : \mu\left( T(\infty\_k) - L, \frac{t}{2} \right) \succ 1 - \varepsilon \quad \text{or} \quad \nu\left( T(\infty\_k) - L, \frac{t}{2} \right) < \varepsilon \right\} \in \mathcal{F}(I).$$

Conversely, let us choose N∈R<sup>c</sup> . Then

$$\mu\left(T(\varkappa\_N) - L, \frac{t}{2}\right) \succ 1 - \varepsilon \quad \text{or} \quad \nu\left(T(\varkappa\_N) - L, \frac{t}{2}\right) < \varepsilon.$$

Now, we want to show that there exists a number N ¼ N xð Þ ; ε; t such that

$$\{k : \mu(T(\mathfrak{x}\_k) - T(\mathfrak{x}\_N), t) \le 1 - \varepsilon \text{ or } \nu(T(\mathfrak{x}\_k) - T(\mathfrak{x}\_N), t) \ge \varepsilon\} \in I.$$

For this, we define for each x∈M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T

$$\mathcal{S} = \{k : \mu(T(\mathfrak{x}\_k) - T(\mathfrak{x}\_N), t) \not\subseteq \mathfrak{1} - \mathfrak{s} \quad \text{or} \quad \nu(T(\mathfrak{x}\_k) - T(\mathfrak{x}\_N), t) \not\supseteq \mathfrak{s}\} \in I.$$

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

$$\mu\left(T(\boldsymbol{\kappa}\_n) - T(\boldsymbol{\kappa}\_N), t\right) \not\subseteq \mathbf{1} - \boldsymbol{s} \quad \text{or} \quad \mu\left(T(\boldsymbol{\kappa}\_n) - L, \frac{t}{2}\right) \succ \mathbf{1} - \boldsymbol{\varepsilon}.$$

In particular <sup>μ</sup> T xð Þ� <sup>N</sup> <sup>L</sup>; <sup>t</sup> 2 >1 � <sup>ε</sup>: Therefore, we have

$$(\mathbf{1} - \varepsilon \underline{\mathbf{x}} \boldsymbol{\mu}(T(\mathbf{x}\_n) - T(\mathbf{x}\_N), t) \underline{\mathbf{x}} \boldsymbol{\mu}(T(\mathbf{x}\_n) - L, \frac{t}{2}) \ast \boldsymbol{\mu}\left(T(\mathbf{x}\_N) - L, \frac{t}{2}\right) \underline{\mathbf{x}} (\mathbf{1} - \boldsymbol{\varepsilon}) \ast (\mathbf{1} - \boldsymbol{\varepsilon}) \ast \mathbf{1} - \boldsymbol{\varepsilon}\_n$$

Proof. Let Bxð Þ r; t ð Þ T; F be an open ball with center x and radius r with respect to

ν T xð Þ� <sup>k</sup> T yk ; t ρ <sup>≥</sup><sup>r</sup>

>1 � <sup>r</sup> and

ρ <sup>&</sup>lt; <sup>r</sup>.

, so we have <sup>r</sup>0>1 � <sup>r</sup>, there exists <sup>s</sup>∈ð Þ <sup>0</sup>; <sup>1</sup> such

<sup>x</sup>ð Þ r; t ð Þ T; F :

ρ <sup>&</sup>gt;r<sup>3</sup> and

ν T yk

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is first countable.

<sup>x</sup>ð Þ r; t ð Þ T; F :

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F : for each x∈A there exists t>0

 � T zð Þ<sup>k</sup> ; <sup>t</sup> � <sup>t</sup><sup>0</sup> ρ 

 � T zð Þ<sup>k</sup> ; <sup>t</sup> � <sup>t</sup><sup>0</sup> ρ 

ρ

ρ

that r0>1 � s>1 � r. For r0>1 � s, we have r1, r2∈ð Þ 0; 1 such that r0∗r1>1 � s and

<sup>y</sup>ð Þ <sup>1</sup> � <sup>r</sup>3; <sup>t</sup> � <sup>t</sup><sup>0</sup> ð Þ <sup>T</sup>; <sup>F</sup> , then <sup>F</sup> <sup>μ</sup> T y ð Þ ð Þ<sup>k</sup> �T zð Þ<sup>k</sup> ;t�t<sup>0</sup>

 ρ

≥ð Þ r0∗r<sup>3</sup> ≥ð Þ r0∗r<sup>1</sup> ≥ð Þ 1 � s ≥ð Þ 1 � r

 ρ <sup>⋄</sup><sup>F</sup>

≤ð Þ 1 � r<sup>0</sup> ⋄ð Þ 1 � r<sup>3</sup> ≤ð Þ 1 � r<sup>0</sup> ⋄ð Þ 1 � r<sup>2</sup> ≤s≤r:

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F . 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

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> on <sup>M</sup><sup>I</sup>

ν T xð Þ� <sup>k</sup> T yk

<sup>y</sup>ð Þ <sup>1</sup> � <sup>r</sup>3; <sup>t</sup> � <sup>t</sup><sup>0</sup> ð Þ <sup>T</sup>; <sup>F</sup> <sup>⊂</sup>B<sup>c</sup>

and r∈ð Þ 0; 1 such that Bxð Þ r; t ð Þ T; F ⊂A:

; t<sup>0</sup>

<sup>∗</sup><sup>F</sup> <sup>μ</sup> T yk

; t<sup>0</sup>

ð Þ 1 � r<sup>0</sup> ⋄ð Þ 1 � r<sup>0</sup> ≤s: Putting r<sup>3</sup> ¼ maxf g r1;r<sup>2</sup> . Now we consider a ball

<sup>y</sup>ð Þ <sup>1</sup> � <sup>r</sup>3; <sup>t</sup> � <sup>t</sup><sup>0</sup> ð Þ <sup>T</sup>; <sup>F</sup> <sup>⊂</sup>B<sup>c</sup>

or F

<sup>x</sup>ð Þ <sup>r</sup>; <sup>t</sup> ð Þ <sup>T</sup>; <sup>F</sup> , then <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ;<sup>t</sup>

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

>1 � <sup>r</sup> and <sup>F</sup> <sup>ν</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ ;t<sup>0</sup>

<sup>&</sup>lt; <sup>r</sup>: Since <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ;<sup>t</sup>

ρ

<sup>∈</sup>ℓ<sup>∞</sup> : <sup>k</sup> : <sup>F</sup> <sup>μ</sup> T xð Þ� <sup>k</sup> T yk

 ; t ρ <sup>≤</sup><sup>1</sup> � <sup>r</sup>

>1 � r, there exists <sup>t</sup>0∈ð Þ <sup>0</sup>; <sup>t</sup> such

 ∈I :

t. That is

Let y∈B<sup>c</sup>

Bc

<sup>F</sup> <sup>ν</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ;<sup>t</sup> ρ

that <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ ;t<sup>0</sup> ρ

Let <sup>z</sup> <sup>¼</sup> ð Þ zk <sup>∈</sup>B<sup>c</sup>

<sup>F</sup> <sup>μ</sup>ð Þ T xð Þ� <sup>k</sup> T zð Þ<sup>k</sup> ; <sup>t</sup> ρ

<sup>F</sup> <sup>ν</sup>ð Þ T xð Þ� <sup>k</sup> T zð Þ<sup>k</sup> ; <sup>t</sup> ρ <sup>≤</sup><sup>F</sup>

<sup>F</sup> <sup>ν</sup> T y ð Þ ð Þ<sup>k</sup> �T zð Þ<sup>k</sup> ;t�t<sup>0</sup> ρ

and

Thus z∈B<sup>c</sup>

Define

Then τ<sup>I</sup>

our future work.

19

Remark 3.2. M<sup>I</sup>

τI

Bxð Þ r; t ð Þ¼ T; F y ¼ yk

DOI: http://dx.doi.org/10.5772/intechopen.82528

Putting <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ ;t<sup>0</sup>

<sup>y</sup>ð Þ 1 � r3; t � t<sup>0</sup> ð Þ T; F . And we prove that

Bc

< 1 � <sup>r</sup>3. Therefore, we have

<sup>≥</sup><sup>F</sup> <sup>μ</sup> T xð Þ� <sup>k</sup> T yk

<sup>x</sup>ð Þ r; t ð Þ T; F and hence, we get

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is an IFNS.

Bc

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> is a topology on <sup>M</sup><sup>I</sup>

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup>; <sup>F</sup> <sup>A</sup>⊂M<sup>I</sup>

Theorem 3.2. The topology τ<sup>I</sup>

which is not possible. On the other hand

$$\nu\left(T(\boldsymbol{\kappa}\_n) - T(\boldsymbol{\kappa}\_N), t\right) \succeq \boldsymbol{s} \quad \text{or} \quad \nu\left(T(\boldsymbol{\kappa}\_n) - L, \frac{t}{2}\right) < \varepsilon.$$

In particular <sup>ν</sup> T xð Þ� <sup>N</sup> <sup>L</sup>; <sup>t</sup> 2 < ε: So, we have

$$\mathfrak{s} \le \mathfrak{\nu} \left( T(\mathfrak{x}\_n) - T(\mathfrak{x}\_N), t \right) \le \mathfrak{\nu} \left( T(\mathfrak{x}\_n) - L, \frac{t}{2} \right) \circ \nu \left( T(\mathfrak{x}\_N) - L, \frac{t}{2} \right) \le \mathfrak{e} \circ \mathfrak{e} < \mathfrak{s}, t$$

which is not possible. Hence <sup>S</sup>⊂R. <sup>R</sup>∈<sup>I</sup> which implies <sup>S</sup>∈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 : ½ Þ! 0; ∞ ½ Þ 0; ∞ , which is continuous, non-decreasing and convex with Fð Þ¼ 0 0,F xð Þ>0 for x>0 and F xð Þ! ∞ as x ! ∞. If the convexity of Orlicz function F is replaced by F xð Þ þ y ≤F xð Þþ F yð Þ, then this function is called modulus function.

Remark 3.1. If F is an Orlicz function, then Fð Þ λx ≤λF xð Þ for all λ with 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:

$$\mathcal{M}\_{(\mu,\nu)}^{I}(T,F) = \left\{ (\mathbf{x}\_{k}) \in \mathcal{E}\_{\infty} \, : \, \{k : F\left(\frac{\mu(T(\mathbf{x}\_{k}) - L, t)}{\rho}\right) \le 1 - \varepsilon \right\}$$

$$\text{or} \quad F\left(\frac{\nu(T(\mathbf{x}\_{k}) - L, t)}{\rho} \ge \varepsilon \right) \in I \right\}; \tag{5}$$

$$\mathcal{M}\_{0(\mu,\nu)}^{I}(T,F) = \left\{ (\mathbf{x}\_{k}) \in \mathcal{E}\_{\infty} \, : \, \{k : F\left(\frac{\mu(T(\mathbf{x}\_{k}), t)}{\rho}\right) \le 1 - \varepsilon \right\}$$

$$\text{or} \quad F\left(\frac{\nu(T(\mathbf{x}\_{k}), t)}{\rho} \ge \varepsilon \right) \in I \right\}. \tag{6}$$

We also define an open ball with center x and radius r with respect to t as follows:

$$\mathcal{B}\_{\mathbf{x}}(r,t)(T,F) = \left\{ (\mathbf{y}\_k) \in \ell\_{\infty} : k : F \left( \frac{\mu\left(T(\mathbf{x}\_k) - T(\mathbf{y}\_k), t\right)}{\rho} \right) \le 1 - \varepsilon \right\} \tag{7}$$

$$\text{or} \quad F\left(\frac{\nu\left(T(\mathbf{x}\_k) - T(\mathbf{y}\_k), t\right)}{\rho}\right) \ge \varepsilon \in I \right\}. \tag{7}$$

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 <sup>B</sup>xð Þ <sup>r</sup>; <sup>t</sup> ð Þ <sup>T</sup>; <sup>F</sup> is an open set in <sup>M</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F . Some Topological Properties of Intuitionistic Fuzzy Normed Spaces DOI: http://dx.doi.org/10.5772/intechopen.82528

Proof. Let Bxð Þ r; t ð Þ T; F be an open ball with center x and radius r with respect to t. That is

Bxð Þ r; t ð Þ¼ T; F y ¼ yk <sup>∈</sup>ℓ<sup>∞</sup> : <sup>k</sup> : <sup>F</sup> <sup>μ</sup> T xð Þ� <sup>k</sup> T yk ; t ρ <sup>≤</sup><sup>1</sup> � <sup>r</sup> or F ν T xð Þ� <sup>k</sup> T yk ; t ρ <sup>≥</sup><sup>r</sup> ∈I : Let y∈B<sup>c</sup> <sup>x</sup>ð Þ <sup>r</sup>; <sup>t</sup> ð Þ <sup>T</sup>; <sup>F</sup> , then <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ;<sup>t</sup> ρ >1 � <sup>r</sup> and <sup>F</sup> <sup>ν</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ;<sup>t</sup> ρ <sup>&</sup>lt; <sup>r</sup>: Since <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ;<sup>t</sup> ρ >1 � r, there exists <sup>t</sup>0∈ð Þ <sup>0</sup>; <sup>t</sup> such that <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ ;t<sup>0</sup> ρ >1 � <sup>r</sup> and <sup>F</sup> <sup>ν</sup> T xð Þ�<sup>k</sup> T yð Þ<sup>k</sup> ð Þ ;t<sup>0</sup> ρ <sup>&</sup>lt; <sup>r</sup>. Putting <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>F</sup> <sup>μ</sup> T xð Þ�<sup>k</sup> T y ð Þ<sup>k</sup> ð Þ ;t<sup>0</sup> ρ , so we have <sup>r</sup>0>1 � <sup>r</sup>, there exists <sup>s</sup>∈ð Þ <sup>0</sup>; <sup>1</sup> such that r0>1 � s>1 � r. For r0>1 � s, we have r1, r2∈ð Þ 0; 1 such that r0∗r1>1 � s and ð Þ 1 � r<sup>0</sup> ⋄ð Þ 1 � r<sup>0</sup> ≤s: Putting r<sup>3</sup> ¼ maxf g r1;r<sup>2</sup> . Now we consider a ball Bc <sup>y</sup>ð Þ 1 � r3; t � t<sup>0</sup> ð Þ T; F . And we prove that Bc <sup>y</sup>ð Þ <sup>1</sup> � <sup>r</sup>3; <sup>t</sup> � <sup>t</sup><sup>0</sup> ð Þ <sup>T</sup>; <sup>F</sup> <sup>⊂</sup>B<sup>c</sup> <sup>x</sup>ð Þ r; t ð Þ T; F : Let <sup>z</sup> <sup>¼</sup> ð Þ zk <sup>∈</sup>B<sup>c</sup> <sup>y</sup>ð Þ <sup>1</sup> � <sup>r</sup>3; <sup>t</sup> � <sup>t</sup><sup>0</sup> ð Þ <sup>T</sup>; <sup>F</sup> , then <sup>F</sup> <sup>μ</sup> T y ð Þ ð Þ<sup>k</sup> �T zð Þ<sup>k</sup> ;t�t<sup>0</sup> ρ <sup>&</sup>gt;r<sup>3</sup> and

$$F\left(\frac{\mu\left(T(\mathbf{y}\_k) - T(\mathbf{z}\_k), t - t\_0\right)}{\rho}\right) < 1 - r\_3. \text{ Therefore, we have}$$

$$F\left(\frac{\mu\left(T(\mathbf{x}\_k) - T(\mathbf{z}\_k), t\right)}{\rho}\right) \ge F\left(\frac{\mu\left(T(\mathbf{x}\_k) - T(\mathbf{y}\_k), t\_0\right)}{\rho}\right) \ast F\left(\frac{\mu\left(T(\mathbf{y}\_k) - T(\mathbf{z}\_k), t - t\_0\right)}{\rho}\right).$$

$$\ge (r\_0 \ast r\_3) \ge (r\_0 \ast r\_1) \ge (1 - s) \ge (1 - r)$$

and

In particular <sup>μ</sup> T xð Þ� <sup>N</sup> <sup>L</sup>; <sup>t</sup>

Fuzzy Logic

In particular <sup>ν</sup> T xð Þ� <sup>N</sup> <sup>L</sup>; <sup>t</sup>

definition in our work.

M<sup>I</sup>

M<sup>I</sup>

18

Bxð Þ r; t ð Þ¼ T; F yk

reader to verify the linearity of these sequence spaces.

or F

Theorem 3.1. Every open ball <sup>B</sup>xð Þ <sup>r</sup>; <sup>t</sup> ð Þ <sup>T</sup>; <sup>F</sup> is an open set in <sup>M</sup><sup>I</sup>

1 � s≥μð Þ T xð Þ� <sup>n</sup> T xð Þ <sup>N</sup> ; t ≥μ T xð Þ� <sup>n</sup> L;

which is not possible. On the other hand

2

2 < ε: So, we have

s≤νð Þ T xð Þ� <sup>n</sup> T xð Þ <sup>N</sup> ; t ≤ν T xð Þ� <sup>n</sup> L;

>1 � <sup>ε</sup>: Therefore, we have

νð Þ T xð Þ� <sup>n</sup> T xð Þ <sup>N</sup> ; t ≥s or ν T xð Þ� <sup>n</sup> L;

which is not possible. Hence <sup>S</sup>⊂R. <sup>R</sup>∈<sup>I</sup> which implies <sup>S</sup>∈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

continuous, non-decreasing and convex with Fð Þ¼ 0 0,F xð Þ>0 for x>0 and F xð Þ! ∞ as x ! ∞. If the convexity of Orlicz function F is replaced by F xð Þ þ y ≤F xð Þþ F yð Þ, then this function is called modulus function.

defined by compact operator and Orlicz function which are as follows:

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup>; <sup>F</sup> ð Þ xk <sup>∈</sup>ℓ<sup>∞</sup> : {<sup>k</sup> : <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ� <sup>k</sup> <sup>L</sup>; <sup>t</sup>

or <sup>F</sup> <sup>ν</sup>ð Þ T xð Þ� <sup>k</sup> <sup>L</sup>; <sup>t</sup> ρ 

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ¼ <sup>T</sup>; <sup>F</sup> ð Þ xk <sup>∈</sup>ℓ<sup>∞</sup> : {<sup>k</sup> : <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ<sup>k</sup> ; <sup>t</sup>

We also define an open ball with center x and radius r with respect to t as follows:

<sup>∈</sup>ℓ<sup>∞</sup> : <sup>k</sup> : <sup>F</sup> <sup>μ</sup> T xð Þ� <sup>k</sup> T yk

ν T xð Þ� <sup>k</sup> T yk ; t ρ 

We shall now consider some theorems of these sequence spaces and invite the

or <sup>F</sup> <sup>ν</sup>ð Þ T xð Þ<sup>k</sup> ; <sup>t</sup> ρ 

Definition 3.1. An Orlicz function is a function F : ½ Þ! 0; ∞ ½ Þ 0; ∞ , which is

Remark 3.1. If F is an Orlicz function, then Fð Þ λx ≤λF xð Þ for all λ with 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

t 2

∗μ T xð Þ� <sup>N</sup> L;

t 2

⋄ν T xð Þ� <sup>N</sup> L;

ρ 

> ≥ε}∈I

ρ 

 ; t ρ 

> ≥ε∈I

≥ε}∈I  ≤1 � ε

≤1 � ε

: (6)

≤1 � ε

: (7)

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F .

; (5)

t 2

> t 2

< ε:

t 2

≤ε⋄ε < s,

≥ð Þ 1 � ε ∗ð Þ 1 � ε >1 � s,

$$\begin{split} F\left(\frac{\nu\left(T(\mathbf{x}\_{k})-T(\mathbf{z}\_{k}),t\right)}{\rho}\right) &\leq F\left(\frac{\nu\left(T(\mathbf{x}\_{k})-T(\mathbf{y}\_{k}),t\_{0}\right)}{\rho}\right)\circ F\left(\frac{\nu\left(T(\mathbf{y}\_{k})-T(\mathbf{z}\_{k}),t-t\_{0}\right)}{\rho}\right) \\ &\leq (1-r\_{0})\circ(1-r\_{3})\preceq(1-r\_{0})\circ(1-r\_{2})\preceq r\leq r. \end{split}$$

Thus z∈B<sup>c</sup> <sup>x</sup>ð Þ r; t ð Þ T; F and hence, we get

$$
\mathcal{B}^c\_\mathbf{y} (\mathbf{1} - r\_3, t - t\_0)(T, F) \subset \mathcal{B}^c\_\mathbf{x} (r, t)(T, F) \dots
$$

Remark 3.2. M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is an IFNS. Define

$$\begin{array}{rl} \tau^{l}\_{(\mu,\nu)}(T,F) = A \mathsf{C} \mathcal{M}^{l}\_{(\mu,\nu)}(T,F): & \text{for each } \mu \in \mathcal{A} \text{ there exists } t \succ 0, \\\ & \text{and } \ r \in (0,1) \text{ such that } \mathcal{B}\_{\mathbf{x}}(r,t)(T,F) \mathsf{C} \mathcal{A}. \end{array}$$

Then τ<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> is a topology on <sup>M</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F .

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 τ<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> on <sup>M</sup><sup>I</sup> <sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is first countable.

Proof. B<sup>x</sup> <sup>1</sup> n ; 1 n � �ð Þ <sup>T</sup>; <sup>F</sup> : <sup>n</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; … � � is a local base at <sup>x</sup>, the topology <sup>τ</sup><sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> on M<sup>I</sup> <sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> is first countable. □

Theorem 3.3. M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> and <sup>M</sup><sup>I</sup> <sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F are Hausdorff spaces. Proof. Let x, y∈M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> such that <sup>x</sup> 6¼ <sup>y</sup>. Then 0 < <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � � < 1 and 0 < F <sup>ν</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � � < 1:

Putting <sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � �, <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>F</sup> <sup>ν</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � � and <sup>r</sup> <sup>¼</sup> maxf g <sup>r</sup>1; <sup>1</sup> � <sup>r</sup><sup>2</sup> : For each r0∈ð Þ r; 1 there exists r<sup>3</sup> and r<sup>4</sup> such that r3∗r4≥r<sup>0</sup> and ð Þ 1 � r<sup>3</sup> ⋄ð Þ 1 � r<sup>4</sup> ≤ð Þ 1 � r<sup>0</sup> .

Putting <sup>r</sup><sup>5</sup> <sup>¼</sup> maxf g <sup>r</sup>3; <sup>1</sup> � <sup>r</sup><sup>4</sup> and consider the open balls <sup>B</sup><sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � � and <sup>B</sup><sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �. Then clearly B<sup>c</sup> <sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �∩B<sup>c</sup> <sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � � <sup>¼</sup> <sup>ϕ</sup>. For if there exists z∈B<sup>c</sup> <sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �∩B<sup>c</sup> <sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �, then

$$\begin{aligned} r\_1 &= F\left(\frac{\mu(T(\mathbf{x}) - T(\mathbf{y}), t)}{\rho}\right) \ge \left(\frac{\mu\left(T(\mathbf{x}) - T(\mathbf{z}), \frac{t}{2}\right)}{\rho}\right) \ast F\left(\frac{\mu\left(T(\mathbf{z}) - T(\mathbf{y}), \frac{t}{2}\right)}{\rho}\right) \\ \ge r\_3 \ast r\_5 \ge r\_3 \ast r\_3 \ge r\_0 \ast r\_1 \end{aligned}$$

and

$$\begin{aligned} r\_2 &= F\left(\frac{\nu(T(\mathbf{x}) - T(\mathbf{y}), t)}{\rho}\right) \nleq\_l \frac{\nu\left(\nu(T(\mathbf{x}) - T(\mathbf{z}), \frac{t}{2})\right)}{\rho} \nleq\_l \left(\frac{\nu\left(T(\mathbf{z}) - T(\mathbf{y}), \frac{t}{2}\right)}{\rho}\right) \\ &\leq (1 - r\_5) \circ (1 - r\_5) \leq (1 - r\_4) \circ (1 - r\_4) \leq (1 - r\_0) < r\_2 \end{aligned}$$

which is a contradiction. Hence, M<sup>I</sup> ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is Hausdorff. Similarly the proof follows for M<sup>I</sup> <sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> . □

## 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.

Author details

21

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

DOI: http://dx.doi.org/10.5772/intechopen.82528

\*Address all correspondence to: vakhanmaths@gmail.com

provided the original work is properly cited.

Department of Mathematics, Aligarh Muslim University, Aligarh, India

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

## Conflict of interest

The authors declare that they have no competing interests.

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces DOI: http://dx.doi.org/10.5772/intechopen.82528

Proof. B<sup>x</sup> <sup>1</sup>

0 < F <sup>ν</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � �

<sup>B</sup><sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup>

and

follows for M<sup>I</sup>

4. Conclusion

z∈B<sup>c</sup>

Theorem 3.3. M<sup>I</sup>

Proof. Let x, y∈M<sup>I</sup>

ð Þ 1 � r<sup>3</sup> ⋄ð Þ 1 � r<sup>4</sup> ≤ð Þ 1 � r<sup>0</sup> .

� �. Then clearly B<sup>c</sup>

<sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ� T yð Þ; <sup>t</sup> ρ � �

<sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>F</sup> <sup>ν</sup>ð Þ T xð Þ� T yð Þ; <sup>t</sup> ρ � �

which is a contradiction. Hence, M<sup>I</sup>

discussed for proper understanding.

Conflict of interest

20

≥r5∗r5≥r3∗r3≥r0>r<sup>1</sup>

2

<sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �∩B<sup>c</sup>

on M<sup>I</sup>

Fuzzy Logic

n ; 1 n

Putting <sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup>

< 1:

ρ � �

<sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �, then

� �ð Þ <sup>T</sup>; <sup>F</sup> : <sup>n</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; … � � is a local base at <sup>x</sup>, the topology <sup>τ</sup><sup>I</sup>

, <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>F</sup> <sup>ν</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup> ρ � �

Putting <sup>r</sup><sup>5</sup> <sup>¼</sup> maxf g <sup>r</sup>3; <sup>1</sup> � <sup>r</sup><sup>4</sup> and consider the open balls <sup>B</sup><sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup>

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> and <sup>M</sup><sup>I</sup>

each r0∈ð Þ r; 1 there exists r<sup>3</sup> and r<sup>4</sup> such that r3∗r4≥r<sup>0</sup> and

<sup>x</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2 � �∩B<sup>c</sup>

≥

≤F

0

B@

≤ð Þ 1 � r<sup>5</sup> ⋄ð Þ 1 � r<sup>5</sup> ≤ð Þ 1 � r<sup>4</sup> ⋄ð Þ 1 � r<sup>4</sup> ≤ð Þ 1 � r<sup>0</sup> < r<sup>2</sup>

The authors declare that they have no competing interests.

0

B@

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> is first countable. □

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> such that <sup>x</sup> 6¼ <sup>y</sup>. Then 0 < <sup>F</sup> <sup>μ</sup>ð Þ T xð Þ�T yð Þ;<sup>t</sup>

<sup>y</sup> <sup>1</sup> � <sup>r</sup>5; <sup>t</sup> 2

> t 2

> > t 2

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup> . □

1

CA ⋄F

1

CA<sup>∗</sup><sup>F</sup>

μ T xð Þ� T zð Þ;

� �

ρ

ν T xð Þ� T zð Þ;

� �

ρ

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

<sup>0</sup>ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F are Hausdorff spaces.

� � <sup>¼</sup> <sup>ϕ</sup>. For if there exists

0

B@

0

B@

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ T; F is Hausdorff. Similarly the proof

ρ � �

and r ¼ maxf g r1; 1 � r<sup>2</sup> : For

μ T zð Þ� T yð Þ;

� �

ρ

ν T zð Þ� T yð Þ;

� �

ρ

2 � � and

ð Þ <sup>μ</sup>;<sup>ν</sup> ð Þ <sup>T</sup>; <sup>F</sup>

< 1 and

t 2

> t 2

1

CA

1

CA

## Author details

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad Department of Mathematics, Aligarh Muslim University, Aligarh, India

\*Address all correspondence to: vakhanmaths@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20(1):87-96

[2] Zadeh LA. Fuzzy sets. Information and Control. 1965;8:338-353

[3] Barros LC, Bassanezi RC, Tonelli PA. Fuzzy modelling in population dynamics. Ecological Modelling. 2000; 128:27-33

[4] Hong L, Sun JQ. Bifurcations of fuzzy non-linear dynamical systems. Communications in Nonlinear Science and Numerical Simulation. 2006;1:1-12

[5] Fradkov AL, Evans RJ. Control of chaos: Methods of applications in engineering. Chaos, Solitons & Fractals. 2005;29:33-56

[6] Giles R. A computer program for fuzzy reasoning. Fuzzy Sets and Systems. 1980;4:221-234

[7] Saddati R, Park JH. On the intuitionistic fuzzy topological spaces. Chaos, Solution and Fractals. 2006;27: 331-344

[8] Karakus S, Demirci K, Duman O. Statistical convergence on intuitionistic fuzzy normed spaces. Chaos, Solitons and Fractals. 2008;35:763-769

[9] Coker D. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems. 1997;88(1): 81-89

[10] Mursaleen M, Lohani QMD. Intuitionistic fuzzy 2-normed space and some related concepts. Chaos, Solution and Fractals. 2009;42:331-344

[11] Khan VA, Ebadullah K, Rababah RKA. Intuitionistic fuzzy zweier Iconvergent sequence spaces. Functional Analysis: Theory, Methods and Applications. 2015;1:1-7

[12] Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum. 1951;2:73-74

[21] Bromwich TJI. An Introduction to the Theory of Infinite Series. New York:

DOI: http://dx.doi.org/10.5772/intechopen.82528

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

[22] Khan VA, Fatima H, Abdullaha SAA, Khan MD. On a new BV<sup>σ</sup> Iconvergent double sequence spaces. Theory and Application of Mathematics and Computer Science. 2016;6(2):

[23] Khan VA, Shafiq M, Guillen BL. On paranorm I-convergent sequence spaces defined by a compact operator. Afrika Matematika, Journal of the African Mathematical Union (Springer). 2014; 25(4):12. DOI: 10.1007/s13370-014-

[24] Kreyszig E. Introductory Functional Analysis with Application. New York, Chicheste, Brisbane, Toronto: John

Wiley and Sons, Inc; 1978

MacMillan Co. Ltd; 1965

187-197

0287-2

23

[13] Fast H. Sur la convergence statistique. Colloquium Mathematicum. 1951;2:241-244

[14] Schoenberg IJ. The integrability of certain functions and related summability methods. American Mathematical Monthly. 1959;66:361-375

[15] Kostyrko P, Salat T, Wilczynski W. I-convergence. Real Analysis Exchange. 2000;26(2):669-686

[16] Alotaibi A, Hazarika B, Mohiuddine SA. On the ideal convergence of double sequences in locally solid Riesz spaces. Abstract and Applied Analysis. 2014. 6 p. Article ID: 396254. http://dx.doi. org/10.1155/2014/396254

[17] Hazarika B, Mohiuddine SA. Ideal convergence of random variables. Journal of Function Spaces and Applications. 2013. Article ID 148249:7

[18] Mohiuddine SA, Lohani QMD. On generalized statistical convergence in intuitionistic fuzzy normed spaces. Chaos, Solitons and Fractals. 2009;41: 142-149

[19] Mursaleen M, Mohiuddine SA, Edely OHH. On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Computers and Mathematics with Application. 2010;59: 603-611

[20] Nabiev A, Pehlivan S, Gürdal M. On I-Cauchy sequence. Taiwanese Journal of Mathematics. 2007;11(2):569-576

Some Topological Properties of Intuitionistic Fuzzy Normed Spaces DOI: http://dx.doi.org/10.5772/intechopen.82528

[21] Bromwich TJI. An Introduction to the Theory of Infinite Series. New York: MacMillan Co. Ltd; 1965

References

Fuzzy Logic

20(1):87-96

128:27-33

2005;29:33-56

331-344

81-89

22

[1] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986;

Analysis: Theory, Methods and Applications. 2015;1:1-7

ordinaire et la convergence asymptotique. Colloquium Mathematicum. 1951;2:73-74

[13] Fast H. Sur la convergence

certain functions and related summability methods. American Mathematical Monthly. 1959;66:361-375

2000;26(2):669-686

org/10.1155/2014/396254

142-149

603-611

1951;2:241-244

statistique. Colloquium Mathematicum.

[14] Schoenberg IJ. The integrability of

[15] Kostyrko P, Salat T, Wilczynski W. I-convergence. Real Analysis Exchange.

[16] Alotaibi A, Hazarika B, Mohiuddine SA. On the ideal convergence of double sequences in locally solid Riesz spaces. Abstract and Applied Analysis. 2014. 6 p. Article ID: 396254. http://dx.doi.

[17] Hazarika B, Mohiuddine SA. Ideal convergence of random variables. Journal of Function Spaces and

Applications. 2013. Article ID 148249:7

[18] Mohiuddine SA, Lohani QMD. On generalized statistical convergence in intuitionistic fuzzy normed spaces. Chaos, Solitons and Fractals. 2009;41:

[19] Mursaleen M, Mohiuddine SA, Edely OHH. On the ideal convergence of double sequences in intuitionistic fuzzy

Mathematics with Application. 2010;59:

[20] Nabiev A, Pehlivan S, Gürdal M. On I-Cauchy sequence. Taiwanese Journal of Mathematics. 2007;11(2):569-576

normed spaces. Computers and

[12] Steinhaus H. Sur la convergence

[2] Zadeh LA. Fuzzy sets. Information

[3] Barros LC, Bassanezi RC, Tonelli PA.

dynamics. Ecological Modelling. 2000;

[4] Hong L, Sun JQ. Bifurcations of fuzzy non-linear dynamical systems. Communications in Nonlinear Science and Numerical Simulation. 2006;1:1-12

[5] Fradkov AL, Evans RJ. Control of chaos: Methods of applications in engineering. Chaos, Solitons & Fractals.

[6] Giles R. A computer program for fuzzy reasoning. Fuzzy Sets and

intuitionistic fuzzy topological spaces. Chaos, Solution and Fractals. 2006;27:

[8] Karakus S, Demirci K, Duman O. Statistical convergence on intuitionistic fuzzy normed spaces. Chaos, Solitons

Systems. 1980;4:221-234

[7] Saddati R, Park JH. On the

and Fractals. 2008;35:763-769

[9] Coker D. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems. 1997;88(1):

[10] Mursaleen M, Lohani QMD.

and Fractals. 2009;42:331-344

Intuitionistic fuzzy 2-normed space and some related concepts. Chaos, Solution

[11] Khan VA, Ebadullah K, Rababah RKA. Intuitionistic fuzzy zweier Iconvergent sequence spaces. Functional

and Control. 1965;8:338-353

Fuzzy modelling in population

[22] Khan VA, Fatima H, Abdullaha SAA, Khan MD. On a new BV<sup>σ</sup> Iconvergent double sequence spaces. Theory and Application of Mathematics and Computer Science. 2016;6(2): 187-197

[23] Khan VA, Shafiq M, Guillen BL. On paranorm I-convergent sequence spaces defined by a compact operator. Afrika Matematika, Journal of the African Mathematical Union (Springer). 2014; 25(4):12. DOI: 10.1007/s13370-014- 0287-2

[24] Kreyszig E. Introductory Functional Analysis with Application. New York, Chicheste, Brisbane, Toronto: John Wiley and Sons, Inc; 1978

Section 3

Adaptive Neuro-Fuzzy

Inference Systems

25

Section 3
