# A note on IFN-spaces

- Reza Saadati
^{1}and - Choonkil Park
^{2}Email author

**2014**:63

https://doi.org/10.1186/1687-1847-2014-63

© Saadati and Park; licensee Springer. 2014

**Received: **26 December 2013

**Accepted: **30 January 2014

**Published: **11 February 2014

## Abstract

Recently, Al-Fhaid and Mohiuddine (Adv. Differ. Equ. 2013:203, 2013) and Mohiuddine and Alghamdi (Adv. Differ. 2012:141, 2012) got some results in intuitionistic fuzzy normed spaces using ideas of intuitionistic fuzzy sets due to Atanassov and fuzzy normed spaces due to Saadati and Vaezpour. In this note, we show that the mentioned results follow directly from well-known theorems in fuzzy normed spaces.

**MSC:**54E40, 54E35, 54H25.

## Keywords

## 1 Introduction

Intuitionistic fuzzy normed spaces were investigated by Saadati and Park [1]. They introduced and studied intuitionistic fuzzy normed spaces based both on the idea of intuitionistic fuzzy sets due to Atanassov [2] and the concept of fuzzy normed spaces given by Saadati and Vaezpour in [3]. Next Deschrijver *et al.* [4] modified the concept of intuitionistic fuzzy normed spaces and introduced the notation of ℒ-fuzzy normed space. Recently, Al-Fhaid and Mohiuddine [5] and Mohiuddine and Alghamdi [6] got some results in intuitionistic fuzzy normed spaces. In this note we prove that the topology ${\tau}_{(\mu ,\nu )}$ generated by an intuitionistic fuzzy normed space $(X,\mu ,\nu ,\ast ,\diamond )$ coincides with the topology ${\tau}_{\mu}$ generated by the generalized fuzzy normed space $(X,\mu ,\ast )$, and thus, the results obtained in [5] and [6] are immediate consequences of the corresponding results for fuzzy normed spaces.

## 2 Preliminaries

*t*-norm if it satisfies the following conditions:

- (a)
∗ is associative and commutative,

- (b)
∗ is continuous,

- (c)
$a\ast 1=a$ for all $a\in [0,1]$,

- (d)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$, for each $a,b,c,d\in [0,1]$.

Two typical examples of continuous *t*-norm are $a\ast b=ab$ and $a\ast b=min(a,b)$.

*t*-conorm if it satisfies the following conditions:

- (a)
⋄ is associative and commutative,

- (b)
⋄ is continuous,

- (c)
$a\diamond 0=a$ for all $a\in [0,1]$,

- (d)
$a\diamond b\le c\diamond d$ whenever $a\le c$ and $b\le d$, for each $a,b,c,d\in [0,1]$.

Two typical examples of a continuous *t*-conorm are $a\diamond b=min(a+b,1)$ and $a\diamond b=max(a,b)$.

In 2005, Saadati and Vaezpour [3] introduced the concept of fuzzy normed spaces.

**Definition 2.1** Let *X* be a real vector space. A function $\mu :X\times \mathbb{R}\to [0,1]$ is called a *fuzzy norm* on *X* if for all $x,y\in X$ and all $s,t\in \mathbb{R}$,

(${\mu}_{1}$) $\mu (x,t)=0$ for $t\le 0$;

(${\mu}_{2}$) $x=0$ if and only if $\mu (x,t)=1$ for all $t>0$;

(${\mu}_{3}$) $\mu (cx,t)=\mu (x,\frac{t}{|c|})$ if $c\ne 0$;

(${\mu}_{4}$) $\mu (x+y,s+t)\ge \mu (x,s)\ast \mu (y,t)$;

(${\mu}_{5}$) $\mu (x,\cdot )$ is a non-decreasing function of ℝ and ${lim}_{t\to \mathrm{\infty}}\mu (x,t)=1$;

(${\mu}_{6}$) for $x\ne 0$, $\mu (x,\cdot )$ is continuous on ℝ.

for all $x,y,z\in X$ and $t>0$. Then *μ* is a (standard) fuzzy normed and $(X,\mu ,\cdot )$ is a fuzzy normed space.

Saadati and Vaezpour showed in [3] that every fuzzy norm $(\mu ,\ast )$ on *X* generates a first countable topology ${\tau}_{\mu}$ on *X* which has as a base the family of open sets of the form $\{{B}_{\mu}(x,r,t):x\in X,r\in (0,1),t>0\}$ where ${B}_{\mu}(x,r,t)=\{y\in X:\mu (x-y,t)>1-r\}$ for all $x\in X$, $r\in (0,1)$ and $t>0$.

## 3 Intuitionistic fuzzy normed spaces

Saadati and Park [1] defined the notion of intuitionistic fuzzy normed spaces with the help of continuous *t*-norms and continuous *t*-conorms as a generalization of fuzzy normed space due to Saadati and Vaezpour [3].

**Definition 3.1**The 5-tuple $(X,\mu ,\nu ,\ast ,\diamond )$ is said to be an

*intuitionistic fuzzy normed space*if

*X*is a vector space, ∗ is a continuous

*t*-norm, ⋄ is a continuous

*t*-conorm, and

*μ*,

*ν*are fuzzy sets on $X\times (0,\mathrm{\infty})$ satisfying the following conditions for every $x,y\in X$ and $t,s>0$:

- (a)
$\mu (x,t)+\nu (x,t)\le 1$,

- (b)
$\mu (x,t)>0$,

- (c)
$\mu (x,t)=1$ if and only if $x=0$,

- (d)
$\mu (\alpha x,t)=\mu (x,\frac{t}{|\alpha |})$ for each $\alpha \ne 0$,

- (e)
$\mu (x,t)\ast \mu (y,s)\le \mu (x+y,t+s)$,

- (f)
$\mu (x,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is continuous,

- (g)
${lim}_{t\to \mathrm{\infty}}\mu (x,t)=1$ and ${lim}_{t\to 0}\mu (x,t)=0$,

- (h)
$\nu (x,t)<1$,

- (i)
$\nu (x,t)=0$ if and only if $x=0$,

- (j)
$\nu (\alpha x,t)=\nu (x,\frac{t}{|\alpha |})$ for each $\alpha \ne 0$,

- (k)
$\nu (x,t)\diamond \nu (y,s)\ge \nu (x+y,t+s)$,

- (l)
$\nu (x,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is continuous,

- (m)
${lim}_{t\to \mathrm{\infty}}\nu (x,t)=0$ and ${lim}_{t\to 0}\nu (x,t)=1$.

In this case $(\mu ,\nu )$ is called an *intuitionistic fuzzy norm*.

**Example 3.2**Let $(X,\parallel \cdot \parallel )$ be a normed space. Denote $a\ast b=ab$ and $a\diamond b=min(a+b,1)$ for all $a,b\in [0,1]$ and let

*μ*and

*ν*be fuzzy sets on $X\times (0,\mathrm{\infty})$ defined as follows:

for all $t\in {\mathbf{R}}^{+}$. Then $(X,\mu ,\nu ,\ast ,\diamond )$ is an intuitionistic fuzzy normed space.

Saadati and Park proved in [1] that every intuitionistic fuzzy norm $(\mu ,\nu )$ on *X* generates a first countable topology ${\tau}_{(\mu ,\nu )}$ on *X* which has as a base the family of open sets of the form $\{{B}_{(\mu ,\nu )}(x,r,t):x\in X,r\in (0,1),t>0\}$ where ${B}_{(\mu ,\nu )}(x,r,t)=\{y\in X:\mu (x-y,t)>1-r,\nu (x-y,t)<r\}$ for all $x\in X$, $r\in (0,1)$ and $t>0$.

**Lemma 3.3** *Let* $(X,\mu ,\nu ,\ast ,\diamond )$ *be an intuitionistic fuzzy normed space*. *Then*, *for each* $x\in X$, $r\in (0,1)$ *and* $t>0$, *we have* ${B}_{(\mu ,\nu )}(x,r,t)={B}_{\mu}(x,r,t)$.

*Proof* It is clear that ${B}_{(\mu ,\nu )}(x,r,t)\subseteq {B}_{\nu}(x,r,t)$.

Hence $\nu (x-y,t)<r$, and consequently $y\in {B}_{(\mu ,\nu )}(x,r,t)$. The proof is finished. □

From Lemma 3.3, we deduce the following.

**Theorem 3.4** *Let* $(X,\mu ,\nu ,\ast ,\diamond )$ *be an intuitionistic fuzzy normed space*. *Then the topologies* ${\tau}_{(\mu ,\nu )}$ *and* ${\tau}_{\mu}$ *coincide on* *X*.

## Declarations

## Authors’ Affiliations

## References

- Saadati R, Park JH: On the intuitionistic fuzzy topological spaces.
*Chaos Solitons Fractals*2006, 27(2):331-344. 10.1016/j.chaos.2005.03.019MATHMathSciNetView ArticleGoogle Scholar - Atanassov KT: Intuitionistic fuzzy sets.
*Fuzzy Sets Syst.*1986, 20(1):87-96. 10.1016/S0165-0114(86)80034-3MATHMathSciNetView ArticleGoogle Scholar - Saadati R, Vaezpour SM: Some results on fuzzy Banach spaces.
*J. Appl. Math. Comput.*2005, 17(1-2):475-484. 10.1007/BF02936069MATHMathSciNetView ArticleGoogle Scholar - Deschrijver G, O’Regan D, Saadati R, Vaezpour SM: ℒ-Fuzzy Euclidean normed spaces and compactness.
*Chaos Solitons Fractals*2009, 42(1):40-45. 10.1016/j.chaos.2008.10.026MATHMathSciNetView ArticleGoogle Scholar - Al-Fhaid AS, Mohiuddine SA: On the Ulam stability of mixed type QA mappings in IFN-spaces.
*Adv. Differ. Equ.*2013., 2013: Article ID 203Google Scholar - Mohiuddine SA, Alghamdi MA: Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces.
*Adv. Differ. Equ.*2012., 2012: Article ID 141Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.