International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 100-105

**Published online** March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.100

© The Korean Institute of Intelligent Systems

Veeraiyan Suganya^{1}, Pitchayan Gomathi Sundari^{2}, and Neelamegarajan Rajesh^{2}

^{1}Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tamilnadu, India^{2}Department of Mathematics, Rajah Serfoji Government College (affliated to Bharathdasan University), Tamilnadu, India

**Correspondence to : **

Neelamegarajan Rajesh (nrajesh_topology@yahoo.co.in)

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, we introduce and investigate the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. Several characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

**Keywords**: Double fuzzy topological space, (*r*, *s*)-fuzzy α-open set, Double fuzzy α-irresolute multifunctions

The fuzzy concept has penetrated nearly all branches of mathematics since the concept has been defined by Zadeh [1]. Fuzzy sets have applications in several fields, such as information [2] and control [3]. The theory of fuzzy topological spaces was first defined and developed by Chang [2], and since then various notions of general topology have been generalized to the fuzzy topological spaces defined by Chang. Šostak [4] and Kubiak [5] developed fuzzy topology as an extension of the fuzzy topology introduced by Chang [2]. It has since been developed in several ways. Šostak [6] published a survey article on the areas of development of fuzzy topological spaces. Topologists often refer to the concept of fuzzy topology developed by Chang as “”and the concept of fuzzy topology proposed by Kubiak-Šostak as “”, where

Throughout this paper, let _{0} = (0, 1] and _{1} = [0, 1). The family of all fuzzy sets in ^{X}^{X}^{X}^{Y}^{−1} : ^{Y}^{X}_{f}_{(}_{x}_{)=}_{y}^{−1}(^{X}^{Y}

Let ^{Y}_{F}

Let

(1) normalized if and only if for each

x ∈X , there exists ay _{0}∈Y such thatG (_{F}x, y _{0}) = 1̄.(2) crisp if and only if

G (_{F}x, y _{0}) = 1̄ for eachx ∈X andy ∈Y .

Let

(1) the lower inverse of

μ ∈I is a fuzzy set^{Y}F (^{l}μ ) ∈I defined by^{X}${F}^{l}(\mu )(x)={\displaystyle \underset{y\in Y}{\bigvee}}({G}_{F}(x,y)\bigwedge \mu (y))$ .(2) the upper inverse of

μ ∈I is a fuzzy set^{Y}F (^{u}μ ) ∈I defined by^{X}${F}^{u}(\mu )(x)={\displaystyle \underset{y\in Y}{\bigwedge}}({G}_{F}^{c}(x,y)\vee \mu (y))$ .

Let

(1)

F (λ _{1}) ≤F (λ _{2}) ifλ _{1}≤λ _{2}.(2)

F (^{l}μ _{1}) ≤F (^{l}μ _{2}) andF (^{u}μ _{1}) ≤F (^{u}μ _{2}) ifμ _{1}≤μ _{2}.(3)

F (^{l}μ ) ≤ (^{c}F (^{u}μ )) .^{c}(4)

F (^{u}μ ) ≤ (^{c}F (^{l}μ )) .^{c}(5)

F (F (^{u}μ )) ≤μ ifF is a crisp.(6)

F (^{u}F (λ )) ≥λ ifF is a crisp.

Let

Let

(1)

H ∘F =F (H ).(2) (

H ∘F ) =^{u}F (^{u}H ).^{u}(3) (

H ∘F ) =^{l}F (^{l}H ).^{l}

Let

(1)

$({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})(\lambda )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}(\lambda )$ .(2)

${({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})}^{l}(\mu )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}^{l}(\mu )$ .(3)

${({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})}^{u}(\mu )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}^{u}(\mu )$ .

A fuzzy point _{t}_{0} at _{t}

A double fuzzy topology on ^{★} : ^{X}

(1)

τ (λ ) ≤ 1̄ –τ ^{★}(λ ) for eachλ ∈I .^{X}(2)

τ (λ _{1}∧λ _{2}) ≥τ (λ _{1}) ∧τ (λ _{2}) andτ ^{★}(λ _{1}∧λ _{2}) ≤τ ^{★}(λ _{1}) ∨τ ^{★}(λ _{2}) for eachλ _{1},λ _{2}∈I .^{X}(3)

$\tau ({\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{\lambda}_{i})\ge {\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigwedge}}\tau ({\lambda}_{i})$ and${\tau}^{\u2605}({\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{\lambda}_{i})\le {\vee}_{i\in \mathrm{\Gamma}}{\tau}^{\u2605}({\lambda}_{i})$ for eachλ ∈_{i}I ^{X}andi ∈ Γ.

The triplet (^{★}) is called a double fuzzy topological space.

A fuzzy set ^{★} (

Let (^{★}) be a double fuzzy topological space. Then, the double fuzzy closure and double fuzzy interior operators of ^{X}_{τ,τ}_{★} (^{X}^{★}(1̄ – _{τ,τ}_{★} (^{X}^{★}(_{0} and _{1} such that

Let (^{★}) be a double fuzzy topological space. For each ^{X}_{0}, and _{1},

(1)

λ is called (r, s )-fuzzyα -open ifλ ≤I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(λ, r, s ),r, s ),r, s ).(2)

λ is called (r, s )-fuzzyα -closed if 1̄ –λ is (r, s )-fuzzyα -open.(3)

α C _{τ,τ}_{★}(λ, r, s ) = ∧{μ ∈I :^{X}μ ≥λ andμ is (r, s )-fuzzyα -closed} is called (r, s )-fuzzyα -closure ofλ .(4)

α I _{τ,τ}_{★}(λ, r, s ) = ∨{μ ∈I :^{X}μ ≤λ andμ is (r, s )-fuzzyα -open} is called (r, s )-fuzzyα -interior ofλ .

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then,

(1) double fuzzy upper

α -continuous at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for eachμ ∈I ,^{Y}σ (μ ) ≥r , andσ ^{★}(μ ) ≤s , there exists an (r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ∧dom (F ) ≤F (^{u}μ ).(2) double fuzzy lower

α -continuous at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for eachμ ∈I ,^{Y}σ (μ ) ≥r andσ ^{★}(μ ) ≤s , there exists an (r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ≤F (^{l}μ ).(3) double fuzzy upper (lower)

α -continuous if it is fuzzy upper (lower)α -continuous at everyx ∈_{t}dom (F ).

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then,

(1) double fuzzy upper

α -irresolute at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for each (r, s )-fuzzyα -open setμ ∈I , there exists an (^{Y}r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ∧dom (F ) ≤F (^{u}μ ).(2) double fuzzy lower

α -irresolute at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for each (r, s )-fuzzyα -open setμ ∈I , there exists an (^{Y}r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ≤F (^{l}μ ).(3) double fuzzy upper (lower)

α -irresolute if it is fuzzy upper (lower)α -irresolute at everyx ∈_{t}dom (F ).

If _{t}_{t}^{u}^{Y}^{X}_{t}^{u}

For a fuzzy multifunction ^{★}) → (^{★}), the following statements are equivalent:

(1)

F is double fuzzy lowerα -irresolute,(2)

F (^{l}μ ) is (r, s )-fuzzyα -open for any (r, s )-fuzzyα -open setμ ∈I ,^{Y}(3)

F (^{u}μ ) is (r, s )-fuzzyα -closed for any (r, s )-fuzzyα -closed setμ ∈I ,^{Y}(4)

C _{τ,τ}_{★}(I _{τ,τ}_{★}(C _{τ,τ}_{★}(F (^{u}μ ),r, s ),r, s ),r, s ) ≤F (^{u}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I ,^{Y}(5)

α C _{τ,τ}_{★}(F (^{u}μ ),r, s ) ≤F (^{u}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I .^{Y}

(1) ⇒ (2): Let _{t}^{Y}_{t}^{l}^{X}_{t}^{l}_{t}_{τ,τ}_{★} (_{τ,τ}_{★} (_{τ,τ}_{★} (^{l}^{l}_{τ,τ}_{★} (_{τ,τ}_{★} (_{τ,τ}_{★} (^{l}^{l}

(2) ⇒ (3): Let ^{Y}^{l}^{u}^{u}

(3) ⇒ (4): Let ^{Y}^{u}_{σ,σ}_{★} (

(4) ⇒ (5): Let ^{Y}

(5) ⇒ (3): Let ^{Y}_{τ,τ}_{★} (^{u}^{u}_{σ,σ}_{★} (^{u}^{u}

(3) ⇒ (2): Let ^{Y}^{u}^{l}^{l}

(2) ⇒ (1): Let _{t}^{Y}_{t}^{l}^{l}

We state the following result without proof in view of the above theorem.

For a fuzzy multifunction ^{★}) → (^{★}), the following statements are equivalent:

(1)

F is double fuzzy upperα -irresolute.(2)

F (^{u}μ ) isr -fuzzyα -open for any (r, s )-fuzzyα -open setμ ∈I .^{Y}(3)

F (^{l}μ ) isr -fuzzyα -closed for any (r, s )-fuzzyα -closed setμ ∈I .^{Y}(4)

C _{τ,τ}_{★}(I _{τ,τ}_{★}(C _{τ,τ}_{★}(F (^{l}μ ),r, s ),r, s ),r, s ) ≤F (^{l}α C _{σ,σ}_{★}(μ ,r, s )) for anyμ ∈I .^{Y}(5)

α C _{τ,τ}_{★}(F (^{l}μ ),r, s ) ≤F (^{l}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I .^{Y}

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then, the following can be postulated:

(1)

F as normalized implies that it is double fuzzy upperα -irresolute at a fuzzy pointx if and only if_{t}x ∈_{t}I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(F (^{u}μ ),r, s ),r, s ),r, s ) for (r, s )-fuzzyα -open setsμ ∈I and^{Y}x ∈_{t}F (^{u}μ ).(2)

F is double fuzzy lowerα -irresolute at a fuzzy pointx if and only if_{t}x ∈_{t}I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(F (^{l}μ ),r, s ),r, s ),r, s ) for (r, s )-fuzzyα -open setsμ ∈I and^{Y}x ∈_{t}F (^{l}μ ).

For a fuzzy multifunction ^{★}) → (^{★}), we obtain the following:

(1) Every double fuzzy upper

α -continuous multifunction is double fuzzy upperα -irresolute.(2) Every double fuzzy lower

α -continuous multifunction is double fuzzy lowerα -irresolute.

The following example demonstrates that the converses of the above propositions are not true.

Let _{1}, _{2}}, _{1}, _{2}, _{3}}, and ^{★}) → (^{★}) be fuzzy multifunctions defined by _{F}_{1}, _{1}) = 0.2, _{F}_{1}, _{2}) = 1̄, _{F}_{1}, _{3}) = 0.3, _{F}_{2}, _{1}) = 0.5, _{F}_{2}, _{2}) = 0.3, and _{F}_{2}, _{3}) = 1̄. We define fuzzy topologies ^{★} : ^{X}^{★} : ^{Y}

Then,

Let ^{★})→(^{★}) and ^{★}) → (^{★}) be fuzzy multifunctions. Thus, the following can be concluded:

(1) If

F andH are double fuzzy lowerα -irresolute, then,H ∘F is also double fuzzy lowerα -irresolute.(2) If

F is double fuzzy lowerα -irresolute andH is double fuzzy lowerα -continuous, then,H ∘F is also double fuzzy lowerα -irresolute.

Follows from the respective definitions.

Let {_{i}

Let ^{Y}_{i}^{Y}

A fuzzy multifunction ^{★}) → (^{★}) is called (_{t}_{t}

Let ^{★}) → (^{★}) be a crisp double fuzzy upper

Let ^{X}_{i}_{i}_{t}_{x}_{t} of Γ such that _{t}^{u}_{t}^{u}_{x}_{t}) and _{x}_{t} is (^{u}_{x}_{t}) is (^{u}_{x}_{t}) : ^{u}_{x}_{t}) is (_{t}

There is no potential conflict of interest relevant to this study.

- Zadeh, LA (1965). Fuzzy sets. Information and Control.
*8*, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X - Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications.
*24*, 182-190. - Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*61*, 137-142. https://doi.org/10.1016/0165-0114(94)90229-1 - AP Sostak (1985). On a fuzzy topological structure. Proceedings of the 13th Winter School on Abstract Analysis. Palermo, Italy: Circolo Matematico di Palermo, pp. 89-103
- Kubiak, T 1985. On fuzzy topologies. PhD dissertation. Adam Mickiewicz University. Poznan, Poland.
- Sostak, AP (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences.
*78*, 662-701. - Coker, D (1996). An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces. Journal of Fuzzy Mathematics.
*4*, 749-764. - Coker, D (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems.
*88*, 81-89. https://doi.org/10.1016/S0165-0114(96)00076-0 - Mondal, TK, and Samanta, SK (2002). On intuitionistic gradation of openness. Fuzzy Sets and Systems.
*131*, 323-336. https://doi.org/10.1016/S0165-0114(01)00235-4 - Garcia, JG, and Rodabaugh, SE (2005). Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets and Systems.
*156*, 445-484. https://doi.org/10.1016/j.fss.2005.05.023 - Abd-Allah, MA, El-Saady, K, and Ghareeb, A (2009). (r, s)-Fuzzy F-open sets and (r, s)-fuzzy F-closed spaces. Chaos, Solitons & Fractals.
*42*, 649-656. https://doi.org/10.1016/j.chaos.2009.01.031 - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2016). Generalized fuzzy b-closed and generalized ★-fuzzy b-closed sets in double fuzzy topological spaces. Egyptian Journal of Basic and Applied Sciences.
*3*, 61-67. - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2014). Somewhat slightly generalized double fuzzy semicontinuous functions. International Journal of Mathematics and Mathematical Sciences.
*2014*. article no. 756376 - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2017). New notions from (alpha,beta)-generalised fuzzy preopen sets. Gazi University Journal of Science.
*30*, 311-331. - Ghareeb, A (2011). Normality of double fuzzy topological spaces. Applied Mathematics Letters.
*24*, 533-540. https://doi.org/10.1016/j.aml.2010.11.008 - Ghareeb, A (2012). Weak forms of continuity in I-double gradation fuzzy topological spaces. SpringerPlus.
*1*. article no. 19 - El-Saady, K, and Ghareeb, A (2012). Several types of (r, s)-fuzzy compactness defined by an (r, s)-fuzzy regular semiopen sets. Annals of fuzzy Mathematics and Informatics.
*3*, 159-169. - Lee, EP (2004). Semiopen sets on intuitionistic fuzzy topological spaces in Sostak’s sense. Journal of the Korean Institute of Intelligent Systems.
*14*, 234-238. https://doi.org/10.5391/JKIIS.2004.14.2.234 - Lee, SO, and Lee, EP (2005). Fuzzy (r, s)-preopen sets. International Journal of Fuzzy Logic and Intelligent Systems.
*5*, 136-139. https://doi.org/10.5391/IJFIS.2005.5.2.136 - Lee, SO, and Lee, EP (2006). Fuzzy strongly (r, s)-semiopen sets. International Journal of Fuzzy Logic and Intelligent Systems.
*6*, 299-303. https://doi.org/10.5391/IJFIS.2006.6.4.299 - Lee, SJ, and Kim, JT (2007). Fuzzy (r, s)-irresolute maps. International Journal of Fuzzy Logic and Intelligent Systems.
*7*, 49-57. https://doi.org/10.5391/IJFIS.2007.7.1.049 - Zahran, AM, Abd-Allah, MA, and Ghareeb, A (2010). Several types of double fuzzy irresolute functions. International Journal of Computational Cognition.
*8*, 19-23. - Abbas, SE, and Taha, IM (2020). On upper and lower contra-continuous fuzzy multifunctions. Punjab University Journal of Mathematics.
*47*, 104-117. - Coker, D, and Demirci, M (1996). An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense. Busefal.
*67*, 67-76. - Lee, EP, and Im, YB (2001). Mated fuzzy topological spaces. Journal of the Korean Institute of Intelligent Systems.
*11*, 161-165. - Suganya, V, Gomathisundari, P, and Rajesh, N (2022). Double fuzzy α-continuous multifunctions. International Journal of Fuzzy Logic and Intelligent Systems. Manuscript submitted

E-mail: nrajesh topology@yahoo.co.in

E-mail: suganjo115sai@gmail.com

E-mail: gomathi.rsg@gmail.com

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 100-105

**Published online** March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.100

Copyright © The Korean Institute of Intelligent Systems.

Veeraiyan Suganya^{1}, Pitchayan Gomathi Sundari^{2}, and Neelamegarajan Rajesh^{2}

^{1}Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tamilnadu, India^{2}Department of Mathematics, Rajah Serfoji Government College (affliated to Bharathdasan University), Tamilnadu, India

**Correspondence to:**Neelamegarajan Rajesh (nrajesh_topology@yahoo.co.in)

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, we introduce and investigate the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. Several characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

**Keywords**: Double fuzzy topological space, (*r*, *s*)-fuzzy &alpha,-open set, Double fuzzy &alpha,-irresolute multifunctions

The fuzzy concept has penetrated nearly all branches of mathematics since the concept has been defined by Zadeh [1]. Fuzzy sets have applications in several fields, such as information [2] and control [3]. The theory of fuzzy topological spaces was first defined and developed by Chang [2], and since then various notions of general topology have been generalized to the fuzzy topological spaces defined by Chang. Šostak [4] and Kubiak [5] developed fuzzy topology as an extension of the fuzzy topology introduced by Chang [2]. It has since been developed in several ways. Šostak [6] published a survey article on the areas of development of fuzzy topological spaces. Topologists often refer to the concept of fuzzy topology developed by Chang as “”and the concept of fuzzy topology proposed by Kubiak-Šostak as “”, where

Throughout this paper, let _{0} = (0, 1] and _{1} = [0, 1). The family of all fuzzy sets in ^{X}^{X}^{X}^{Y}^{−1} : ^{Y}^{X}_{f}_{(}_{x}_{)=}_{y}^{−1}(^{X}^{Y}

Let ^{Y}_{F}

Let

(1) normalized if and only if for each

x ∈X , there exists ay _{0}∈Y such thatG (_{F}x, y _{0}) = 1̄.(2) crisp if and only if

G (_{F}x, y _{0}) = 1̄ for eachx ∈X andy ∈Y .

Let

(1) the lower inverse of

μ ∈I is a fuzzy set^{Y}F (^{l}μ ) ∈I defined by^{X}${F}^{l}(\mu )(x)={\displaystyle \underset{y\in Y}{\bigvee}}({G}_{F}(x,y)\bigwedge \mu (y))$ .(2) the upper inverse of

μ ∈I is a fuzzy set^{Y}F (^{u}μ ) ∈I defined by^{X}${F}^{u}(\mu )(x)={\displaystyle \underset{y\in Y}{\bigwedge}}({G}_{F}^{c}(x,y)\vee \mu (y))$ .

Let

(1)

F (λ _{1}) ≤F (λ _{2}) ifλ _{1}≤λ _{2}.(2)

F (^{l}μ _{1}) ≤F (^{l}μ _{2}) andF (^{u}μ _{1}) ≤F (^{u}μ _{2}) ifμ _{1}≤μ _{2}.(3)

F (^{l}μ ) ≤ (^{c}F (^{u}μ )) .^{c}(4)

F (^{u}μ ) ≤ (^{c}F (^{l}μ )) .^{c}(5)

F (F (^{u}μ )) ≤μ ifF is a crisp.(6)

F (^{u}F (λ )) ≥λ ifF is a crisp.

Let

Let

(1)

H ∘F =F (H ).(2) (

H ∘F ) =^{u}F (^{u}H ).^{u}(3) (

H ∘F ) =^{l}F (^{l}H ).^{l}

Let

(1)

$({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})(\lambda )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}(\lambda )$ .(2)

${({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})}^{l}(\mu )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}^{l}(\mu )$ .(3)

${({\displaystyle \underset{i\in \mathrm{\Gamma}}{\cup}}{F}_{i})}^{u}(\mu )={\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{F}_{i}^{u}(\mu )$ .

A fuzzy point _{t}_{0} at _{t}

A double fuzzy topology on ^{★} : ^{X}

(1)

τ (λ ) ≤ 1̄ –τ ^{★}(λ ) for eachλ ∈I .^{X}(2)

τ (λ _{1}∧λ _{2}) ≥τ (λ _{1}) ∧τ (λ _{2}) andτ ^{★}(λ _{1}∧λ _{2}) ≤τ ^{★}(λ _{1}) ∨τ ^{★}(λ _{2}) for eachλ _{1},λ _{2}∈I .^{X}(3)

$\tau ({\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{\lambda}_{i})\ge {\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigwedge}}\tau ({\lambda}_{i})$ and${\tau}^{\u2605}({\displaystyle \underset{i\in \mathrm{\Gamma}}{\bigvee}}{\lambda}_{i})\le {\vee}_{i\in \mathrm{\Gamma}}{\tau}^{\u2605}({\lambda}_{i})$ for eachλ ∈_{i}I ^{X}andi ∈ Γ.

The triplet (^{★}) is called a double fuzzy topological space.

A fuzzy set ^{★} (

Let (^{★}) be a double fuzzy topological space. Then, the double fuzzy closure and double fuzzy interior operators of ^{X}_{τ,τ}_{★} (^{X}^{★}(1̄ – _{τ,τ}_{★} (^{X}^{★}(_{0} and _{1} such that

Let (^{★}) be a double fuzzy topological space. For each ^{X}_{0}, and _{1},

(1)

λ is called (r, s )-fuzzyα -open ifλ ≤I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(λ, r, s ),r, s ),r, s ).(2)

λ is called (r, s )-fuzzyα -closed if 1̄ –λ is (r, s )-fuzzyα -open.(3)

α C _{τ,τ}_{★}(λ, r, s ) = ∧{μ ∈I :^{X}μ ≥λ andμ is (r, s )-fuzzyα -closed} is called (r, s )-fuzzyα -closure ofλ .(4)

α I _{τ,τ}_{★}(λ, r, s ) = ∨{μ ∈I :^{X}μ ≤λ andμ is (r, s )-fuzzyα -open} is called (r, s )-fuzzyα -interior ofλ .

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then,

(1) double fuzzy upper

α -continuous at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for eachμ ∈I ,^{Y}σ (μ ) ≥r , andσ ^{★}(μ ) ≤s , there exists an (r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ∧dom (F ) ≤F (^{u}μ ).(2) double fuzzy lower

α -continuous at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for eachμ ∈I ,^{Y}σ (μ ) ≥r andσ ^{★}(μ ) ≤s , there exists an (r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ≤F (^{l}μ ).(3) double fuzzy upper (lower)

α -continuous if it is fuzzy upper (lower)α -continuous at everyx ∈_{t}dom (F ).

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then,

(1) double fuzzy upper

α -irresolute at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for each (r, s )-fuzzyα -open setμ ∈I , there exists an (^{Y}r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ∧dom (F ) ≤F (^{u}μ ).(2) double fuzzy lower

α -irresolute at a fuzzy pointx ∈_{t}dom (F ) ifx ∈_{t}F (^{u}μ ) for each (r, s )-fuzzyα -open setμ ∈I , there exists an (^{Y}r, s )-fuzzyα -open setλ ∈I and^{X}x ∈_{t}λ such thatλ ≤F (^{l}μ ).(3) double fuzzy upper (lower)

α -irresolute if it is fuzzy upper (lower)α -irresolute at everyx ∈_{t}dom (F ).

If _{t}_{t}^{u}^{Y}^{X}_{t}^{u}

For a fuzzy multifunction ^{★}) → (^{★}), the following statements are equivalent:

(1)

F is double fuzzy lowerα -irresolute,(2)

F (^{l}μ ) is (r, s )-fuzzyα -open for any (r, s )-fuzzyα -open setμ ∈I ,^{Y}(3)

F (^{u}μ ) is (r, s )-fuzzyα -closed for any (r, s )-fuzzyα -closed setμ ∈I ,^{Y}(4)

C _{τ,τ}_{★}(I _{τ,τ}_{★}(C _{τ,τ}_{★}(F (^{u}μ ),r, s ),r, s ),r, s ) ≤F (^{u}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I ,^{Y}(5)

α C _{τ,τ}_{★}(F (^{u}μ ),r, s ) ≤F (^{u}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I .^{Y}

(1) ⇒ (2): Let _{t}^{Y}_{t}^{l}^{X}_{t}^{l}_{t}_{τ,τ}_{★} (_{τ,τ}_{★} (_{τ,τ}_{★} (^{l}^{l}_{τ,τ}_{★} (_{τ,τ}_{★} (_{τ,τ}_{★} (^{l}^{l}

(2) ⇒ (3): Let ^{Y}^{l}^{u}^{u}

(3) ⇒ (4): Let ^{Y}^{u}_{σ,σ}_{★} (

(4) ⇒ (5): Let ^{Y}

(5) ⇒ (3): Let ^{Y}_{τ,τ}_{★} (^{u}^{u}_{σ,σ}_{★} (^{u}^{u}

(3) ⇒ (2): Let ^{Y}^{u}^{l}^{l}

(2) ⇒ (1): Let _{t}^{Y}_{t}^{l}^{l}

We state the following result without proof in view of the above theorem.

^{★}) → (^{★}), the following statements are equivalent:

(1)

F is double fuzzy upperα -irresolute.(2)

F (^{u}μ ) isr -fuzzyα -open for any (r, s )-fuzzyα -open setμ ∈I .^{Y}(3)

F (^{l}μ ) isr -fuzzyα -closed for any (r, s )-fuzzyα -closed setμ ∈I .^{Y}(4)

C _{τ,τ}_{★}(I _{τ,τ}_{★}(C _{τ,τ}_{★}(F (^{l}μ ),r, s ),r, s ),r, s ) ≤F (^{l}α C _{σ,σ}_{★}(μ ,r, s )) for anyμ ∈I .^{Y}(5)

α C _{τ,τ}_{★}(F (^{l}μ ),r, s ) ≤F (^{l}α C _{σ,σ}_{★}(μ, r, s )) for anyμ ∈I .^{Y}

Let ^{★}) → (^{★}) be a fuzzy multifunction. Then, the following can be postulated:

(1)

F as normalized implies that it is double fuzzy upperα -irresolute at a fuzzy pointx if and only if_{t}x ∈_{t}I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(F (^{u}μ ),r, s ),r, s ),r, s ) for (r, s )-fuzzyα -open setsμ ∈I and^{Y}x ∈_{t}F (^{u}μ ).(2)

F is double fuzzy lowerα -irresolute at a fuzzy pointx if and only if_{t}x ∈_{t}I _{τ,τ}_{★}(C _{τ,τ}_{★}(I _{τ,τ}_{★}(F (^{l}μ ),r, s ),r, s ),r, s ) for (r, s )-fuzzyα -open setsμ ∈I and^{Y}x ∈_{t}F (^{l}μ ).

For a fuzzy multifunction ^{★}) → (^{★}), we obtain the following:

(1) Every double fuzzy upper

α -continuous multifunction is double fuzzy upperα -irresolute.(2) Every double fuzzy lower

α -continuous multifunction is double fuzzy lowerα -irresolute.

The following example demonstrates that the converses of the above propositions are not true.

Let _{1}, _{2}}, _{1}, _{2}, _{3}}, and ^{★}) → (^{★}) be fuzzy multifunctions defined by _{F}_{1}, _{1}) = 0.2, _{F}_{1}, _{2}) = 1̄, _{F}_{1}, _{3}) = 0.3, _{F}_{2}, _{1}) = 0.5, _{F}_{2}, _{2}) = 0.3, and _{F}_{2}, _{3}) = 1̄. We define fuzzy topologies ^{★} : ^{X}^{★} : ^{Y}

Then,

Let ^{★})→(^{★}) and ^{★}) → (^{★}) be fuzzy multifunctions. Thus, the following can be concluded:

(1) If

F andH are double fuzzy lowerα -irresolute, then,H ∘F is also double fuzzy lowerα -irresolute.(2) If

F is double fuzzy lowerα -irresolute andH is double fuzzy lowerα -continuous, then,H ∘F is also double fuzzy lowerα -irresolute.

Follows from the respective definitions.

Let {_{i}

Let ^{Y}_{i}^{Y}

A fuzzy multifunction ^{★}) → (^{★}) is called (_{t}_{t}

Let ^{★}) → (^{★}) be a crisp double fuzzy upper

Let ^{X}_{i}_{i}_{t}_{x}_{t} of Γ such that _{t}^{u}_{t}^{u}_{x}_{t}) and _{x}_{t} is (^{u}_{x}_{t}) is (^{u}_{x}_{t}) : ^{u}_{x}_{t}) is (_{t}

- Zadeh, LA (1965). Fuzzy sets. Information and Control.
*8*, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X - Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications.
*24*, 182-190. - Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*61*, 137-142. https://doi.org/10.1016/0165-0114(94)90229-1 - AP Sostak (1985). On a fuzzy topological structure. Proceedings of the 13th Winter School on Abstract Analysis. Palermo, Italy: Circolo Matematico di Palermo, pp. 89-103
- Kubiak, T 1985. On fuzzy topologies. PhD dissertation. Adam Mickiewicz University. Poznan, Poland.
- Sostak, AP (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences.
*78*, 662-701. - Coker, D (1996). An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces. Journal of Fuzzy Mathematics.
*4*, 749-764. - Coker, D (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems.
*88*, 81-89. https://doi.org/10.1016/S0165-0114(96)00076-0 - Mondal, TK, and Samanta, SK (2002). On intuitionistic gradation of openness. Fuzzy Sets and Systems.
*131*, 323-336. https://doi.org/10.1016/S0165-0114(01)00235-4 - Garcia, JG, and Rodabaugh, SE (2005). Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets and Systems.
*156*, 445-484. https://doi.org/10.1016/j.fss.2005.05.023 - Abd-Allah, MA, El-Saady, K, and Ghareeb, A (2009). (r, s)-Fuzzy F-open sets and (r, s)-fuzzy F-closed spaces. Chaos, Solitons & Fractals.
*42*, 649-656. https://doi.org/10.1016/j.chaos.2009.01.031 - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2016). Generalized fuzzy b-closed and generalized ★-fuzzy b-closed sets in double fuzzy topological spaces. Egyptian Journal of Basic and Applied Sciences.
*3*, 61-67. - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2014). Somewhat slightly generalized double fuzzy semicontinuous functions. International Journal of Mathematics and Mathematical Sciences.
*2014*. article no. 756376 - Mohammed, FM, Noorani, MSM, and Ghareeb, A (2017). New notions from (alpha,beta)-generalised fuzzy preopen sets. Gazi University Journal of Science.
*30*, 311-331. - Ghareeb, A (2011). Normality of double fuzzy topological spaces. Applied Mathematics Letters.
*24*, 533-540. https://doi.org/10.1016/j.aml.2010.11.008 - Ghareeb, A (2012). Weak forms of continuity in I-double gradation fuzzy topological spaces. SpringerPlus.
*1*. article no. 19 - El-Saady, K, and Ghareeb, A (2012). Several types of (r, s)-fuzzy compactness defined by an (r, s)-fuzzy regular semiopen sets. Annals of fuzzy Mathematics and Informatics.
*3*, 159-169. - Lee, EP (2004). Semiopen sets on intuitionistic fuzzy topological spaces in Sostak’s sense. Journal of the Korean Institute of Intelligent Systems.
*14*, 234-238. https://doi.org/10.5391/JKIIS.2004.14.2.234 - Lee, SO, and Lee, EP (2005). Fuzzy (r, s)-preopen sets. International Journal of Fuzzy Logic and Intelligent Systems.
*5*, 136-139. https://doi.org/10.5391/IJFIS.2005.5.2.136 - Lee, SO, and Lee, EP (2006). Fuzzy strongly (r, s)-semiopen sets. International Journal of Fuzzy Logic and Intelligent Systems.
*6*, 299-303. https://doi.org/10.5391/IJFIS.2006.6.4.299 - Lee, SJ, and Kim, JT (2007). Fuzzy (r, s)-irresolute maps. International Journal of Fuzzy Logic and Intelligent Systems.
*7*, 49-57. https://doi.org/10.5391/IJFIS.2007.7.1.049 - Zahran, AM, Abd-Allah, MA, and Ghareeb, A (2010). Several types of double fuzzy irresolute functions. International Journal of Computational Cognition.
*8*, 19-23. - Abbas, SE, and Taha, IM (2020). On upper and lower contra-continuous fuzzy multifunctions. Punjab University Journal of Mathematics.
*47*, 104-117. - Coker, D, and Demirci, M (1996). An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense. Busefal.
*67*, 67-76. - Lee, EP, and Im, YB (2001). Mated fuzzy topological spaces. Journal of the Korean Institute of Intelligent Systems.
*11*, 161-165. - Suganya, V, Gomathisundari, P, and Rajesh, N (2022). Double fuzzy α-continuous multifunctions. International Journal of Fuzzy Logic and Intelligent Systems. Manuscript submitted

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