# On FuzzyF∗-Simply Connected Spaces in Fuzzy F∗-Homotopy

Authors: V. Madhuri; Omar Bazighifan; Ali Hasan Ali; A. El-Mesady
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, the notions of fuzzyF∗-simply connected spaces and fuzzy F∗-structure homeomorphisms are introduced, and further fuzzy F∗-structure homeomorphism between fuzzy F∗-path-connected spaces are studied. Also, it is shown that every fuzzy F∗-structure subspace of fuzzy F∗-simply connected space is fuzzy F∗-simply connected subspace. Further, the concepts of fuzzy F∗-contractible spaces and fuzzy F∗-retracts are introduced, and it is proved that every fuzzy F∗-contractible space is fuzzy F∗-simply connected.


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## Body

## 1. Introduction

Homotopy theory is the main part of algebraic topology which studies topological objects up to homotopy equivalence. Homotopy equivalence is a weaker relation than topological equivalence. The homotopy theory is one among the foremost branches of algebraic topology. The thoughts and techniques of homotopy theory have pervaded many components of topology. Several topologists like Massey [1], Munkres [2], and Hatcher [3], introduced and studied concepts of homotopy theory and fundamental groups. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [4]. The fuzzy homotopy theory was introduced by Culvacioglu and Citil [5]. Salleh and Tap [6, 7] introduced the concept of the fundamental group to fuzzy topological spaces based on the definition of fuzzy topology introduced by Chang [8]. Based on Chang’s [8] description of fuzzy topology, Salleh and Tap [6, 7, 9] introduced the concept of basic fundamental group of fuzzy topological spaces. Several recent papers contain extensive investigations and applications about fuzzy topological space and fuzzy homotopy theory; see [10–12].Later on, Guner [13] intensively investigated the concept of fuzzy contractible spaces. Rodabaugh [14, 15] introduced the concept of L-fuzzy retract in 1981. Further fuzzy homotopy theory was intensively developed by Aras [16], Citil and Cuvalcioglu [17], Palmeira and Bedregal [18], etc. and gave many interesting results on fuzzy homotopy theory. In this connection, Madhuri and Amudhambigai [19, 20] introduced the concepts of fuzzy F∗-homotopy and F∗-fundamental group. Also the concept of fuzzy F∗-structure isomorphisms between F∗-fundamental groups are studied in [19, 20]. In this recent work, the concept of fuzzy F∗-structure homeomorphism between fuzzy F∗-path-connected spaces are studied. The notion of fuzzy F∗-simply connected space is introduced, and it is shown that every fuzzy F∗-structure subspace of fuzzy F∗-simply connected space is fuzzy F∗-simply connected. Further, the concepts of fuzzy F∗-contractible spaces and fuzzy F∗-retracts are introduced and it is proved that every fuzzy F∗-contractible space is fuzzy F∗-simply connected. Also in fuzzy F∗-contractible space, each fuzzy F∗-loop based at any fuzzy point is equivalent to the constant fuzzy F∗-loop.


## 2. Preliminaries

In this section, some basic concepts of fuzzy homotopy have been recalled from the previous literature. Also related results and propositions are studied from various research articles. Some definitions and preliminary results are presented in this section.Definition 1.
LetX,τ be a fuzzy topological space. A fuzzy set μ∈IX is called fuzzy irreducible if μ≠0X, and for all fuzzy closed sets γ,δ∈IX with μ≤γ∨δ, it follows that either μ≤γ or μ≤δ.Remark 2.
LetX,τ be a fuzzy topological space. Any λ∈IX is said to be fuzzy irreducible closed if it is both fuzzy irreducible and fuzzy closed.Definition 3.
Suppose thatX,τ be a fuzzy topological space and assume that α∈IX be a fuzzy open set in X,τ. Then the collection F=σ∈IX:αqσ and 1−σ is a fuzzy irreducible closed set in X,τ}. Then the collection F which is finer than the fuzzy topology τ on X is said to be a F-structure on X. A nonempty set X with a F-structure denoted by X,F is said to be fuzzy F-structure space. Each member of F is said to be fuzzy F-structure open set, and the complement of each fuzzy F-structure open set is said to be fuzzy F-structure closed. A F-structure on a nonempty set X together with 0X is said to be fuzzy F∗-structure. Then X,F∗ is called a fuzzy F∗-structure space generated by τ.Definition 4.
Suppose thatX,τ be any topological space and assume that X,F∗ be fuzzy F∗-structure space. Assume also that U⊂X and χU denotes the so-called fuzzy characteristic function of the subset U. Then the fuzzy ξ~-structure given by τ is F∗τ=χU:U∈τ, and the pair X,F∗τ is called a fuzzy ξ~-structure space given by X,τ.Note 5.
LetI be the unit interval. Let ζ be an Euclidean topology on I, and then I,F∗ζ is a fuzzy F∗-structure space introduced by the (usual) topological space I,ζ.Definition 6.
Letπ1X,F∗,xλ and π1X,F∗,xμ′ be any two fuzzy F∗-fundamental groups of X,F∗ at xλ and xμ′, respectively. A function f:π1X1,F1∗,xλ⟶π2Y,F2∗,xμ′ is called a fuzzy F∗-structure homomorphism if fθ∘η=fθ∘fη for every θ,η∈π1X1,F1∗,xλ. Moreover the fuzzy F∗-structure homomorphism is called a fuzzy F∗-structure isomorphism if it is bijective.Proposition 7.
LetX,F∗ be any fuzzy F∗-structure space and let xλ∈FPX. Let η0,η1,θ0,θ1∈YX,F∗,xλ be any fuzzy F∗-loops in X,F∗. If η0≅Pη1 and θ0≅Pθ1, then η1∗θ1≅Pη0∗θ0.Proposition 8.
LetX,F∗ be any fuzzy F∗-structure space. Let α,β,γ∈π1X,F∗,xλ, where xλ is a fuzzy point in X. Then α∘β∘γ=α∘β∘γ.Proposition 9.
LetX,F∗ be any fuzzy F∗-structure space and let I,F∗ζ be any fuzzy F∗-structure space introduced by I,ζ. Also, let e:I,F∗ζ⟶X,F∗ be the fuzzy F∗-path defined by etζ=xλ for each tζ in I,F∗ζ, and xλ is fuzzy point in X. Then α∘e=e∘α=α, for each α∈π1X,F∗,xλ.Proposition 10.
LetX,F∗ be any fuzzy F∗-structure space and xλ be fuzzy point in X. Let α∈π1X,F∗,xλ. Then there exists a α¯∈π1X,F∗,xλ such that α∘α¯=α¯∘α=e.Remark 11.
From Proposition8, Proposition 9, and Proposition 10, it is seen that π1X,F∗,xλ forms a group under an operation (namely, multiplication). It is called F∗-fundamental group of X,F∗ based at xλ.Definition 12. (see [21]).
Letf,g:X⟶Y be fuzzy continuous maps and f≅g. If g is a constant, then f is called a fuzzy nulhomotopic map.Definition 13.
LetA⊂X. A is an L-fuzzy retract of X in X,T (abbreviated F-retract) if there is an F-continuous r:X,T⟶A,TA such that rx=x for each x∈A.Definition 14.
Let1X:X,τ⟶X,τ be an identity mapping. If 1X is fuzzy homotopic to a constant, then X,τ is called a fuzzy contractible space.Proposition 15.
LetX1,I1, X2,I2, and X3,I3 be any three fuzzy F∗-structure spaces. If φ:X1,I1⟶X2,I2 and ϕ:X2,I2⟶X3,I3 are fuzzy F∗-structure continuous functions, then ϕ∘φ:X1,I1⟶X3,I3 is a fuzzy F∗-structure continuous function.Remark 16.
LetX,I be any fuzzy F∗-structure space and let xλ,yμ∈FPX. Let β be any fuzzy F∗-path joining xλ with yμ. Also, let lxλ,lyμ:I,Iζ⟶X,I be any two fuzzy F∗-loops defined such that
(1)lxλtρ=xλforalltρ∈FPIinI,Iζ,lyμtρ=yμforalltρ∈FPIinI,Iζ.
Then(i)
lxλ∘β=lxλ∗β=β, on replacing e by lxλ in the proof of Proposition 9(ii)
β∘lyμ=β∗lyμ=β, on replacing e by lyμ in the proof of Proposition 9(iii)
β∘β¯=β∗β¯=lxλ. where β¯t=β1−t, on replacing e by lxλ in the proof of Proposition 10(iv)
β¯∘β=β¯∗β=lyμ, on replacing e by lyμ in the proof of Proposition 10(v)
Ifl1≅Ll2, then β¯∗l1∗β≅Lβ¯∗l2∗β.
Therefore,lxλ serves as the left identity and lyμ serves as the right identity for any l.Proposition 17.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces. Let ψ,φ:X1,I1⟶X2,I2 be any two fuzzy F∗-structure continuous functions, where Q is fuzzy F∗-structure homotopy between ψ and φ. Let I,Iζ be any fuzzy F∗-structure space introduced by I,ζ. Also if β:I,Iζ⟶X2,I2 is a fuzzy F∗-path joining ψxλ with φxλ defined by βt=Qxλ,t, where t∈I and xλ∈FPX1, then the following Figure 1 of induced fuzzy F∗-structure homomorphisms is commutative.Figure 1


## 3. Properties of FuzzyF∗-Simply Connected Spaces

Throughout this paper, the collection of all fuzzy pointsxt, where 0<t≤1 over X, is denoted by FPX, and the set of all fuzzy points over I is denoted by FPI. In this section, the concepts of fuzzy F∗-simply connected spaces, fuzzy F∗-null-homotopic functions, fuzzy F∗-contractible spaces, and fuzzy F∗-retractions are introduced, and some interesting properties are studied.Definition 18.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces. Let ϕ,φ:X1,I1⟶X2,I2 be any two fuzzy F∗-structure continuous functions and ϕ≅φ. If φ is a constant function, then ϕ is said to be a fuzzy F∗-null-homotopic function.Definition 19.
Any fuzzyF∗-structure space X,I is said to be a fuzzy F∗-simply connected space if it is fuzzy F∗-path-connected and every fuzzy F∗-loop in X,I is a fuzzy F∗-null-homotopic function.
A fuzzyF∗-structure space which is not fuzzy F∗-simply connected is said to be fuzzy F∗-multiply connected.Definition 20.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces. If the bijective function φ:X1,I1⟶X2,I2 and its inverse function are fuzzy F∗-structure continuous functions, then the function φ is said to be a fuzzy F∗-structure homeomorphism.Proposition 21.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces and let φ:X1,I1⟶X2,I2 be a fuzzy F∗-structure homeomorphism. If X1,I1 is fuzzy F∗-path-connected, then X2,I2 is also a fuzzy F∗-path-connected space.Proof.
LetX1,I1 be fuzzy F∗-path-connected and φ:X1,I1⟶X2,I2 be a fuzzy F∗-structure homeomorphism. Let I,Iζ be a fuzzy F∗-structure space introduced by I,ζ. For t1,t2∈I, let xt1,yt2∈FPX be any two fuzzy points. Since X1,I1 is fuzzy F∗-path connected, there exists a fuzzy F∗-path α:I,τζ⟶X1,I1 such that α0=φ−1xt1 and α1=φ−1yt2. Thus, φ∘α:I,τζ⟶X2,I2 is such that
(2)φ∘α0=φα0=φφ−1xt1=xt1,asφisonto.
Also,φ∘α1=φα1=φφ−1yt2=yt2, as φ is onto.
Therefore,φ∘α is a fuzzy F∗-path in X2,I2 joining from xt1 to yt2. Hence, X2,I2 is a fuzzy F∗-path-connected space.Proposition 22.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces and I,Iζ be a fuzzy F∗-structure space introduced by I,ζ. Let φ,ψ:I,Iζ⟶X1,I1 be any two fuzzy F∗-paths joining from xt1 to yt2 such that φ≅Pψ, where xt1,yt2∈FPX1. If ϕ:X1,I1⟶X2,I2 is a fuzzy F∗-structure continuous function, then ϕ∘φ,ϕ∘ψ:I,Iζ⟶X2,I2 are fuzzy F∗-structure continuous functions and ϕ∘φ≅Pϕ∘ψ.Proof.
LetI,Iζ be any fuzzy F∗-structure space introduced by I,ζ. Since ϕ,φ,ψ are fuzzy F∗-structure continuous functions and by Proposition 15, ϕ∘φ, ϕ∘ψ are also fuzzy F∗-structure continuous functions. Furthermore, φ≅Pψ implies that there exists a fuzzy F∗-structure continuous function G:I,Iω×I,Iζ⟶X1,I1 such that
(3)G0,st=xt1andG1,st=yt2,foreachst∈FPIinI,Iζ,Gpt,0=φptandGpt,1=ψpt,foreachpt∈FPIinI,Iω,where xt1,yt2∈FPX1. Now, H:I,Iω×I,Iζ⟶X2,I2 is such that Hpt,st=ϕGpt,st, for each fuzzy point pt∈FPI in I,Iω and st∈FPI in I,Iζ. Since ϕ and G are fuzzy F∗-structure continuous functions, by Proposition 15, ϕ∘G=H is also a fuzzy F∗-structure continuous function. Moreover, H satisfies the following conditions.
(4)H0,st=ϕG0,st=ϕxt1,H1,st=ϕG1,st=ϕyt2,Hpt,0=ϕGpt,0=ϕφpt=ϕ∘φpt,Hpt,1=ϕGpt,1=ϕψpt=ϕ∘ψpt,for each fuzzy point pt∈FPI in I,Iω and st∈FPI in I,Iζ. Hence, ϕ∘φ≅Pϕ∘ψ.Proposition 23.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces. Let ϕ:X1,I1⟶X2,I2 be any fuzzy F∗-structure homeomorphism. Then X1,I1 is fuzzy F∗-simply connected if and only if X2,I2 is fuzzy F∗-simply connected.Proof.
LetI,Iζ be any fuzzy F∗-structure space introduced by I,ζ. Since X1,I1 is fuzzy F∗-simply connected, by Definition 19, X1,I1 is fuzzy F∗-path-connected. Let ϕ:X1,I1⟶X2,I2 be a fuzzy F∗-structure homeomorphism. Then by Proposition 21, X2,I2 is fuzzy F∗-path-connected. Thus, it is enough to prove that every fuzzy F∗-loop in X2,I2 is fuzzy F∗-null-homotopic. Let l:I,τζ⟶X2,I2 be a fuzzy F∗-loop in X2,I2. Since X1,I1 is fuzzy F∗-simply connected, ϕ¯∘l:I,τζ⟶X1,I1 is a fuzzy F∗-loop in X1,I1 which is fuzzy F∗-path-homotopic to some constant fuzzy F∗-loop β:I,τζ⟶X1,I1, i.e., ϕ¯∘l≅Pβ.
Sinceϕ is a fuzzy F∗-structure continuous function and by Proposition 22(5)ϕ∘ϕ¯∘l≅Pϕ∘β,which implies that ϕ∘ϕ¯∘l≅Pϕ∘β, by using associative property, and so IX2∘l≅Pϕ∘β, as IX2 is an identity function.
Thus,l≅Pϕ∘β.
Also,l is fuzzy F∗-path-homotopic to some constant fuzzy F∗-loop ϕ∘β. Thus, every fuzzy F∗-loop l in X2,I2 is fuzzy F∗-null-homotopic. Thus, X2,I2 is fuzzy F∗-simply connected.
The converse part is also proved in the reverse direction ofϕ and ϕ¯.Proposition 24.
LetX,I be a fuzzy F∗-structure space. Then, any fuzzy F∗-path-connected space X,I is fuzzy F∗-simply connected if and only if any two fuzzy F∗-paths in X,I having same endpoints are fuzzy F∗-path-homotopic.Proof.
LetI,Iζ be any fuzzy F∗-structure space introduced by I,ζ. Let α,β:I,Iζ⟶X,I be any two fuzzy F∗-paths in X,I such that
(6)α0=β0=xt1andα1=β1=yt2,where xt1,yt2∈FPX. Thus, α∗β¯ is a fuzzy F∗-loop based at xt1. Since X,I is fuzzy F∗-simply connected, every fuzzy F∗-loop based at xt1 in X,I is fuzzy F∗-null-homotopic, that is
(7)α∗β¯≅Plxt1,which implies that α∗β¯=lxt1, then α∗β¯∗β=lxt1∗β.
Thus,α∗β¯∗β=β, as in (i) of Remark 16, α∗lyt2=β, as in (iv) of Remark 16, α=β,as in (ii) of Remark 3.
Hence,α≅Pβ.
Therefore, any two fuzzyF∗-paths in X,I having same endpoints are fuzzy F∗-path-homotopic.
Conversely, letα,β:I,Iζ⟶X,I be any two fuzzy F∗-paths in X,I such that α≅Pβ. Thus, α≅Pβ, α=β, and α∗lyt2=β, as in (ii) of Remark 16.
Thus,α∗β¯∗β=β, as in (iv) of Remark 16, and thenα∗β¯∗β=lxt1∗β, as in (i) of Remark 16, which implies thatα∗β¯=lxt1.
Thus,α∗β¯≅Plxt1.
Therefore, for every fuzzyF∗-loop based at xt1 in X,I is fuzzy F∗-null-homotopic. Thus, X,I is fuzzy F∗-simply connected.Definition 25.
LetX,I be a fuzzy F∗-structure space and Y⊆X. Let Y,IY be a fuzzy F∗-structure subspace and R:X,I⟶Y,IY be any fuzzy F∗-structure continuous function. Then Y,IY is said to be fuzzy F∗-retract of X,I if there exists a fuzzy F∗-structure continuous function R such that Rxt=xt, for all xt∈FPY. Also, the function R is called as fuzzy F∗-retraction.Proposition 26.
LetX,I be a fuzzy F∗-structure space and Y⊆X. If X,I is fuzzy F∗-simply connected and Y,IY is the fuzzy F∗-retract of X,I, then Y,IY is also a fuzzy F∗-simply connected space.Proof.
LetX,I be fuzzy F∗-simply connected and Y,IY be the fuzzy F∗-retract of X,I. Then R:X,I⟶Y,IY is a fuzzy F∗-retraction function. If xt1,yt2∈FPY are any two fuzzy points, then there is a fuzzy F∗-path α:I,Iζ⟶X,I such that α0=xt1 and α1=yt2. Then R∘α:I,τζ⟶Y,IY is a fuzzy F∗-path in Y,IY such that
(8)R∘α0=Rα0=Rxt1=xt1,R∘α1=Rα1=Ryt2=yt2,where xt1,yt2∈FPY. Thus, R∘α is a fuzzy F∗-path in Y,IY. Therefore, Y,IY is fuzzy F∗-path-connected.
LetJ:Y,IY⟶X,I be the inclusion function. Thus, R∘J:Y,IY⟶Y,IY, that is, R∘J=IY where IY is an identity function on Y,IY. Also, let l:I,τζ⟶Y,IY is a fuzzy F∗-loop in Y,IY at xt∈FPY. Thus, J∘l:I,τζ⟶X,I is a fuzzy F∗-loop in X,I at xt. Since X,I is fuzzy F∗-simply connected, J∘l is fuzzy F∗-path homotopic in X,I to the constant fuzzy F∗-loop C:I,τζ⟶X,I such that Cpt1=st2, for all pt1∈FPI and st2∈FPX and hence J∘l≅PC. Also, by Proposition 22(9)R∘J∘l≅PR∘C,implies that R∘J∘l≅_PR∘C, and so IY∘l≅PR∘C.
Thus,l≅PR∘C and R∘C is a constant fuzzy F∗-loop (i.e.,) R∘Cpt1=RCpt1=Rst2=st2, for all pt1∈FPI. Therefore l is fuzzy F∗-null-homotopic in Y,IY. Hence, Y,IY is a fuzzy F∗-simply connected space.Definition 27.
LetX1,I1 and X2,I2 be any two fuzzy F∗-structure spaces and ϕ:X1,I1⟶X2,I2. Any fuzzy F∗-structure continuous function ϕ is said to be a fuzzy F∗-homotopy equivalence if there exists a fuzzy F∗-structure continuous function φ:X2,I2⟶X1,I1 such that φ∘ϕ is fuzzy F∗-homotopic to IX1 where IX1 is an identity function on X1,I1 and ϕ∘φ is fuzzy F∗-homotopic to IX2 where IX2 is an identity function on X2,I2.
Also, any two fuzzyF∗-structure spaces X1,I1 and X2,I2 are said to be of the same fuzzy F∗-homotopy type if there exists a fuzzy F∗-structure continuous function ϕ:X1,I1⟶X2,I2 which is fuzzy F∗-homotopy equivalence.Definition 28.
LetX,I be a fuzzy F∗-structure space and IX:X,I⟶X,I be an identity function. If IX is a fuzzy F∗-null-homotopic function, then X,I is said to be a fuzzy F∗-contractible space.Proposition 29.
LetX,I be a fuzzy F∗-structure space. If X,I is fuzzy F∗-contractible, then each fuzzy F∗-loop α based at any fuzzy point xt∈FPX is equivalent to the constant fuzzy F∗-loop lxt at xt.Proof.
LetX,I be a fuzzy F∗-contractible space. Then IX:X,I⟶X,I is an identity function and it is fuzzy F∗-homotopic to the constant fuzzy F∗-loop lxt by a fuzzy F∗-homotopy H. Thus there is a fuzzy F∗-loop δ at xt which is defined as δs=Hxt,s, where s∈I. Let IX#:π1X,I,xt⟶π1X,I,xt and lxt#:π1X,I,xt⟶π1X,I,xt be any two induced fuzzy F∗-structure homomorphisms. Also, let γ^δ:π1X,I,xt⟶π1X,I,xt be defined as γ^δα=δ¯∗α∗δ, where α∈π1X,I,xt. Then by Proposition 17, the following Figure 2 shows that γ^δ∘IX#=lxt#.
For eachα∈π1X,I,xt,γ^δ∘IX#α=lxt#α,γ^δIX∘α=lxt∘α,andδ¯∗IX∘α∗δ=lxt, as in (i) of Remark16;δ¯∗α∗δ=lxt,δ¯∘α∘δ=lxt,α∘δ¯∘δ=lxt,andα∘lyt=lxt, as in (iv) of Remark16; andα=lxt, as in (ii) of Remark16,α≅lxt.
Therefore, each fuzzyF∗-loop α is equivalent to the constant fuzzy F∗-loop lxt.Proposition 30.
Every fuzzyF∗-contractible space is fuzzy F∗-simply connected.Proof.
LetX,I be a fuzzy F∗-contractible space and let lxt be the constant fuzzy F∗-loop at xt∈FPX. Since X,I is fuzzy F∗-contractible, IX:X,I⟶X,I is a fuzzy F∗-null-homotopic function by a fuzzy F∗-homotopy H, so H:IX≅lxt. Thus, the fuzzy F∗-homotopy H is defined as:
(10)Hyt,0=yt,Hyt,1=xt,Hxt,s=xt,where xt,yt∈FPX and s∈I. Let α:I,Iζ⟶X,I be a fuzzy F∗-path such that αs=Hyt,s. Thus, α0=Hyt,0=yt and α1=Hyt,1=xt. Thus, α is a fuzzy F∗-path from yt to xt. Similarly, β is also fuzzy F∗-path from xt to zt where zt∈FPX. Therefore, α∗β is also a fuzzy F∗-path from yt to zt. Thus, X,I is fuzzy F∗-path connected.
Letα∈π1X,I,xt and also let us define a fuzzy F∗-homotopy Gs,t=Hαs,t. Thus, Gs,0=Hαs,0=αs and Gs,1=Hαs,1=lxt=xt, where lxt is a constant fuzzy F∗-loop at xt. Thus, G:α≅lxt. Thus, α=lxt. Hence, every fuzzy F∗-loop in X,I is fuzzy F∗-null-homotopic. Therefore, X,I is fuzzy F∗-simply connected. Hence, every fuzzy F∗-contractible space is fuzzy F∗-simply connected.Figure 2


## 4. Conclusion

In this paper, the notions of fuzzyF∗-simply connected spaces and fuzzy F∗-contractible spaces are introduced, and some important characterizations related to fuzzy F∗-homotopy are discussed. Also it is proved that in fuzzy F∗-contractible space, each fuzzy F∗-loop based at any fuzzy point is equivalent to the constant fuzzy F∗-loop. This is just a beginning of studying fuzzy F∗-simply connected spaces. There is a huge scope of further study in extending the results of fuzzy F∗-homotopy to fuzzy F∗-covering spaces. Further using fuzzy compact and fuzzy Lindelof spaces, types of fuzzy F∗-homotopy can be developed.



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