Fuzzy sets have many applications in engineering, economics, and clinical science. Since then, numerous authors have contributed to the development of a new mathematical field known as “fuzzy mathematics.” Fuzzy topological space theory is a branch of mathematics. In mathematics, the topology provides the most natural framework for fuzzy units to excel. Chang [1] introduced and developed the concept of a fuzzy topological space. The main problem in fuzzy mathematics is applying ordinary concepts to fuzzy cases. Madhuri and Amudhambigai [2] introduced the concept of fuzzy ℑ^*-structural spaces related to fuzzy topological spaces. The concept of a compact-open topology is crucial in defining the functional spaces in standard topologies. We aim to introduce the concept of a fuzzy ℑ^*-structure compact-open topology and contribute to several theories and effects related to this concept. The concept of the exponential law can be used for the KU algebra [3, 4].

In this study, a new concept of a fuzzy ℑ^*-structure locally compact space and fuzzy ℑ^*-structure evaluation maps are introduced, and a fuzzy ℑ^*-structure covering the map between two fuzzy ℑ^*-structure spaces is studied. In addition, the concept of the fuzzy ℑ^*-structure exponential law and other theories have been proven. Finally, a fuzzy ℑ^*-structure compact-open topology is introduced, and some results of the fuzzy ℑ^*-structure compact-open topology are proven. In this study, we established the pivotal concepts of fuzzy local compactness and fuzzy product topology.

In terms of notation, we use letters such as A, B, C, U, V, and W to represent the fuzzy sets. The set of all fuzzy sets on a nonempty set X is denoted by I^X. Constant fuzzy sets with values 0 and 1 on X are denoted as 0_X and 1_X, respectively. The fuzzy closure, fuzzy interior, and fuzzy complement of A ∈ I^X are denoted as Ã, A^o, and A′, respectively. For future use, we require the following definitions and findings:

Definition 2.1([5])

A function from a nonempty set X to a unit interval I = [0, 1] is called a fuzzy set λ. I^X denotes the entire fuzzy set family.

Definition 2.2([5])

LetX be a set and τ be a family of fuzzy subsets of X. Subsequently, τ is called a fuzzy topology on X if it satisfies the following conditions:

• (i) 0_X, 1_X ∈ τ

• (ii) If λ, μ ∈ τ, then λ ∧ μ ∈ τ

• (iii) If λ_i ∈ τ for all I, then ∨λ_i ∈ τ.

The ordered pair (X, τ) is called a fuzzy topological space (FTS). Moreover, members of τ are considered to be fuzzy open sets, and their complements are to be fuzzy closed sets.

Definition 2.3([2])

Let (X, τ) be the fuzzy topological space. A fuzzy set λ ∈ I^X is called fuzzy-irreducible if λ ≠ 0_X and for all fuzzy closed sets γ and δ ∈ I^X with λ ≤ (γ ∨ δ), it follows that either λ ≤ γ or λ ≤ δ.

Let (X, τ) be the fuzzy topological space. Any α ∈ I^X is considered to be fuzzy irreducible closed if both fuzzy irreducible and fuzzy are closed. The complement of a fuzzy irreducible closed set is called a fuzzy irreducible open set. For two fuzzy sets, we write αqμ such that α is quasi-coincident (q-coincident) with μ; that is, there exists a minimum of one point x ∈ X such that α(x) + μ(x) > 1.

Definition 2.4([2])

Let (X, τ) be a fuzzy topological space and let α ∈ I^X be a fuzzy open set in (X, τ). Subsequently, collection ℑ = {σ ∈ I^X: α q σ and 1 − σ is a fuzzy irreducible closed set in (X, τ)}. Subsequently, the collection ℑ which is finer than the fuzzy topology τ on X is considered an ℑ-structure on X. A nonempty set X with an ℑ-structure, denoted by (X, ℑ) is considered a fuzzy ℑ-structure space. An ℑ-structure on nonempty sets X and 0_X is said to be a fuzzy ℑ^*structure. Subsequently, (X, ℑ^*) is called the fuzzy ℑ^*-structure space generated by τ. Each member of ℑ^* is called a fuzzy ℑ^*-structure open set, and the complement of each fuzzy ℑ^*-structure open set is called a be fuzzy ℑ^*-structure closed.

Example 2.5

Let X = {c, d} and let λ₁, λ₂ ∈ I^X defined as λ₁(c) = 0.3, λ₁(d) = 0.5, λ₂(c) = 0.6 and λ₂(d) = 0.7. Subsequently, τ = {0_X, 1_X, λ₁, λ₂}. Clearly, τ is a fuzzy topology and (X, τ) is a fuzzy topological space. Subsequently, 0_X, 1_X, 1_X − λ₁, 1_X − λ₂ are the possible fuzzy irreducible closed sets in (X, τ). Let α = (0.6, 0.7) ∈ I^X be a fuzzy open set in (X, τ). Subsequently, ℑ∪0_X forms a fuzzy ℑ^*-structure on X. Thus, (X, ℑ^*) is a fuzzy ℑ^*-structural space generated by τ.

Definition 2.6([2])

Let (X, Ix*) and (Y, Iy*) be two fuzzy ℑ^*structure spaces generated by τ_x and τ_y. A function f:(X,Ix*)→(Y,Iy*) is said to be a fuzzy ℑ^*-structure continuous function if for each fuzzy ℑ^*-structure open set λ ∈ I_Y, f⁻¹(λ) is a fuzzy Ix* -structure open set in (X, Ix*).

Definition 2.7([6])

A fuzzy topological space is called a fuzzy Hausdorff space if and only if (iff), for any two distinct fuzzy points p, q of X, there exist disjoint open sets U and V of τ with p ∈ U and q ∈ V.

Definition 2.8([2])

A mapping f:(X,Ix*)→(Y,Iy*) is said to be a fuzzy ℑ^*-structure homeomorphism iff f is fuzzy ℑ^*-structure bijective, fuzzy ℑ^*-structure continuous and fuzzy ℑ^*-structure open.

Definition 2.9

Let (X, ℑ^*) be any fuzzy ℑ^*-structure space generated by τ. Let U be a subset of X and χ_U be the fuzzy characteristic function of U. Subsequently, the fuzzy ℑ^*-structure introduced by τ is ℑ^*(τ) = {χ_U: U ⊆ X} and the pair (X, ℑ^*) is considered to be the fuzzy ℑ^*-structure space introduced by (X, τ).

Let I be a unit interval. Let δ be a Euclidean topology on I then, (I, ℑ^*(ζ)) is a fuzzy ℑ^*-structure space introduced by the (usual) topological space (I, ζ).

Definition 2.10

If f_i: X_i → Y_i, i = 1, 2, Subsequently, f₁ × f₂: X₁ × X₂ → Y₁ × Y₂ is defined as follows: (f₁ × f₂)(x₁, x₂) = (f₁(x₁), f₂(x₂)) for each (x₁, x₂) ∈ X₁ × X₂.

Definition 3.1

Let (X, ℑ^*) be a fuzzy ℑ^*-structured topological space. The family of fuzzy sets A of X is said to be a cover of fuzzy set B in X iff B = ∨A. If each member of the cover is a member of ℑ^*, then the cover is said to be an open cover of B. The fuzzy ℑ^*-structure topological space (X, ℑ^*) is called a fuzzy ℑ^*-structure compact if every fuzzy ℑ^*-structure open cover of I^X contains a finite fuzzy ℑ^*-structure open subcover.

Example 3.2

Let (X, τ) be the unit interval I with ℑ^*-structure topology, and τ = {0_X, 1_X, λ_X, μ_X}. We consider λ_X(x) = {x ∈ I: 0 ≤ x ≤ 0.7} and λ_X(y) = {y ∈ I: 0.7 ≤ y ≤ 1}. Clearly, {λ_X(x), λ_X(y)} are open covers of X. Now define ∀x ∈ I, x ≠ 0, x ≠ 1, let μ_X(x) = 1 and ∀y∈[0,x2]∪[1+x2,1], μ_X(y) = 0. Thus, {μ_X(x), μ_X(y)} is a subcover of the open cover of X. Thus, (X, τ) is a fuzzy ℑ^*-structured compact space.

Definition 3.3

A fuzzy ℑ^*-structure space (X, ℑ^*) is called a fuzzy ℑ^*-structure Hausdorff space if any two distinct fuzzy points, that is, x_t and y_t, exist in fuzzy ℑ^*-structure open sets U and V such that x_t ∈ U, y_t ∈ V U ∧ V = 0_X

Example 3.4

Let X = {x, y} and λ ∈ I_X defined as λ(x) = 0.5 and λ(y) = 0.6. Subsequently, τ = {0_X, 1_X, λ}. Here, τ ∪0_X forms a fuzzy ℑ^*-structure on X: Clearly, (X, ℑ^*) is a fuzzy ℑ^*-structured Hausdorff space generated by τ.

Definition 3.5

Let p:(Xx,Ix*)→(Y,Iy*) be a fuzzy ℑ^*-structure covering map and let f:(X˜,I˜*)→(Y,Iy*) be a fuzzy ℑ^*-structure continuous function. Subsequently, a map f˜:(X˜,I˜*)→(X1,I1*) is said to be a fuzzy ℑ^*-structure lifted on map f if p ∘ f̃ = f.

Definition 3.6

The mapping e: Y^X ×X → Y is defined as follows: using e(f, x_t) = f(x_t) for each fuzzy point x_t, where f ∈ Y^x is the fuzzy ℑ^*-structure evaluation map.

Definition 3.7

The product of two fuzzy sets A and B in a fuzzy ℑ^*-structure space (X, ℑ^*) is defined as (A×B)(x, y) = min(A(x), B(y)) for all (x, y) ∈ X × Y.

Theorem 3.8

A fuzzy ℑ^*-structure compact subspace of a fuzzy ℑ^*-structure Hausdorff space is fuzzy ℑ^*-structure closed.

Theorem 3.9

The following criteria are equivalent in a fuzzy ℑ^*-structured Hausdorff space:

• (a) X is fuzzy ℑ^*-structure locally compact.

• (b) For every fuzzy point x_t ∈ X, there exists a fuzzy ℑ^*-structure open set U in X such that x_t ∈ U and Ū is fuzzy ℑ^*-structure compact.

Theorem 3.10

The fuzzy ℑ^*-structured Hausdorff space X is a fuzzy ℑ^*-structure that is locally compact at fuzzy point x_t in X if and only if there exists a fuzzy ℑ^*-structure open set U containing x_t, there exists a fuzzy ℑ^*-structure open set V such that x_t ∈ V, V̄ is a fuzzy ℑ^*-structure compact, and V̄ ⊆ U.

Let (X, Ix*) and (Y, Iy*) be any two fuzzy ℑ^*-structural spaces generated by τ_x and τ_y. Let

We give this class Y ^X a topology called the fuzzy ℑ^*-structure compact-open topology as follows:

Let κ : {K : I → X : K is fuzzy ℑ*-structure compact in X}

For any K ∈ κ and U ∈ η, let W(K,U) = {ω ∈ Y^X: ω(K) ⊆ U}. The collection

can be as a fuzzy sub base to generate a fuzzy ℑ^*-structure topology on the class Y^X, called the fuzzy ℑ^*-structure compact-open topology. The class Y^X with this topology is called a fuzzy ℑ^*-structure compact-open topological space. Unless otherwise stated, Y^X will constantly have the fuzzy ℑ^*-structured compact-open topology.

We now consider the fuzzy ℑ^*-structure product topological space Y^X ×C and define a fuzzy ℑ^*-structure continuous map from Y^X × X into Y.

Definition 4.1

A fuzzy ℑ^*-structure evaluation map is a mapping f′: Z^W ×W → Z defined by f′ (f, λ_t) = f(λ_t) for each fuzzy point λ_t ∈ W and f ∈ Z^W.

Theorem 4.2

Let (W,ℑ^*) be a fuzzy ℑ^*-structure in locally compact Hausdorff space. Susbequently, the fuzzy ℑ^*-structure evaluation map f′: Z^W × W → Z is fuzzy ℑ^*-structure continuous.

Definition 4.3

(W,Iw*), (Z,Iz*), (L,Il*) are any of the three ℑ^*-structures generated by τ_x, τ_z and τ_l. Let f:(L,Il*)× (W,Iw*)→ (Z,Iz*) denote the function. For fuzzy point wt∈(W,Iw*) and lr′∈(L,Il*) the induced map f̂: W → Z^l′ is defined by (f(wt))(lr′)=f(lr′,wt) for fuzzy points. For the function f̂: W → Z^l′ a comparable function f can be found using the same rule.

f̂’s continuity can be described in terms of f’s continuity, and vice versa. For this purpose, the following outcomes are required.

Theorem 4.4

Consider two fuzzy ℑ^*-structures, W and Z, with Z fuzzy ℑ^*-structure compact. Let O be a fuzzy ℑ^*-structure open set in the fuzzy ℑ^*-structure product space and w_t be any fuzzy point in W. w_t × Z is contained in W × Z, Subsequently, there exists a fuzzy ℑ^*-structure neighborhood S of w_t in W such that {w_t} × Z ⊆ S × Z ⊆ O.

Theorem 4.5

Let (X,Ix*),(Y,Iy*) be an arbitrary fuzzy ℑ^*-structure topological space and Z be a fuzzy ℑ^*-structure locally compact Hausdorff space. Subsequently, a map α × X → Y is fuzzy ℑ^*-structure continuous iff g: X → Y^Z is a fuzzy ℑ^*-structure, where g is defined by the rule g(λ_t)(z_s) = g(z_s, λ_t).

We explored some of the features of the exponential law using induced maps.

Theorem 5.1

Let (X, Ix*) and (Z, Iz*) be the fuzzy ℑ^*-structured Hausdorff spaces generated by τ_x and τ_y respectively, which are locally fuzzy and compact. The function e: Y ^Z×X → (Y ^Z)^X defined by e(α) = g, that is, e(α) = α̂(e(α)(λ_t)(z_u) = α(z_u, λ_t) = (α̂(λ_t))(z_u) for all α: Z × X → Y is a fuzzy ℑ^*-structure homeomorphism for every fuzzy ℑ^*-structure topological space Y.

Definition 5.2

The fuzzy ℑ^*-structure map e in Theorem 5.1 is known as the fuzzy ℑ^*-structure exponential law.

Corollary 5.3

Let X, Y, Z be the locally fuzzy ℑ^*-structured compact Hausdorff spaces. Subsequently, the map u: Y^X × Z^Y → Z^X defined by u(f, g) = g ∘ f is fuzzy ℑ^*-structure continuous.

In this study, the concepts of fuzzy ℑ^*-structure compactopen topology and fuzzy ℑ^*-structure exponential law were introduced, along with some basic theories. This work is foundational for further research on fuzzy ℑ^*-structured compact-open topologies. We aim to build the concept of fuzzy higher homotopy groups and a fuzzy universal covering space using this concept. Furthermore, this topic can be expanded to fuzzy category theory in the future.