The concept of mixing two topologies to obtain a new topology is not a new concept. In 1958, Alexiewicz and Semadeni [1] introduced and investigated a mixed topology on a set via norm spaces. Thereafter, many mathematicians worldwide investigated mixed topologies and discovered some interesting and applicable results regarding mixed topologies. In this regard, some remarkable results have been reported by Cooper [2], Buck [3], Wiweger [4], and many others. A new era began after the inception of the fuzzy set theory in 1965 by Zadeh [5]. Since the inception of the notion of fuzzy sets and fuzzy logic, fuzziness has been applied to studies in almost all branches of science and technology. The notion of fuzziness has been applied to topology for the first time by Chang [6], and fuzzy topological spaces have been introduced and then investigated extensively. The concept of strong separation and strong countability in fuzzy topological spaces was introduced and investigated by Stadler and de Prada Vicente [7]. In 1995, the concept of mixed fuzzy topological spaces was investigated from different perspectives by Das and Baishya [8]. Recently, in 2012, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology introduced by Das and Baishya [8], by replacing fuzzy points with fuzzy sets. Tripathy and Ray [9] introduced the concept of countability base on a mixed fuzzy topology and proved the existence of three types of countability. However, this concept of mixed fuzzy topology is not a generalization of the concept of a crispy mixed topology (or simply mixed topology) by Wiweger [4], Alexiewicz and Semadeni [1], Cooper [2], and others. Ghanim et. al [10] investigated separation axioms. Recently, W. F. Al-Omeri [11] investigated a mixed b-fuzzy topology using the definition by Das and Baishya [8].

Most recently, Al-Shami and his colleagues [12–21] investigated separation axioms in topological spaces and obtained some important findings. Rashid and Ali [22] introduced separation axioms in a mixed fuzzy topology using the concept of mixed fuzzy topology defined by Das and Baishya [8]. Recently, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology by Das and Baishya [8]. In this study, we redefined and investigated separation axioms in mixed fuzzy topological spaces introduced by Tripathy and Ray [9]. We investigated a few properties of separation axioms that varied slightly, as introduced by Rashid and Ali [22]. We have previously proven some results and constructed a few concrete numerical examples of T₀, T₁, T₂(Hausdorff), and regular and normal mixed fuzzy topological spaces such that the spaces can be visualized more effectively. Herein, we provide the basic definitions in Section 2, although most of the concepts, definitions, and notations used herein are standard ones.

Definition 1 [23]

Let X be a non-empty set, and I be the unit interval [0, 1]. A fuzzy set A in X is characterized by a function μ_A : X → I, where μ_A is known as the membership function of A. μ_A(x) representing the membership grade of x in A. The empty fuzzy set is defined as μ_φ(x) = 0 for all x ∈ X. Additionally, X can be regarded as a fuzzy set in itself, defined μ_X(t) = 1 for all t ∈ X. Furthermore, an ordinary subset A of X can be regarded as a fuzzy set in X if its membership function is regarded as the usual characteristic function of A, i.e., μ_A(x) = 1, for all x ∈ X and μ_A(x) = 0 for all x ∈ X − A. The two fuzzy sets A and B are equal if μ_A = μ_B. A fuzzy set A is contained in a fuzzy set B, written as A ⊆ B, if μ_A ≤ μ_B. The complement of a fuzzy set A in X is a fuzzy set A in X, defined as μ_A^c = 1− μ_A. We write A_c = coA to avoid confusion.

The union and intersection of a set {A_i : i ∈ I} of fuzzy sets in X to be written as ∪i=1Ai and ∩i=1Ai, separately, are defined as follows:

and

Definition 2 [6]

Let I = [0, 1], X be a non-empty set, and I_X be a set of all mappings from X to I, i.e., the class of all fuzzy sets in X.

A fuzzy topology on X is a family τ of member I_X, such that

• 1̄, 0̄ ∈ τ.

• For any finite subset B={Bi}i=1n of members of τ, ∩i=1n{Bi}∈τ.

• For any arbitrary set Δ of members of τ, ∪_B∈Δ {B} ∈ τ.

The pair (X, τ ) is known as a fuzzy topological space (fts), and the members of τ are known as τ-open fuzzy sets.

Definition 3 [24]

The closure of a fuzzy set A in an fts (X, τ ) is defined as the intersection of all closed supersets of A, i.e.,

Definition 4 [23]

A fuzzy set in X is known as a fuzzy point if and only if it assumes the value 0 for all y ∈ X except one, e.g., x ∈ X. If its value at x is α (0 <≤ 1), then this point is denoted by x, and we refer to point x as its support.

Definition 5 [23]

A fuzzy point x_λ is contained in a fuzzy set A or belongs to A, denoted by x_α ∈ A, if and only if α ≤ A(x).

Definition 6 [23]

Two fuzzy sets A and B in X are intersecting if a point x ∈ X exists such that (A ∩ B)(x) ≠ 0. In this case, A and B intersect at x.

Definition 7 [23]

A fuzzy set A in a fts (X, τ ) is known as the neighborhood of a fuzzy point x_λ ∈ X if and only if B ∈ τ exists such that x_λ ∈ B ⊆ A; a neighborhood A is known as an open neighborhood if and only if A is fuzzy open. A family comprising all the neighborhoods of x_λ is known as a system of neighborhoods of x_λ.

Definition 8 [23]

A fuzzy point x_α is quasi-coincident with a fuzzy set A, denoted by x_αqA, if and only if α + A(x) > 1 or α > (A(x))^c.

Definition 9 [23]

A fuzzy set A is quasi-coincident with B and is denoted by AqB if and only if x ∈ X exists such that A(x) + B(x) > 1.

It is clear that if A and B are quasi-coincident at x, then both A(x) and B(x) are non-zero at x; hence, A and B intersect at x.

Definition 10 [23]

A fuzzy set A in an fts (X, τ ) is known as a quasi-neighborhood (Q-neighborhood) of x_λ if and only if ∃ A₁ ∈ τ, such that A₁ ⊆ A and x_λqA₁. A family U_xλ comprising all the Q-neighborhoods of x is known as the Q-neighborhood system of x. The intersection of two Q-neighborhoods of x_λ is a quasi-neighborhood.

Theorem 1 [8]

Let (X, τ₁) and (X, τ₂) be two fuzzy topological spaces. Consider fuzzy sets τ₁(τ₂) = {A ∈ I_X: For any xqA, a τ₂-Q-neighborhood A of x exists such that τ₁-closure Ā_|α ⊆ A}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Definition 11 [10]

A fts (X, τ ) is known as a fuzzy T₀-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V exist in τ such that x_α ∈ U ⊆ co(y_β) or y_β ∈ V ⊆ co(x_α) in τ.

Definition 12 [10]

A fts (X, τ ) is known as a fuzzy T₁-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V exist in τ such that x_α ∈ U and y_β∉ U, and x_α ∉ V and y_β ∈ V.

Definition 13 [10]

A fts (X, τ ) is known as the fuzzy Hausdorff or fuzzy T₂-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V exist such that x_α ∈ U and y_β ∈ V, and U ∩ V = 0̄.

Definition 14 [10]

A fts (X, τ ) is known as fuzzy regular if and only if for any fuzzy point x and a closed fuzzy set B with x ∈ B^c, open fuzzy sets U and V exist in τ such that x_α ∈ U and B ⊆ V, and U ⊆ 1 − V.

Proposition 1 [25]

fts (X, τ ) is known as a fuzzy regular space if and only if for any fuzzy point x_α in I_X and A ∈ τ withα < A(x), B ∈ τ exists such that α < B(x) and B̄ ⊆ A.

Definition 15 [22]

A fts (X, τ ) is known as a fuzzy normal space if and only if for each closed set B and an open fuzzy set U in (X, τ ) where B ⊆ U, V ∈ τ exists such that B ⊆ V ⊆ V ⊆ U.

Lemma 1 [8]

Let τ₁ and τ₂ be two fuzzy topological spaces on set X. If every τ₁-Q-neighbourhood of x is a τ₂-Q-neighbourhood of x for all fuzzy points x, then τ₁ is coarser than τ₂.

Theorem 2 [9]

Let (X, τ₁) and (X, τ₂) be two fuzzy topological spaces. Consider fuzzy sets τ₁(τ₂) = {A ∈ I_X: For any fuzzy set B in X with AqB, a τ₂-open set A₁ exists such that A₁qB and τ₁-closure Ā₁ ⊆ A}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Theorem 3 [8]

Let τ₁ and τ₂ be two fuzzy topological spaces on set X. Subsequently, the mixed fuzzy topology τ₁(τ₂) is coarser than τ₂. This is represented as τ₁(τ₂) ⊆ τ₂.

In this section, we analogously define the basic definition of T₀, T₁, T₂ (or Hausdorff), as well as regular and normal mixed fuzzy topological spaces (X, τ₁(τ₂)), and then provide some results in mixed fuzzy topological spaces. The definitions used herein are as follows:

Definition 16

A mixed fts (X, τ₁(τ₂)) is known as a fuzzy T₀-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V exist in τ₁(τ₂) such that x_α ∈ U ⊆ co(y_β) or y_β ∈ V ⊆ co(x_α) in τ₁(τ₂).

Definition 17

A mixed fts (X, τ₁(τ₂)) is known as a mixed fuzzy T₁-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V in τ₁(τ₂) exist such that x_α ∈ U and y_β ∉ U as well as x_α ∉ V and y_β ∈ V.

Definition 18

A mixed fts (X, τ₁(τ₂)) is known as a fuzzy Hausdorff or fuzzy T₂-space if and only if for any pair of fuzzy points x_α and y_β in I_X with x ≠ y, open fuzzy sets U and V exist in τ₁(τ₂) such that x_α ∈ U and y_β ∈ V, and U ∩ V = 0̄.

Definition 19

A mixed fts (X, τ₁(τ₂)) is known as a fuzzy regular space if and only if for any fuzzy point x_α in I_X and a closed set B in τ₁(τ₂) with x ∈ B^c, open fuzzy sets U and V exist in τ₁(τ₂) such that x_α ∈ U and B ⊆ V, and U ⊆ 1 − V.

Definition 20

A mixed fts (X, τ₁(τ₂)) is known as a fuzzy normal space if and only if for each closed fuzzy set B and an open fuzzy set U in τ₁(τ₂) where B ⊆ U, V ∈ τ₁(τ₂) exists such that B ⊆ V ⊆ V̄ ⊆ U.

Example 1

Let us consider the following example of mixed fuzzy topology, which comprises fuzz T₀, fuzzy T₁ and fuzzy T₂.

Let X = {x, y} and x ≠ y; therefore

and

are fuzzy Hausdorff topological spaces. We constructed τ₁(τ₂) from these topologies. In addition, τ₁(τ₂) ⊆ τ₂. Therefore we verified the belongingness of {(x, 0.3), (y, 0.3)}, {(x, 0.5), (y, 0.5)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x, 0.5), (y, 0)}, {(x, 0), (y, 0.5)}, {(x, 1), (y, 0)}, and {(x, 0), (y, 1)} in τ₁(τ₂). As an example, = {(x, 0.3), (y, 0.3)} ∈ τ₂ for A₁ = {(x, δ): δ > 0.7}; AqA₁ and the τ₁ closure of (A = {(x, 0.3), (y, 0.3)}) = A ⊆ A. Because {(x, 0.7), (y, 0.7)} is an open set in τ₁, { (x, 0.3), (y, 0.3)} is a closed set in τ₁. (Similarly, for A₂ = {(y, δ): δ > 0.7} or A₃ = {(x, γ), (y, δ) | γ, δ ∈ (0.7, 1]}; AqA₂ and AqA₃ are also a τ₁ closure of (A = {(x, 0.3), (y, 0.3)}) = A ⊆ A.)

Therefore A ∈ τ₁(τ₂). However, for B = {(x, 0.5), (y, 0.5)} ∈ τ₂ and A₂ = {(x, δ): δ > 0.5} BqA₂ and the τ₁ closure of (B = {(x, 0.5), (y, 0.5)}) = 1̄ ⊄ B.

Therefore, B ∉ τ₁(τ₂). Similarly, as in the case of A = {(x, 0.3), (y, 0.3)} ∈ τ₂, {(x, 1), (y, 0)} and {(x, 0), (y, 1)} will be members of τ₁(τ₂). Furthermore, τ₁ is the closure of {(x, 1), (y, 0)} = {(x, 1), (y, 0)}, as {(x, 0), (y, 1)} is an open set in τ₁; therefore, the complement {(x, 1), (y, 0)} is closed. In addition, the τ₁ closure of {(x, 0), (y, 1)} = {(x, 0), (y, 1)} as {(x, 1), (y, 0)} is an open set in τ₁.

The complement of {(x, 0.3), (y, 0)} in τ₁ is {(x, 0.7), (y, 1)}, which is an open set such that {(x, 0.3), (y, 0)} is a closed set in τ₁. Therefore, as in the case of A = {(x, 0.3), (y, 0)} ∈ τ₁(τ₂), we have {(x, 0), (y, 0.3)} ∈ τ₁(τ₂).

However, the complement of {(x, 0.5), (y, 0)} in τ₁ is {(x, 0), (y, 0.5)} and not fuzzy open in τ₁; therefore, {(x, 0.5), (y, 0)} is not fuzzy closed, and the τ₁-closure of {(x, 0.5), (y, 0)} ⊄ {(x, 0.5), (y, 0)}.

Similarly, {(x, 0.5), (y, 0)}∉ τ₁(τ₂) and {(x, 0), (y, 0.5)} ∉ τ₁(τ₂).

Therefore, τ₁(τ₂) = {0̄, 1̄, {(x, 0.3), (y, 0.3)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x 1), (y, 0)}, {(x, 0), (y, 1)}}.

Next, we prove that τ₁(τ₂) is fuzzy T₀, fuzzy-T₁, and fuzzy-T₂.

For x_α and y_β in I_X, x_α ∈ {(x, 1), (y, 0)} ⊆ co(y_β) and y_β ∈ {(x, 0), (y, 1)} ⊆ co(x_α).

Additionally, {(x, 1), (y, 0)} ∩ {(x, 0), (y, 1)} = 0̄.

Therefore, τ₁(τ₂) qualifies as fuzzy-T₀, fuzzy-T₁, and fuzzy-T₂.

Example 2

Let X = {x, y}; subsequently, τ₁ = {1̄, 0̄, {(x, 0), (y, 1)}} and τ₂ = {1̄, 0̄, {(x, 1), (y, 0)}, {(x, 0.7), (y, 0.7)}, {(x, 0.7), (y, 0)}, {(x, 1), (y, 0.7)}} are fuzzy topologies. Therefore, τ₁(τ₂) = {1̄, 0̄, {(x, 1), (y, 0)}}, since {(x, 1), (y, 0)} is a closed fuzzy set in τ₁. For any fuzzy set B in X with {(x, 1), (y, 0)}qB, we have a τ₂-open fuzzy set {(x, 1), (y, 0)} such that {(x, 1), (y, 0)}qB and τ₁-closure ({(x,1), (y,0)}¯)⊆{(x,1),(y,0)}.

Clearly, τ₁(τ₂) is mixed fuzzy T₀ but not mixed fuzzy T₁, as x_α ∈ {(x, 1), (y, 0)} ⊆ co(y_β) and ∄ A ∈ τ₁(τ₂) such that y_α ∈ A ⊆ co(x_β).

Next, we consider a mixed fuzzy topological space that is fuzzy T₁ but not fuzzy T₂.

Example 3

Let X = {x, y, z}; subsequently, τ₁ = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}, {(x, 0.7), (y, 0)(z, 0)}, {(x, 0.7), (y, 1)(z, 0)}, {(x, 0.7), (y, 0)(z, 1)}, {(x, 0.7), (y, 1)(z, 1)}} and τ₂ = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1)(y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}} are fuzzy topologies.

Therefore, τ₁(τ₂) = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}}.

Because every open fuzzy set in τ₂ is a closed fuzzy set in τ₁. Therefore, every open fuzzy set in τ₂ is an open fuzzy set in τ₁(τ₂), as shown in Example 2.

Next, we demonstrate that for every pair x_α, y_β, y_α, z_β, or z_α, x_β, open fuzzy sets U and V exist in τ₁(τ₂) such that x_α ∈ U ⊆ co(y_β) and y_β ∈ V ⊆ co(x_α). We will verify this only for the pair x_α, y_β; similarly, other cases can be completed. We have open fuzzy sets {(x, 1), (y, 0)(z, 1)} and {(x, 0), (y, 1)(z, 1)} in τ₁(τ₂) such that x_α ∈ {(x, 1), (y, 0)(z, 1)} ⊆ co(y_β) and y_β ∈ {(x, 0), (y, 1)(z, 1)} ⊆ co(x_α).

Therefore, τ₁(τ₂) is fuzzy T₁.

However, {(x, 1), (y, 0)(z, 1)} ⊄ co({(x, 0), (y, 1)(z, 1)}).

Therefore, τ₁(τ₂) is not a mixed fuzzy T₂ space.

Next, we establish the following theorems:

Theorem 4

If τ₁ ⊆ τ₂ and τ₁ is fuzzy regular, then τ₁ ⊆ τ₁(τ₂).

Theorem 5

If τ₁ ⊆ τ₂ and τ₁ are fuzzy regular, then τ₁(τ₂) is a fuzzy-T₀, fuzzy-T₁, and fuzzy-T₂ (or Hausdorff) space.

Theorem 6

If τ₁(τ₂) is a fuzzy T₀-space, then τ₂ is also a fuzzy T₀-space.

Theorem 7

If τ₁(τ₂) is a fuzzy T₁-space, then τ₂ is also a fuzzy T₁-space.

Theorem 8

If τ₁(τ₂) is a fuzzy T₂-space (or Hausdorff), then τ₂ is also a fuzzy T₂-space.

Theorem 9

Let τ₁ ⊇ τ₂ and τ₂ be fuzzy regular; therefore, τ₁(τ₂) is also fuzzy regular.

Corollary 1

For any non-empty crisp set X, if τ₂ ⊆ τ₁(τ₂), then τ₂ = τ₁(τ₂) because τ₁(τ₂) ⊆ τ₂.

Next, we consider a mixed fuzzy topology comprising fuzzy regular and fuzzy normal.

Example 4

Let X = {x, y} and x ≠ y. Let τ₁ be the fuzzy topology generated by the base and τ₂ be the fuzzy topology generated by the base , where α is a rational number} ∪ {y_β | β ∈ (0, 1] and β is a rational number} ∪ {∅︀}. Therefore, τ₁ and τ₂ are fuzzy regular and fuzzy normal spaces, respectively. Next, we show that the mixed fuzzy topology τ₁(τ₂) is the same as the fuzzy topology τ₂. We know that τ₁(τ₂) ⊆ τ₂. Moreover, for any element in the base, e.g., B = {x_α | α ∈ (0, 1], where α is a rational number}, the fuzzy set C = {(x, 1 − α), (y, 1)} is an open fuzzy set in τ₁ and a complement of B in τ₁. Therefore B is a closed fuzzy set in τ₁, and τ₁-closure of B = B. Therefore, for any fuzzy set A with AqB ∃ τ₂ open fuzzy set B such that AqB and τ₁-closure of B = B. Therefore, B ∈ τ₁(τ₂). Similarly, any base element {y_β | β ∈ (0, 1], where β is a rational number} is also in τ₁(τ₂); this implies τ₂ ⊆ τ₁(τ₂). Therefore, τ₁(τ₂) = τ₂.

Next, we prove the fuzzy normality and fuzzy regularity of τ₁(τ₂).

For the fuzzy regularity, let x be any fuzzy point and B be a closed fuzzy set in τ₁(τ₂) such that

If B(x) = 1 − λ, then 1 − λ is a rational number.

B ⊆ {(x, 1 − λ), (y, 1)} = V is an open fuzzy set in τ₁(τ₂) and x ∈ U = {(x, λ), (y, 0)} ∈ τ₁(τ₂) such that U = {(x, λ), (y, 0)} ⊆ 1 − V = {(x, λ), (y, 0)}. Therefore, in this case, τ₁(τ₂) is fuzzy regular.

Next, if B(x) < 1−, then ∃ is a rational number δ such that B(x) < δ < 1 − λ.

Therefore, B ⊆ {(x, δ), (y, 1)} = V is an open fuzzy set in τ₁(τ₂).

Furthermore,

⇒ {(x, r), (y, 0)} = U is an open set in τ₁(τ₂) such that x ∈ U and V^c = {(x, 1 − δ), (y, 0)}, resulting in U ⊆ V^c; therefore, τ₁(τ₂) is fuzzy regular in both cases.

Next, we prove that τ₁(τ₂) is fuzzy normal.

Let B be any closed fuzzy set and U be an open fuzzy set such that B ⊆ U.

⇒ B(x) ≤ U(x), ∀ x ∈ X and U is an open fuzzy set in τ₁(τ₂). B(x) = 1 − r for some rational number

⇒ r∈(0,1]⇒B(x)≤B(x)+U(x)2≤U(x) and B(x)+U(x)2 is a rational number in (0, 1].

Therefore, V(x)=B(x)+U(x)2 and V(y)=B(y)+U(y)2 defines a fuzzy open set in τ₁(τ₂).

Furthermore, B(x) ≤ V (x) ≤ U(x) ∀ x ∈ X. Similarly, 1−V yields rational values for both x and y; therefore, 1−V is open in τ₁(τ₂) ⇒ V is fuzzy closed as well as ⇒ V (x) = V̄ (x) ∀ x ∈ X.

Therefore, τ₁(τ₂) is a normal mixed fuzzy topological space.

Theorem 10

If τ₁ ⊇ τ₂ and τ₂ are fuzzy normal, then τ₁(τ₂) is also fuzzy normal.

Theorem 11

A fuzzy T₂ mixed fts (X, τ₁(τ₂)) is a mixed fuzzy T₁-space, and a mixed fuzzy T₁ space is a mixed fuzzy T₀-space.

In this study, we defined fuzzy-T₀, fuzzy-T₁, fuzzy-T₂ (or Hausdorff), fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. We established the relationships among fuzzy-T₀, fuzzy-T₁, fuzzy-T₂, fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. Only a few theorems have been established to investigate separation axioms in mixed fuzzy topological spaces. More properties and theorems pertaining to separation axioms will be established in the near future using these established results in the context of mixed fuzzy topological spaces.