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ψ*closed sets in fuzzy topological spaces
Journal of the Egyptian Mathematical Society volume 28, Article number: 38 (2020)
Abstract
In this paper, we introduce a new class of fuzzy sets, namely, fuzzy ψ*closed sets for fuzzy topological spaces, and some of their properties have been proved. Further, we introduce fuzzy ψ*continuous, fuzzy ψ*irresolute functions, and fuzzy ψ*closed (open) functions, as applications of these fuzzy sets, fuzzy T_{1/5}spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)spaces, and fuzzy ^{ψ*}T_{1/5}spaces.
Introduction
Zadeh [1] introduced the fundamental concept of fuzzy sets and fuzzy set operations in 1965. Fuzzy topology was introduced by Chang [2] in 1965. Subsequently, many researchers have worked on various basic concepts from general topology using fuzzy sets and developed the theory of fuzzy topological spaces [3,4,5,6,7]. Muthukumaraswamy and Devi [8] introduced fuzzy generalized α–closed and fuzzy α–generalized closed (briefly fgαclosed and fαgclosed) sets in fuzzy topological space in 2004. Abd Allah and Nawar [9] introduced and studied ψ*closed sets in topological space in 2014. In this paper, we introduced another new notion of fuzzy generalized closed set called fuzzy ψ*closed sets, which is properly placed in between the class of fuzzy αclosed sets and the class of fuzzy generalized αclosed sets. The structure of the rest of this paper is as follows. The “Preliminaries” section introduces the necessary definitions of fuzzy αclosed sets and fuzzy generalized αclosed sets. In the “Fuzzy ψ*closed sets in fts” section, we introduce the definition of fuzzy ψ*closed sets in fuzzy topological spaces and proved some of their properties. In the “Fuzzy ψ*continuous and fuzzy ψ*irresolute functions in fts” section, we identify the concept of fuzzy ψ*continuous and fuzzy ψ*irresolute functions and fuzzy ψ*closed (open) functions and introducing some of their properties. Further, new classes of spaces, namely, fuzzy T_{1/5}spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)spaces, and fuzzy ^{ψ*}T_{1/5}spaces, are introduced in the “Applications of Fψ*closed sets” section.
Preliminaries
Throughout this paper, (G, τ) and (H, σ) (or simply, G and H) always mean fuzzy topological spaces. The members of τ are called fuzzy open sets, and their complements are fuzzy closed sets. And φ : (G, τ) → (H, σ) (or simply, φ: G → H) denotes a mapping ϕ from fts G to fts H.
For a fuzzy set D of (G, τ), fuzzy closure and fuzzy interior of D denoted by cl(D) and int(D), respectively and are defined by cl(D) = ∧{E : E is fuzzy closed set of G, E ≥ D, 1 − E ∈ τ} and int(D) = v{S : S is fuzzy open set of G, S ≤ D, S ∈τ} [10].
Definition 2.1 A fuzzy set D of a fts G is called fuzzy αopen (briefly, Fαopen) if D ≤ int(cl(int(D))) and a fuzzy αclosed (briefly, Fαclosed) if D ≥ cl(int(cl(D))) [4]; the intersection of all fuzzy αclosed sets of (G, τ) containing D is called fuzzy αclosure of a fuzzy subset D of G and is denoted by αcl(D).
Definition 2.2 A fuzzy set D of a fts G is called fuzzy generalized αclosed (briefly, Fgαclosed) [8] if αcl(D) ≤ U whenever D ≤ U and U is fuzzy αopen in (G, τ). The complement of Fgαclosed set is called Fgαopen set.
Definition 2.3 Let (G, τ) and (H, σ) be two fuzzy topological spaces. A function ϕ : (G, τ) → (H, σ) is called as follows:

(i)
Fαcontinuous [10] if ϕ^{−1}(V) is Fαclosed in G, for each V ∈ FC (H);

(ii)
Fgαcontinuous [8] if ϕ^{−1}(V) is Fgαclosed in G, for each V ∈ FC (H);

(iii)
Firresolute [11] if ϕ^{−1}(V) is Fclosed in G, for each V ∈ FC (H).
Definition 2.4 A function φ : (G, τ) → (H, σ) is said to be fuzzyopen (fuzzyclosed) [2] if the image of every fuzzy open (fuzzyclosed) set in G is fuzzyopen (fuzzyclosed) set in H.
Fuzzy ψ*closed sets in fts
In this section, we introduce fuzzy ψ*closed sets in fuzzy topological space and discuss some of its characterizations and relationships with other notions.
Definition 3.1 A fuzzy set D in (G, τ) is called fuzzy ψ*closed (Fψ*closed) if αcl(D) ≤ U whenever D ≤ U and U is Fgαopen in (G, τ). The complement of Fψ*closed set is called Fψ*open set.
The class of fuzzy ψ*closed sets of fts (G, τ) is denoted by Fψ*C(G).
Proposition 3.1 Every fuzzy αclosed set is fuzzy ψ*closed.
Proof Let D be a Fαclosed set in (G, τ), and since every Fαclosed set is Fgαclosed. Then, αcl(D) ≤ U whenever D ≤ U and U is Fαopen in (G, τ), and since every Fαopen set is Fgαopen. So, αcl(D) ≤ U whenever D ≤ U and U is Fgαopen in (G, τ). Thus, D is Fψ*closed.
The converse of Proposition 3.1 needs not be true as seen from the following example.
Example 3.1 Let G = {a, b, c} with fuzzy topology τ = {0, 1, {a_{0.5}, b_{0.2}, c_{0.7}}, {a_{0.7}, b_{0.8}, c_{0.3}}, {a_{0.5}, b_{0.2}, c_{0.3}}, {a_{0.7}, b_{0.8}, c_{0.7}}}. The fuzzy subset D = {a_{0.4}, b_{0.8}, c_{0.7}} is Fψ*closed set in (G, τ) but not Fαclosed set since cl(int(cl(D))) = {a_{0.5}, b_{0.8}, c_{0.7}}.
Proposition 3.2 Every fuzzy ψ*closed set is fuzzy gαclosed set.
Proof Follows from the fact that every Fαopen set is Fgαopen.
The converse of Proposition 3,2 needs not be true as seen from the following example.
Example 3.2 In Example 3.1, the fuzzy subset D = {a_{0.5}, b_{0.3}, c_{0.7}} is Fgαclosed set in (G, τ) but not Fψ*closed set.
Proposition 3.3 If D and E are Fψ*closed sets in (G, τ), then D∪ E is also Fψ*closed set in (G, τ).
Proof If D v E ≤ U and U are Fgαopen, then D ≤ and E ≤ U. Since D and E are Fψ*closed, αcl(D) ≤ U and αcl(E) ≤ U, and hence αcl(D v E) = αd(D) v αd(E) ≤ U. Thus, D v E is Fψ*closed set in (G, τ).
Proposition 3.4 If D is Fgαopen set and fuzzy ψ*closed set in (G, τ), then D is fuzzy αclosed set in (G, τ).
Proof Since D ≤ D and D is Fgαopen set and Fψ*closed, then αcl(D) ≤ D. Since D ≤ αcl(D), then D = αcl(D), and thus D is Fαclosed set in (G, τ).
Proposition 3.5 Every fuzzy ψ*open set is fuzzy gαopen.
Proof Let D ∈ Fψ*O(G). Then, 1 – D ∈ Fψ*C(G) and hence Fgαclosed set in (G, τ) by Proposition 3.2. This implies that D is Fgαopen set in (G, τ). Hence, every Fψ*open set in G is Fgαopen set in G.
Proposition 3.6 If D is Fψ*closed set in (G, τ) and D ≤ E ≤ αcl(A), then E is Fψ*closed set of (G, τ).
Proof Let U be a Fgαopen subset of (G, τ) such that E ≤ U. Then, D ≤ U and since D ∈ Fψ*C(G), then αcl(D) ≤ U. Now, αcl(E) ≤ αcl(D) ≤ U. Then, E ∈ Fψ*C(G).
Corollary 3.1 If D is Fψ*open set in (G, τ) and αint(D) ≤ E ≤ D, then E is Fψ*open set.
Proof Let D ∈ Fψ*O(G), and αint(D) ≤ E ≤ D. Then, 1 – D ∈ Fψ*C(G), and 1 – D ≤ 1 – E ≤ αcl(1 – D). By Proposition 3.6, 1 – B ∈ Fψ*C(G). Hence, E ∈ Fψ*O(G).
Definition 3.2 For any fuzzy set D in a fts G, we have the fuzzy ψ*interior of D (briefly ψ*int(D)) is the union of all fuzzy ψ*open sets of G contained in D. That is, ψ* − int (D) = v { E : E ≤ D, E is Fψ* − open in G }.
Definition 3.3 Let (G, τ) be a fuzzy topological space. Then, for a fuzzy subset D of G, the fuzzy ψ*closure of D (briefly ψ*cl(D)) is the intersection of all fuzzy ψ*closed sets of G containing D. That is, ψ* − cl(D) = ∧ {E : E ≥ D, E is fuzzy ψ* − closed in G }.
Proposition 3.7 For any fuzzy sets D and B in a fts G, we have as follows:
Proof (i) Follows from Definition 3.3.
(ii) Let D ∈ Fψ*O(G). Then, D ≤ ψ* − int (D). By using (i), we get D = ψ* − int (D). Conversely, assume that D = ψ* − int (D). By using Definition 3.3, D ∈ Fψ*O(G).
(iii) By using (ii), we get ψ* − int (ψ* − int (D)) = ψ* − int (D).
(iv) Since D ≤ E by using (i), ψ* − int (D) ≤ D ≤ E. That is, ψ* − int (D) ≤ E. By (iii), ψ* − int (ψ* − int (D)) ≤ ψ* − int (E). Thus, ψ* − int (D) ≤ ψ* − int (E).
Proposition 3.8 For any fuzzy sets D and E in a fts G, we have as follows:
Proof (i) Since D ∧ E ≤ D and D ∧ E ≤ E, by using Proposition 3.7 (iv), we get ψ* − int (D ∧ E) ≤ ψ* − int (D) and ψ* − int (D ∧ E) ≤ ψ* − int (E). Thus,
By using Proposition 3.7 (i), we have ψ* − int (D) ≤ D and ψ* − int (E) ≤ E. This implies that ψ* − int (D) ∧ ψ* − int (E) ≤ D ∧ E. Now applying Proposition 3.7 (iv), we get ψ* − int (ψ* − int (D) ∧ ψ* − int (E)) ≤ ψ* − int (D ∧ E). By (1), ψ* − int (ψ* − int (D) ∧ ψ* − int (ψ* − int (E)) ≤ ψ* − int (D ∧ E). By using Proposition 3.7 (iii),
Forms (1) and (2), ψ* − int (D ∧ E) = ψ* − int (D) ∧ ψ* − int (E).
(ii) Since D ≤ D v E and E ≤ D v E, by using Proposition 3.7 (iv), we have ψ* − int (D) ≤ ψ* − int (D v E) and ψ* − int (E) ≤ ψ* − int (D v E). Thus, ψ* − int (D) v ψ* − int (E) ≤ ψ* − int (D v E).
The equality in Proposition 3.8 (ii) need not be hold as seen from the following example.
Example 3.3 In Example 3.1, consider D = {a_{0.4}, b_{0.8}, c_{0.7}}, and E = {a_{0.6}, b_{0.8}, c_{0.5}}. Then, ψ* − int (D) = 0, and ψ* − int (E) = {a_{0.6}, b_{0.2}, c_{0.3}}. That implies ψ* − int (D) v ψ* − int (E) = {a_{0.6}, b_{0.2}, c_{0.3}}. Now, D v E = {a_{0.6}, b_{0.8}, c_{0.7}}; it follows that ψ* − int (D v E) = {a_{0.6}, b_{0.2}, c_{0.7}}. Then, ψ* − int (D v E) ≠ ψ* − int (D) v ψ* − int (E).
Proposition 3.9 For any fuzzy set D in a fts G, we have as follows:
Proof (i) By using Definition 3.3, ψ* − int (D) = v { E : E ≤ D, E ∈ Fψ*O(G)}. Taking complement on both sides, we get as follows:
Replacing E^{c} by C, we get
(ψ* int(D))^{c} = ∧ {C : C ≥ D^{c}, C is Fψ* ‐ closed in G }. By Definition 3.4, (ψ* ‐ int(D))^{c} = ψ* ‐ cl(D^{c}).
(ii) By using (i), (ψ* ‐ int(D^{c}))^{c} = ψ* ‐ cl(D^{c})^{c} = ψ* ‐ cl(D). Taking complement on both sides, we get ψ* ‐ int(D^{c}) = (ψ* ‐ cl(D))^{c}.
Proposition 3.10 Let D be a fuzzy set in a fts G. Then, D ∈ Fψ*C(G) if and only if D^{c} is Fψ*open.
Proposition 3.11
For any fuzzy sets D and E in a fts G, we have as follows:
Proof (i) Follows from Definition 3.4.
(ii) Let D ∈ Fψ*C(G). By using Proposition 3.10, D^{c} ∈ Fψ*O(G). By using Proposition 3.9 (ii), ψ* ‐ int(D^{c}) = D^{c} ⇔ (ψ* ‐ cl(D))^{c} = D^{c} ⇔ ψ* ‐ cl(D) = D.
(iii) By using (ii), we get ψ* − cl(ψ* − cl(D)) = ψ* − cl(D).
(iv) If D ∧ E ≤ D and D ∧ E ≤ E By using Proposition 3.7 (iv), ψ* ‐ int(E^{c}) ≤ ψ* ‐ int(D^{c}). Taking complement on both sides, we get (ψ* ‐ int(E^{c}))^{c} ≥ (ψ* ‐ int(D^{c}))^{c}. By using Proposition 3.9 (ii), ψ* − cl(E) ≥ ψ* − cl(D).
Proposition 3.12
Let D be a fuzzy set in a fts G. Then, int(D) ≤ α − int (D) ≤ ψ* − int (D) ≤ D ≤ ψ* − cl(D) ≤ α − cl(D ) ≤ cl(D).
Proof It follows from the definition of corresponding operators.
Proposition 3.13 For any fuzzy sets D and E in a fts G, we have as follows:
Proof (i) Since ψ* ‐ cl(D ∨ E) = ψ* ‐ cl((D ∨ E)^{c})^{c}, by using Proposition 3.9 (i), we have ψ* ‐ cl(D ∨ E) = (ψ* ‐ int(D ∨ E)^{c})^{c} = (ψ* ‐ int(D^{c} ∧ E^{c}))^{c}. By using Proposition 3.8 (i), we have ψ* ‐ cl(D ∨ E) = (ψ* ‐ int(D^{c}) ∧ ψ* ‐ int(E^{c}))^{c} = (ψ* ‐ int(D^{c}))^{c} ∨ (ψ ∗ ‐ int(E^{c}))^{c}.
By using Proposition 3.9 (i), we have ψ* ‐ cl(D ∨ E) = ψ* ‐ cl(D^{c})^{c} ∨ ψ* ‐ cl(E^{c})^{c} = ψ* ‐ cl(D) ∨ ψ* ‐ cl(E).
(ii) Since D ∧ E ≤ D and D ∧ E ≤ E, by using Proposition 3.11 (iv), we have ψ* − cl(D ∧ E) ≤ ψ* − cl(D) and ψ* − cl(D ∧ E) ≤ ψ* − cl(E). This implies that ψ* − cl(D ∧ E) ≤ ψ* − cl(D) ∧ ψ* − cl(E).
Proposition 3.14 For any fuzzy sets D and E in a fts G, we have as follows:
Proof (i) By Proposition 3.11 (i), D ≤ ψ* − cl(A) . Again, using Proposition 3.7 (i), ψ* − int (D) ≤ D. Then, ψ* − cl(ψ* − int (D)) ≤ ψ* − cl(D) .
Then, we have D v ψ* − cl(ψ* − int (D)) ≤ ψ* − cl(D).
(ii) By Proposition 3.7 (i), ψ* − int (D) ≤ D. Again, using Proposition 3.11(i), D ≤ ψ* − cl(D). Then, ψ* − int (D) ≤ ψ* − int (ψ* − cl(D)). Then, we have ψ* − int (D) ≤ D v ψ* − int (ψ* − cl(D)).
(iii) By Proposition 3.12, ψ* − cl(D) ≤ cl(D). We get int(ψ* − cl(D)) ≤ int (cl(D)).
(iv) By (i), ψ* − cl(D) ≥ D v ψ* − cl(ψ* − int (D)). Then, we have int(ψ* − cl(D)) ≥ int (D v ψ* − cl(ψ* − int (D))). Since int(D v E) ≥ int (D) v int (E), int (ψ* − cl(D)) ≥ int (D) v int (ψ* − cl(ψ* − int (D))) ≥ int (ψ* − cl(ψ* − int (D))).
Fuzzy ψ*continuous and fuzzy ψ*irresolute functions in FTS
As application of fuzzy ψ*closed set, we identify some types of fuzzy functions and introducing some of their properties.
Definition 4.1 A function φ : (G, τ) → (H, σ) is said to be fuzzy ψ*continuous (Fψ*continuous) if ϕ^{−1}(V) is Fψ*closed in G, for each fuzzy closed set V in H.
Proposition 4.1 Every Fαcontinuous function is Fψ*continuous.
Proof Let V ∈ FC (H). Since φ is Fαcontinuous, then ϕ^{−1}(V) is Fαclosed in G. Since every Fαclosed set is Fψ*closed set, then ϕ^{−1}(V) ∈ Fψ*C(G). Thus, φ is Fψ*continuous.
The converse of Proposition 4.1 need not be true as seen from the following example.
Example 4.1 Suppose that G = {a, b, c} with fuzzy topology τ = {0, 1, {a_{0.5}, b_{0.2}, c_{0.7}}, {a_{0.7}, b_{0.8}, c_{0.3}}, {a_{0.5}, b_{0.2}, c_{0.3}}, {a_{0.7}, b_{0.8}, c_{0.7}} and H = {x, y, z} with fuzzy topology σ = {0, 1, {x_{0.8}, y_{0.2}, z_{0.3}}}. Let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fψ*continuous function, but it is not a Fαcontinuous function, since V = {x_{0.2}, y_{0.8}, z_{0.7}} ∈ FC(H) but ϕ^{−1}(V) ∉ FαC(G).
Proposition 4.2 Every Fψ*continuous function is Fgαcontinuous.
Proof Let V ∈ FC (H). Since φ is Fψ*continuous, then ϕ^{−1}(V) ∈ Fψ*C(G). By Proposition 3.1, every Fψ*closed set is Fgαclosed set; then, ϕ^{−1}(V) is Fgαclosed. Thus, φ is Fgαcontinuous.
The converse of Proposition 4.2 need not be true as seen from the following example.
Example 4.2 Suppose that G = {a, b, c} with fuzzy topology τ = {0, 1, {a_{0.5}, b_{0.2}, c_{0.7}}, {a_{0.7}, b_{0.8}, c_{0.3}}, {a_{0.5}, b_{0.2}, c_{0.3}}, {a_{0.7}, b_{0.8}, c_{0.7}} and H = {x, y, z} with fuzzy topology σ = {0, 1, {x_{0.5}, y_{0.6}, z_{0.3}}}. Let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fgαcontinuous function, but it is not a Fψ*continuous function, since V = {x_{0.5}, y_{0.4}, z_{0.7}} ∈ FC(Y) but ϕ^{−1}(V) ∉ F ψ*C(X).
Definition 4.2 A function φ : (G, τ) → (H, σ) is said to be Fψ*irresolute (Fψ*irresolute) if ϕ^{−1}(V) ∈ Fψ*C(G), for each Fψ*closed set V in H.
Proposition 4.3 Every Fψ*irresolute function is Fψ*continuous.
Proof It follows from the definitions.
The converse of Proposition 4.3 need not be true as seen from the following example.
Example 4.3 In the Example 4.1, Let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fψ*continuous function, but it is not a Fψ*irresolute function, since V = {x_{0.2}, y_{0.7}, z_{0.4}} ∈ Fψ*C(H) but ϕ^{−1}(V) ∉ F ψ*C(G).
Proposition 4.4 Let φ : G → H and γ : H → W be any two functions. Then, as follows:
(i) γ o φ is Fψ*continuous if g is fuzzy continuous, and φ is Fψ*continuous.
(ii) γ o φ is Fψ*irresolute if both φ and g are Fψ*irresolute.
(iii) γ o φ is Fψ*continuous if g is Fψ*continuous, and φ is Fψ*irresolute.
Proof Let V ∈ FC(W). Since γ is fuzzy continuous, then γ^{−1}(V) ∈ FC(H). Since φ is Fψ*continuous, then we have ϕ^{−1}(γ^{−1}(V)) ∈ Fψ*C(G). Consequently, γ o φ is Fψ*continuous.
(ii)  (iii) By similarity.
Applications of Fψ*closed sets
As applications of Fψ*closed sets, three fuzzy spaces, namely, fuzzy T_{1/5}spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)spaces, and fuzzy ^{ψ*}T_{1/5}spaces are introduced.
We introduce the following definitions.
Definition 5.1 A fuzzy topological space (G, τ) is called as follows:
(i) Fuzzy T_{1/5}space if every Fgαclosed set in G is a Fαclosed set in G.
(ii) Fuzzy \( {T}_{1/5}^{\psi \ast } \)space if every Fψ*closed set in G is a Fαclosed set in G.
(iii) Fuzzy ^{ψ*}T_{1/5}space if every Fgαclosed set in G is a Fψ*closed set in G.
Proposition 5.1 If φ : G → H is Fψ*continuous and G is fuzzy \( {T}_{1/5}^{\psi \ast } \)space; then φ is Fαcontinuous.
Proof Let V ∈ FC (H); since f is Fψ*continuous, then ϕ^{−1}(V) ∈ Fψ*C(G). Since G is F\( {T}_{1/5}^{\psi \ast } \)space, then ϕ^{−1}(V) is Fαclosed set in G. Thus, φ is Fαcontinuous.
Proposition 5.2 If φ : G → H is Fψ*irresolute and G is fuzzy \( {T}_{1/5}^{\psi \ast } \)space, then φ is Fαcontinuous.
ProofBy Theorem 5.1.
Proposition 5.3 If φ : G → H is Fgαcontinuous and G is fuzzy ^{ψ*}T_{1/5}space, then φ is Fψ*continuous.
Proof Let V ∈ FC (H); since φ is Fgαcontinuous, then ϕ^{−1}(V) is Fgαclosed set in G. Since G is F^{ψ*}T_{1/5}space, then ϕ^{−1}(V) ∈ Fψ*C(G). Thus, φ is Fψ*continuous.
Proposition 5.4 Let φ : G → H be onto Fψ*irresolute and Fαclosed. If G is fuzzy \( {T}_{1/5}^{\psi \ast } \)space, then H is also a fuzzy \( {T}_{1/5}^{\psi \ast } \)space.
Proof Let V ∈ Fψ*C(H); since f is Fψ*irresolute, then ϕ^{−1}(V) ∈ Fψ*C(G). Since G is F\( {T}_{1/5}^{\psi \ast } \)space, then ϕ^{−1}(V) is Fαclosed set in G. Since φ is Fαclosed and onto, then we have V is Fαclosed. Therefore, H is also a F\( {T}_{1/5}^{\psi \ast } \)space.
Proposition 5.5 Let G, H, and W be ftss, and φ : G → H, γ : H → W and γ o φ : G → W be functions, then if φ is Fαirresolute function and γ is Fψ*continuous function, such that H is fuzzy \( {T}_{1/5}^{\psi \ast } \)space. Then, γ o φ is Fαcontinuous function.
Proof Let U ∈ FC (W); since γ is Fψ*continuous, then γ^{−1}(U) is ∈ Fψ*C(H). Since H is fuzzy \( {T}_{1/5}^{\psi \ast } \)space, then γ^{−1}(U) is Fαclosed set in H. But φ is Fαirresolute function, then ϕ^{−1}(γ^{−1}(U)) is Fαclosed set in H. But ϕ^{−1}(γ^{−1}(U)) = (γ o ϕ)^{−1}(U). Therefore, γ o φ is Fαcontinuous function.
Definition 5.2 A map φ : (G, τ) → (H, σ) is said to be Fψ*open (Fψ*closed) if the image of every open (closed) fuzzy set in G is Fψ*open (closed) set in H.
Proposition 5.6 Every fuzzyopen map is fuzzy ψ*open map.
Proof The proof follows from the Definition 5.2.
The converse of Proposition 5.6 need not be true as seen from the following example.
Example 5.1 Suppose that G = {a, b, c} with fuzzy topology τ = {0, 1, {a_{0.8}, b_{0.2}, c_{0.3}}}, and H = {x, y, z} with fuzzy topology σ = {0, 1, {x_{0.5}, y_{0.2}, z_{0.7}}, {x_{0.7}, y_{0.8}, z_{0.3}}, {x_{0.5}, y_{0.2}, z_{0.3}}, {x_{0.7}, y_{0.8}, z_{0.7}}. Let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fψ*open map, but it is not a Fopen map, since G = {a_{0.8}, b_{0.2}, c_{0.3}} ∈ FO(G) but ϕ(G) ∉ FO(H).
Proposition 5.7 Every fuzzyclosed map is Fψ*closed map.
Proof The proof follows from the Definition 5.2.
The converse of Proposition 5.7 need not be true as seen from the following example.
Example 5.2 In the Example 5.1, let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fψ*closed map, but it is not an Fclosed map, since V = {a_{0.2}, b_{0.8}, c_{0.7}} ∈ FC(G) but ϕ(V) ∉ F C(H).
Proposition 5.8 If φ : G → H is Fclosed map and γ : H → W is Fψ*closed map, then γ o φ : G → W is Fψ*closed map.
Conclusion
In this paper, we have defined a new class of fuzzy sets, namely, fuzzy ψ*closed sets for fuzzy topological spaces, which is properly placed in between the class of fuzzy αclosed sets and the class of fuzzy generalized αclosed sets. We have also investigated some properties of these fuzzy sets. Fuzzy ψ*continuous, fuzzy ψ*irresolute functions, and fuzzy ψ*closed (open) functions have been introduced. We have proved that every Fψ*continuous function is Fgαcontinuous, but the converse need not be true, and the composition of two Fψ*irresolute functions is Fψ*irresolute. Fuzzy T_{1/5}spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)spaces, and fuzzy ^{ψ*}T_{1/5}spaces have been established as applications of fuzzy ψ*closed set. In the future, we will generalize this class of fuzzy sets in fuzzy bitopological spaces, and some applied examples should be given.
Availability of data and materials
All data generated or analyzed during this study are included in this published article.
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Allah, M.A.A., Nawar, A.S. ψ*closed sets in fuzzy topological spaces. J Egypt Math Soc 28, 38 (2020). https://doi.org/10.1186/s42787020000873
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Keywords
 Fuzzy ψ*closed sets
 Fuzzy T_{1/5}spaces
 Fuzzy ψ*continuous
 Fuzzy ψ*closed functions
Mathematical subject classification
 54 C 10
 54 A 40
 54 A 05