ψ*-closed sets in fuzzy topological spaces

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 T1/5-spaces, fuzzy T1/5ψ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_{1/5}^{\psi \ast } $$\end{document}-spaces, and fuzzy ψ*T1/5-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 fgα-closed and fαg-closed) 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 α-closed sets and the class of fuzzy generalized α-closed sets. The structure of the rest of this paper is as follows. The "Preliminaries" section introduces the necessary definitions of fuzzy α-closed sets and fuzzy generalized α-closed 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 ψÃ 1=5 -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].
) and a fuzzy α-closed (briefly, Fα-closed) if D ≥ cl(int(cl(D))) [4]; the intersection of all fuzzy α-closed sets of (G, τ) containing D is called fuzzy α-closure 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 α-closed (briefly,

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.
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 Proposition 3.7 For any fuzzy sets D and B in a fts G, we have as follows: . 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 Proposition 3.8 For any fuzzy sets D and E in a fts G, we have as follows: By using Proposition 3.
Forms (1) and (2), The equality in Proposition 3.8 (ii) need not be hold as seen from the following example.
Proposition 3.9 For any fuzzy set D in a fts G, we have as follows: Taking complement on both sides, we get as follows: (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 3.11 For any fuzzy sets D and E in a fts G, we have as follows: (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.
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. Applications of Fψ*-closed sets As applications of Fψ*-closed sets, three fuzzy spaces, namely, fuzzy T 1/5 -spaces, fuzzy T ψÃ 1=5 -spaces, and fuzzy ψ* T 1/5 -spaces are introduced.
Allah and Nawar Journal of the Egyptian Mathematical Society (2020) 28:38 Page 6 of 8 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 Fgα-closed set in G is a Fα-closed set in G.
(ii) Fuzzy T ψÃ 1=5 -space if every Fψ*-closed set in G is a Fα-closed set in G. (iii) Fuzzy ψ* T 1/5 -space if every Fgα-closed set in G is a Fψ*-closed set in G.
Proof By Theorem 5.1.
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 Fα-closed. If G is fuzzy Since φ is Fα-closed and onto, then we have V is Fα-closed. Therefore, H is also a FT The converse of Proposition 5.6 need not be true as seen from the following example. The converse of Proposition 5.7 need not be true as seen from the following example. = y, and φ (c) = z. φ is Fψ*-closed map, but it is not an F-closed 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 F-closed 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 α-closed 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 Fgα-continuous, 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 ψÃ 1=5 -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.