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ψ*-closed sets in fuzzy topological spaces

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 T1/5-spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)-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 T1/5-spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)-spaces, and fuzzy ψ*T1/5-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: GH) 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, ED, 1 − E τ} and int(D) = v{S : S is fuzzy open set of G, SD, S τ} [10].

Definition 2.1 A fuzzy set D of a fts G is called fuzzy α-open (briefly, Fα-open) if D ≤ int(cl(int(D))) 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, Fgα-closed) [8] if αcl(D) ≤ U whenever D ≤ U and U is fuzzy α-open in (G, τ). The complement of Fgα-closed set is called Fgα-open 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:

  1. (i)

    Fα-continuous [10] if ϕ−1(V) is Fα-closed in G, for each VFC (H);

  2. (ii)

    Fgα-continuous [8] if ϕ−1(V) is Fgα-closed in G, for each VFC (H);

  3. (iii)

    F-irresolute [11] if ϕ−1(V) is F-closed in G, for each VFC (H).

Definition 2.4 A function φ : (G, τ) → (H, σ) is said to be fuzzy-open (fuzzy-closed) [2] if the image of every fuzzy open (fuzzy-closed) set in G is fuzzy-open (fuzzy-closed) 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 DU and U is Fgα-open 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 α-closed set is fuzzy ψ*-closed.

Proof Let D be a Fα-closed set in (G, τ), and since every Fα-closed set is Fgα-closed. Then, αcl(D) ≤ U whenever DU and U is Fα-open in (G, τ), and since every Fα-open set is Fgα-open. So, αcl(D) ≤ U whenever DU and U is Fgα-open 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, {a0.5, b0.2, c0.7}, {a0.7, b0.8, c0.3}, {a0.5, b0.2, c0.3}, {a0.7, b0.8, c0.7}}. The fuzzy subset D = {a0.4, b0.8, c0.7} is Fψ*-closed set in (G, τ) but not Fα-closed set since cl(int(cl(D))) = {a0.5, b0.8, c0.7}.

Proposition 3.2 Every fuzzy ψ*-closed set is fuzzy gα-closed set.

Proof Follows from the fact that every Fα-open set is Fgα-open.

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 = {a0.5, b0.3, c0.7} is Fgα-closed 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 Fgα-open, 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 Fgα-open set and fuzzy ψ*-closed set in (G, τ), then D is fuzzy α-closed set in (G, τ).

Proof Since DD and D is Fgα-open set and Fψ*-closed, then αcl(D) ≤ D. Since D ≤ αcl(D), then D = αcl(D), and thus D is Fα-closed set in (G, τ).

Proposition 3.5 Every fuzzy ψ*-open set is fuzzy gα-open.

Proof Let DFψ*O(G). Then, 1 – DFψ*C(G) and hence Fgα-closed set in (G, τ) by Proposition 3.2. This implies that D is Fgα-open set in (G, τ). Hence, every Fψ*-open set in G is Fgα-open 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 Fgα-open subset of (G, τ) such that E ≤ U. Then, D ≤ U and since DFψ*C(G), then αcl(D) ≤ U. Now, αcl(E) ≤ αcl(D) ≤ U. Then, EFψ*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 DFψ*O(G), and αint(D) ≤ E ≤ D. Then, 1 – DFψ*C(G), and 1 – D ≤ 1 – E ≤ αcl(1 – D). By Proposition 3.6, 1 – BFψ*C(G). Hence, EFψ*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:

$$ \left(\mathrm{i}\right)\ {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\le D. $$
$$ \left(\mathrm{ii}\right)\ D\ \mathrm{is}\ F{\psi}^{\ast }-\mathrm{open}\iff {\psi}^{\ast }-\mathit{\operatorname{int}}(D)=D. $$
$$ \left(\mathrm{iii}\right)\ {\psi}^{\ast }-\mathit{\operatorname{int}}\left({\psi}^{\ast }-\mathit{\operatorname{int}}\ (D)\right)={\psi}^{\ast }-\mathit{\operatorname{int}}(D). $$
$$ \left(\mathrm{iv}\right)\mathrm{If}\ D\le B,\mathrm{then}\ {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\le {\psi}^{\ast }-\mathit{\operatorname{int}}(B). $$

Proof (i) Follows from Definition 3.3.

(ii) Let DFψ*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, DFψ*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:

$$ \left(\mathrm{i}\right){\psi}^{\ast }-\mathit{\operatorname{int}}\left(D\wedge E\right)={\psi}^{\ast }-\operatorname{int}\left(\mathrm{D}\right)\wedge {\psi}^{\ast }-\mathit{\operatorname{int}}(E). $$
$$ \left(\mathrm{ii}\right)\ {\psi}^{\ast }-\mathit{\operatorname{int}}\left(D\ \mathrm{v}\ E\right)\ge {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\mathrm{v}{\psi}^{\ast }-\mathit{\operatorname{int}}(E). $$

Proof (i) Since DE ≤ D and DE ≤ E, by using Proposition 3.7 (iv), we get ψ* −  int (DE) ≤ ψ* −  int (D) and ψ* −  int (DE) ≤ ψ* −  int (E). Thus,

$$ {\psi}^{\ast }-\operatorname{int}\left(D\wedge E\right)\le {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\wedge {\psi}^{\ast }-\operatorname{int}(E). $$
(1)

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) ≤ DE. Now applying Proposition 3.7 (iv), we get ψ* −  int (ψ* −  int (D) ψ* −  int (E)) ≤ ψ* −  int (DE). By (1), ψ* −  int (ψ* −  int (D) ψ* −  int (ψ* −  int (E)) ≤ ψ* −  int (DE). By using Proposition 3.7 (iii),

$$ {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\wedge {\psi}^{\ast }-\operatorname{int}(E)\le {\psi}^{\ast }-\operatorname{int}\left(D\wedge E\right). $$
(2)

Forms (1) and (2), ψ* −  int (DE) = 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 = {a0.4, b0.8, c0.7}, and E = {a0.6, b0.8, c0.5}. Then, ψ* −  int (D) = 0, and ψ* −  int (E) = {a0.6, b0.2, c0.3}. That implies ψ* −  int (D) v ψ* −  int (E) = {a0.6, b0.2, c0.3}. Now, D v E = {a0.6, b0.8, c0.7}; it follows that ψ* −  int (D v E) = {a0.6, b0.2, c0.7}. 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:

$$ \left(\mathrm{i}\right)\ {\left({\psi}^{\ast}\hbox{-} \operatorname{int}(D)\right)}^{\mathrm{c}}={\psi}^{\ast}\hbox{-} \mathrm{cl}\left({D}^{\mathrm{c}}\right) $$
$$ \left(\mathrm{ii}\right)\ {\left({\psi}^{\ast}\hbox{-} \mathrm{cl}(D)\right)}^{\mathrm{c}}={\psi}^{\ast}\hbox{-} \operatorname{int}\left({D}^{\mathrm{c}}\right) $$

Proof (i) By using Definition 3.3, ψ* −  int (D) =  v { E : E ≤ D,  EFψ*O(G)}. Taking complement on both sides, we get as follows:

$$ {\displaystyle \begin{array}{l}{\left({\psi}^{\ast}\operatorname{int}(D)\right)}^{\mathrm{c}}={\left(\sup\ \left\{\ E:B\le \kern0.5em D,\kern0.5em E\kern0.37em \mathrm{is}\ \mathrm{F}{\psi}^{\ast}\hbox{-} \mathrm{open}\ \mathrm{in}\ G\ \right\}\right)}^{\mathrm{C}}\\ {}\kern3.239999em =\operatorname{inf}\ \left\{{E}^c:{E}^c\ge \kern0.5em {D}^c,\kern0.5em {E}^c\kern0.37em \mathrm{is}\ \mathrm{F}\ {\psi}^{\ast}\hbox{-} \mathrm{closed}\ \mathrm{in}\ G\ \right\}.\kern1em \end{array}} $$

Replacing Ec by   C, we get

(ψ* int(D))c =   {C : C ≥  Dc,  C   is Fψ*  ‐ closed in G }. By Definition 3.4, (ψ*  ‐ int(D))c = ψ*  ‐ cl(Dc).

(ii) By using (i), (ψ*  ‐ int(Dc))c = ψ*  ‐ cl(Dc)c = ψ*  ‐ cl(D). Taking complement on both sides, we get ψ*  ‐ int(Dc) = (ψ*  ‐ cl(D))c.

Proposition 3.10 Let D be a fuzzy set in a fts G. Then, DFψ*C(G) if and only if Dc is Fψ*-open.

Proposition 3.11

For any fuzzy sets D and E in a fts G, we have as follows:

$$ \left(\mathrm{i}\right)\ D\le {\psi}^{\ast }- cl(D) $$
$$ \left(\mathrm{ii}\right)\ D\ \mathrm{is}\ F{\psi}^{\ast }-\mathrm{closed}\iff {\psi}^{\ast }- cl(D)=D $$
$$ \left(\mathrm{iii}\right)\ {\psi}^{\ast }- cl\left({\psi}^{\ast }- cl(D)\right)={\psi}^{\ast }- cl(D) $$
$$ \left(\mathrm{iv}\right)\mathrm{If}\ D\le E,\mathrm{then}\ {\psi}^{\ast }- cl(D)\le {\psi}^{\ast }- cl(E) $$

Proof (i) Follows from Definition 3.4.

(ii) Let DFψ*C(G). By using Proposition 3.10, DcFψ*O(G). By using Proposition 3.9 (ii), ψ*  ‐ int(Dc) = Dc (ψ*  ‐ cl(D))c = Dcψ*  ‐ cl(D) = D.

(iii) By using (ii), we get ψ* − cl(ψ* − cl(D)) = ψ* − cl(D).

(iv) If DE ≤ D and DE ≤ E By using Proposition 3.7 (iv), ψ*  ‐ int(Ec) ≤ ψ*  ‐ int(Dc). Taking complement on both sides, we get (ψ*  ‐ int(Ec))c ≥ (ψ*  ‐ int(Dc))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:

$$ \left(\mathrm{i}\right)\ {\psi}^{\ast }- cl\left(D\ \mathrm{v}\ E\right)={\psi}^{\ast }-d(D)\mathrm{v}{\psi}^{\ast }- cl(E) $$
$$ \left(\mathrm{ii}\right)\ {\psi}^{\ast }- cl\left(D\wedge E\right)\le {\psi}^{\ast }-d(D)\wedge {\psi}^{\ast }- cl(E) $$

Proof (i) Since  ψ*  ‐ cl(DE) = ψ*  ‐ cl((DE)c)c, by using Proposition 3.9 (i), we have ψ*  ‐ cl(DE) = (ψ*  ‐ int(DE)c)c = (ψ*  ‐ int(DcEc))c. By using Proposition 3.8 (i), we have ψ*  ‐ cl(DE) = (ψ*  ‐ int(Dc) ψ*  ‐ int(Ec))c = (ψ*  ‐ int(Dc))c (ψ  ‐ int(Ec))c.

By using Proposition 3.9 (i), we have ψ*  ‐ cl(DE) = ψ*  ‐ cl(Dc)cψ*  ‐ cl(Ec)c = ψ*  ‐ cl(D) ψ*  ‐ cl(E).

(ii) Since DE ≤ D and DE ≤ E, by using Proposition 3.11 (iv), we have ψ* − cl(DE) ≤ ψ* − cl(D) and ψ* − cl(DE) ≤ ψ* − cl(E). This implies that ψ* − cl(DE) ≤ ψ* − cl(D) ψ* − cl(E).

Proposition 3.14 For any fuzzy sets D and E in a fts G, we have as follows:

$$ \left(\mathrm{i}\right)\ {\psi}^{\ast }- cl(D)\ge D\ \mathrm{v}\ {\psi}^{\ast }- cl\left({\psi}^{\ast }-\mathit{\operatorname{int}}(D)\right) $$
$$ \left(\mathrm{ii}\right)\ {\psi}^{\ast }-\mathit{\operatorname{int}}(D)\le D\ \mathrm{v}\ {\psi}^{\ast }-\mathit{\operatorname{int}}\left({\psi}^{\ast }- cl(D)\right) $$
$$ \left(\mathrm{iii}\right)\ \mathit{\operatorname{int}}\left({\psi}^{\ast }- cl(D)\right)\le \mathit{\operatorname{int}}\left( cl(D)\right) $$
$$ \left(\mathrm{iv}\right)\ \mathit{\operatorname{int}}\left({\psi}^{\ast }- cl(D)\right)\ge \mathit{\operatorname{int}}\left({\psi}^{\ast }- cl\left({\psi}^{\ast }-\mathit{\operatorname{int}}(D)\right)\right) $$

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 Fα-continuous function is Fψ*-continuous.

Proof Let VFC (H). Since φ is Fα-continuous, then ϕ−1(V) is Fα-closed in G. Since every Fα-closed set is Fψ*-closed set, then ϕ−1(V) *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, {a0.5, b0.2, c0.7}, {a0.7, b0.8, c0.3}, {a0.5, b0.2, c0.3}, {a0.7, b0.8, c0.7} and H = {x, y, z} with fuzzy topology σ = {0, 1, {x0.8, y0.2, z0.3}}. 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 Fα-continuous function, since V = {x0.2, y0.8, z0.7} FC(H) but ϕ−1(V) FαC(G).

Proposition 4.2 Every Fψ*-continuous function is Fgα-continuous.

Proof Let VFC (H). Since φ is Fψ*-continuous, then ϕ−1(V) Fψ*C(G). By Proposition 3.1, every Fψ*-closed set is Fgα-closed set; then, ϕ−1(V) is Fgα-closed. Thus, φ is Fgα-continuous.

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, {a0.5, b0.2, c0.7}, {a0.7, b0.8, c0.3}, {a0.5, b0.2, c0.3}, {a0.7, b0.8, c0.7} and H = {x, y, z} with fuzzy topology σ = {0, 1, {x0.5, y0.6, z0.3}}. Let φ : (G, τ) → (H, σ) be defined by φ (a) = x, φ (b) = y, and φ (c) = z. φ is Fgα-continuous function, but it is not a Fψ*-continuous function, since V = {x0.5, y0.4, z0.7} 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 = {x0.2, y0.7, z0.4} 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 T1/5-spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)-spaces, and fuzzy ψ*T1/5-spaces are introduced.

We introduce the following definitions.

Definition 5.1 A fuzzy topological space (G, τ) is called as follows:

(i) Fuzzy T1/5-space if every Fgα-closed set in G is a Fα-closed set in G.

(ii) Fuzzy \( {T}_{1/5}^{\psi \ast } \)-space if every Fψ*-closed set in G is a Fα-closed set in G.

(iii) Fuzzy ψ*T1/5-space if every Fgα-closed set in G is a Fψ*-closed set in G.

Proposition 5.1 If φ : GH is Fψ*-continuous and G is fuzzy \( {T}_{1/5}^{\psi \ast } \)-space; then φ is Fα-continuous.

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 Fα-closed set in G. Thus, φ is Fα-continuous.

Proposition 5.2 If φ : G → H is Fψ*-irresolute and G is fuzzy \( {T}_{1/5}^{\psi \ast } \)-space, then φ is Fα-continuous.

ProofBy Theorem 5.1.

Proposition 5.3 If φ : G → H is Fgα-continuous and G is fuzzy ψ*T1/5-space, then φ is Fψ*-continuous.

Proof Let VFC (H); since φ is Fgα-continuous, then ϕ−1(V) is Fgα-closed set in G. Since G is Fψ*T1/5-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 \( {T}_{1/5}^{\psi \ast } \)-space, then H is also a fuzzy \( {T}_{1/5}^{\psi \ast } \)-space.

Proof Let VFψ*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 Fα-closed set in G. Since φ is Fα-closed and onto, then we have V is Fα-closed. 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 Fα-irresolute function and γ is Fψ*-continuous function, such that H is fuzzy \( {T}_{1/5}^{\psi \ast } \)-space. Then, γ o φ is Fα-continuous function.

Proof Let UFC (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 Fα-closed set in H. But φ is Fα-irresolute function, then ϕ−1(γ−1(U)) is Fα-closed set in H. But ϕ−1(γ−1(U)) = (γ o ϕ)−1(U). Therefore, γ o φ is Fα-continuous 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 fuzzy-open 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, {a0.8, b0.2, c0.3}}, and H = {x, y, z} with fuzzy topology σ = {0, 1, {x0.5, y0.2, z0.7}, {x0.7, y0.8, z0.3}, {x0.5, y0.2, z0.3}, {x0.7, y0.8, z0.7}. 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 F-open map, since G = {a0.8, b0.2, c0.3} FO(G) but ϕ(G) FO(H).

Proposition 5.7 Every fuzzy-closed 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 F-closed map, since V = {a0.2, b0.8, c0.7} 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 α-closed 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 T1/5-spaces, fuzzy \( {T}_{1/5}^{\psi \ast } \)-spaces, and fuzzy ψ*T1/5-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.

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The authors are grateful to the referees for their valuable suggestions.

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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/s42787-020-00087-3

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Keywords

  • Fuzzy ψ*-closed sets
  • Fuzzy T1/5-spaces
  • Fuzzy ψ*-continuous
  • Fuzzy ψ*-closed functions

Mathematical subject classification

  • 54 C 10
  • 54 A 40
  • 54 A 05