Levine [1] developed a new and important concept in general topology 1970. Specifically, Generalized closed set is an important in the study of topological spaces. In the literature we have seen the relation between closed set and generalized closed set. In particular, every closed set is a generalized closed set. In this present treatise, a new type of generalized closed set a fuzzy generalized closed set. Chang [2] gave the concept of fuzzy generalized closed is a fuzzy topological space X̃, T̃ briefly f. g-closed, if Ã̄≤Ũ whenever Ã≤Ũ and is Ũ open in (X̃, T̃) many fuzzy topologists have studied this notion completely in recent years. Some new and intriguing concepts have emerged from the study of fuzzy generalized closed sets. 

The aim of this paper is to introduce the concept of generalized regular closed sets and study some of its properties. For instance, in the present study, we define the regular generalized closed (r-g-closed) sets, study the properties of g- regular closed sets and a fuzzy symmetric space (f. sym. s.) with respect to of union, intersection and subspaces and provide an example of each and fuzzy Urysohn space (f.ury.s.) and a few theorems or corollaries are introduced.

Basic Concepts of Fuzzy Set

Definition 1.1 [3]: Let X be a nonempty set, a fuzzy set à in X is characteristic by a membership function, µÃ: X→[0,1] and this fuzzy collection can be defined as;

à = {(x, μÃ(x)): x ∈ X, μÃ(x)≤1}

All fuzzy set sets in X will be denoted by the formula Ix, where I^x = {Ã: Ã is fuzzy set in X}. 

Definition 1.2 [3]

A fuzzy set's support (A) is the set of all the members of it, x∈X such that μÃ(x)>0 and is denoted by A(Ã).

Definition 1.3 [3]

Unknown point P̃^r_xin X is a distinct fuzzy set with a defined membership function. 

P̃^r_x = r, if x=y | 0, if x ≠y

Where,

0<r≤1

y is the support of Prix(x). 

Definition 1.4 [4]

A fuzzy set à is said to by finite fuzzy set if S(Ã) is a finite set.

Definition 1.5 [5]

A fuzzy set à in (X̃, T̃) is called;

• Fuzzy generalized closed set (briefly, fg-closed set) if μÃ̄≤μà whenever μÃ≤à and à is fuzzy open

• Fuzzy semi-generalized closed set [6] (briefly, fsg-closed set) if s μÃ̄≤μà whenever μÃ≤à and à is fuzzy semi open

• Fuzzy θ-generalized closed set [7] (briefly, f-θg-closed set) if (μÃ̄)ᶿ≤μà and à is fuzzy open. The complement of fuzzy generalized closed (resp. fuzzy semi-generalized closed, fuzzy θ-generalized closed) set is fuzzy generalized open (resp. fuzzy semi-generalized open, fuzzy θ-generalized open) set

Definition 1.6 [5]

Let à and B̃ be a fuzzy sets of a universal set X then,

• Ã≤B̃ if and only if μÃ(x)≤μB̃(x), ∀ x ∈ X

• à = B̃ if and only if μÃ(x) = μB̃(x), ∀ x ∈ X

• Ã^C is the complement of a fuzzy set à with membership function μÃ^C = 1-μÃ(x)

• C̃ = à ∪ B̃ if and only if μC̃(x) = Max {μÃ(x), μB̃(x)}, ∀ x ∈ X

• D̃= à ∩ B̃ if and only if μD̃(x)= Min {μÃ(x), μB̃(x)}, ∀ x ∈ X

• More generally, for a family of fuzzy sets {Ã: α ∈ ˄ where ˄ is the any index set} the union C̃= ∪ Ãα

While the intersection D̃ = ∩ Ãα and defined respectively by: 

• μC̃(x) = sup {μÃα(x), x ∈ X}

• μD̃(x)= inf{μÃα(x): x ∈ X}

Fuzzy Generalized Closed Set

Definition 2.1 [8]: A subset à of (X̃, T̃) is a fuzzy regular open set if à = Ã̄° and a fuzzy regular closed set if à = .

Definition 2.2 [8]

A subset à of (X̃, T̃) is called a fuzzy generalized closed set (briefly fg-closed) if Ã̄≤Ũ whenever μÃ≤μŨ and Ũ is a fuzzy open set and (X̃, T̃) is called a fuzzy regular generalized closed (briefly f. r-g closed) set if Ã̄≤Ũ whenever μÃ≤μŨ and Ũ is a fuzzy regular open set.

Theorem 2.3

A subset à of (X̃, T̃) is a fuzzy generalized closed set if it is a fuzzy generalized regular closed set.

Proof

It is obvious from the definition since Ã̄≤.

Remark 2.4

Converse of the above theorem need not be true. It follows from the following example:

Example 2.5

Let 

X̃ = {a, b, c} 

and the corresponding fuzzy topological space be;

T̃ = {X̃, ∅̃, {(a,0.4), (b,0.6)}, {(b,0.6), (c,0.2)}, {(b,0.6)}}

Let

à = {(c,0.2)} 

Obviously à is a fuzzy generalized closed subset of X̃ but not a fuzzy generalized regular closed subset of X̃.

Definition 2.6 [9]

A Fuzzy topological space (X̃, T̃) is said to be a Fuzzy generalized regular space(f.g.r.s.) if for every f.g.r.s. F̃ in X̃ and for every μP̃^r_x>μF̃(x) there exist two f.o.s. Ũ and Ṽ such that μF̃(x)≤μṼ(x), μP̃rx^r<μŨ(x) and min {μŨ(x), μṼ(x)} = 0.

Definition 2.7 [1]

A fuzzy topological space (X̃, T̃) is said to be fuzzy generalized -T3 space (f.g.T3.s.) if it is a fuzzy generalized regular space (f.g.r.s.) and fuzzy T1 space.

Example 2.8

Let

X̃ = {a, b, c} and 

T̃ = {∅, X̃, {(a,0.1)}, {(b,0.4), (c,0.0)}} 

be a fuzzy topology on X̃, then (X̃, T̃) is a f.r.s. but is not f.g.r.s. since There Are two f.o.s. contain a fuzzy point C̃. While;

B̃ = {(a, 0.1), (b,0.4), (c, 0.0)} (fuzzy sets) and 

C̃ = {(a, 0.0), (b, 0.0), (c, 0.6)} (fuzzy point) 

Definition 2.9

Let (X̃, T̃) be a fuzzy topological space and μÃ(x)≤μX̃(x), then a fuzzy θ-closure of Ã, (f.(Ã̄ᶿ)) defined by

f.(Ã̄ᶿ) = {μP̃^r_x≤μX̃(x) : min {μŨ(x)μÃ(x)}≠ 0,

Ũ is a f.o.s., μP̃^r_x≤μŨ(x)}

Example 2.10

Let X̃ = {a, b, c} and T̃ = {∅, X̃, {(a,0.4)}} topology on X̃. If

à = {(a, 0.4), (b, 0.6)} then 

f. Ãᶿ = {(a, 0.4), (b, 0.6), (c,0.8)}

We introduce the following lemma to prove the next theorem.

Lemma 2.11

Let (X̃, T̃) be a f.t.s. and μÃ(x)≤μX̃(x), μ.(Ã̄)ᶢ≤μ f.(Ã̄)ᶿ.

Proof

Let P̃^r_x be a fuzzy point such that μ P̃^r_x≤μ f.(Ã̄)ᶢ, let μP̃^r_x>μ f.(Ã̄)ᶿ, then there exist a f.o.s. Ũ contain P̃^r_x such that min{μŨ(x), μÃ(x)} = 0. Then μÃ(x)≤μŨc(x), since Ũc is a f.c.s., then Ũ is a f.g.c.s. and by definition f(Ã̄)ᶢ we get μP̃^r_x≤μŨc(x) and this is a contradiction. So= μP̃^r_x≤μ f.(Ã̄)ᶿ, therefore μf.(Ã̄)ᶢ≤μ f.(Ã̄)ᶿ.

Theorem 2.12

Let (X̃, T̃) be a f.t.s. then the following statements are equivalent:

• (X̃, T̃) is a f.g.r.s. 

• μ f.(Ã̄)ᶿ = μ f.(Ã̄)ᶢ. for all a fuzzy set à of X̃.

• μ f.(Ã̄)ᶿ = μÃ(x), for all f.g.c.s. à of X̃.

Proof

(1) → (2): For any a fuzzy set à of X̃, we get (by lemma 2.11)

μÃ(x)≤μ f.(Ã̄)ᶢ≤μ f.(Ã̄)ᶿ

Suppose that 

μP̃^r_x≤μ f.(Ã̄)ᶢ)^c = min {F̃ : μÃ(x)≤μF̃(x), F̃ is a f.g.c.s}

So there exist a f.g.c.s. F̃ such that 

μP̃^r_x≤μ(F̃(x))^c, μÃ(x)≤μF̃(x)

Since (X̃, T̃) is a f.g.r.s.by (1), then there exist two f.o.s. Ũ and Ṽ in X̃.

Such that 

μP̃^r_x ≤μŨ(x), μF̃(x)≤μṼ(x) and min {μŨ(x), μṼ(x)} = 0

So, we get: 

μP̃^r_x≤μŨ(x)≤μ(Ũ̄(x))≤μ(Ṽ(x))^c≤μF̃(x)≤μ(Ã(x))^c

So, min {μ(Ũ̄(x)), μÃ(x)} = 0, there exists:

μP̃^r_x≤μ f.(Ã̄)ᶿ)^c

So:

μf.(Ã̄)ᶿ≤μf.(Ã̄)ᶢ

From (1) and (2) we get μf.(Ã̄)ᶿ = μf.(Ã̄)ᶢ.

(2) → (3)

Let à be a f.g.c.s. in X̃, then

μÃ(x) = μ f.(Ã̄)ᶢ 

Since μ f.(Ã̄)ᶿ = μ f.(Ã̄)ᶢ by (2), then:

μ f.(Ã̄)ᶿ = μÃ(x)

(3) → (1) 

Suppose F̃ be a f.g.c.s. in X̃ and μP̃^r_x≤μ(F̃(x))^c, then μP̃^r_x≤μ(F̃̄(x))ᶿ (by (3)), so there exist a f.o.s. Ũ such that μP̃^r_x≤μŨ(x) and min {{μŨ(x), μF̃(x)} = 0. We get μF̃(x)≤μ(Ũ̄(x))^c also min {μŨ(x), μ(Ũ̄(x))^c} = 0 and hence (X̃, T̃) is a f.g.r.s.

Theorem 2.13

A Fuzzy topological space (X̃, T̃) is a f.g.r.s. if and only if for every a f.g.c.s. and a fuzzy point μP̃^r_x> μF̃(x), there exist two f.o.s. Ũ and Ṽ in X̃ such that:

μP̃rx^r≤μŨ(x), μF̃(x)≤μṼ(x) and min {μŨ(x), μṼ(x)} = 0

Proof 

First Side: Let F̃ be a f.g.c.s. in X̃ and μP̃^r_x≤μ(F̃(x))^c, then there exist two f.o.s. Ũ◦ and Ṽ in X̃ such that μP̃^r_x≤μŨ◦(x), μF̃(x)≤μṼ(x) and min {μŨ(x), μṼ(x)} = 0, μṼ(x)≤μŨ◦c(x), where Ũ◦c is a f.c.s., then μcl(Ṽ(x))≤μŨ◦c(x), thus;

min {μŨ◦(x), μ(Ṽ̄(x))} = 0

Since (X̃, T̃) is a f.g.r.s. by hypothesis then there exist two f.o.s. G̃ and H̃ in X̃, such that μ(Ṽ̄(x))≤μH̃(x), μP̃^r_x≤μG̃(x), min {μG̃(x), μH̃(x)} = 0.

Then μG̃(x)≤μ(H̃(x))^c where H̃c is a f.c.s., so μ(G̃̄(x)≤μ(H̃(x))^c and then min {μ(G̃̄(x)), μH̃(x)} = 0.

Let μŨ◦(x) = min{μŨ◦(x), μG̃(x)} then Ũ and Ṽ are two f.o.s. in X̃, such that μP̃^r_x≤μŨ(x), μF̃(x)≤μṼ(x), 

With thimin {μ(Ũ̄(x), μcl(Ṽ(x))} = min{({μŨ(x),μG̃(x)},μ(Ṽ̄(x))) ̅}}≤min {min {μŨ◦(x), μG̃(x)}, μ(Ṽ̄(x)}} = 0, so min {μ(Ũ̄(x), μ(Ṽ̄(x)} = 0.

Other Side

Let F̃ be a f.s.c.s. in X̃ and μP̃^r_x≤μF̃C(x), so there exist two f.o.s. Ũ and Ṽ in X̃, such that μP̃^r_x≤μŨ(x), μF̃(x)≤μṼ(x) and min {μ(Ṽ̄(x)), μ(Ũ̄(x))} = 0 (by hypothesis). Since μṼ(x)≤μ(Ṽ̄(x)), μŨ(x)≤μ(Ũ̄(x)), then min {μŨ(x), μṼ(x)} = 0, therefore (X̃, T̃) is a f.g.r.s.

Definition 2.14

A Fuzzy topological space (X̃, T̃) is said to be a fuzzy symmetric space(f.sym.s.) if P̃^r1_x1 and P̃^r2_x2 are any two fuzzy point in X̃ and μP̃^r1_x1≤ μP̃^r2_x2 then μP̃^r2_x2≤μP̃^r1_x1. 

Theorem 2.15

A Fuzzy topological space (X̃, T̃) is a f.sym.space if and only if P̃^r_x is a f.g.c.s. in X̃, for every P̃^r_x in X̃.

Proof 

First Side: Let μP̃^r_x≤μÕ(x), where Õ is a f.o.s. in X̃, but μ μP̃^r1_x1>μÕ(x), then min {μ ((P̃^r_x) ̅), μÕ(x))^c} ≠0.

Now take a fuzzy point P̃^r2_x2 such that μP̃^r2_x2≤min {μP̃^r1_x1, μÕ(x))^c}, then μP̃^r1_x1≤μ((P̃^r2_x2) ̅)≤μÕ(x))^c (since (X̃,T̃) is a f.sym.space).

So μP̃^r1_x1>μÕ(x) and this a contradiction, then μ((P̃^r1_x1) ̅)≤μÕ(x) and thus μP̃^r1_x1 is a f.g.c.s. in X̃.

Other Side

Let μP̃^r1_x1≤μ(P̃^r2_x2) ̅ but μ(P̃^r2_x2) ̅>μ(P̃^r1_x1) ̅, then μ((P̃^r2_x2 ̅)≤μ(P̃^r1_x1) ̅)^c, so μP̃^r1_x1≤μ((P̃^r1_x1) ̅) and this contradiction.

The μP̃^r2_x2≤μcl(P̃^r1_x1) and thus (X̃, T̃) is a f. sym. space. 

Theorem 2.16

A Fuzzy topological space (X̃, T̃) is a f. r. g. c. s. if for every f.o.s. Ũ in X̃ and μP̃^r_x≤μŨ(x), there exist a f.g.o.s.in X̃, such that μP̃^r_x ≤μṼ(x)≤μ(Ṽ̄(x))ᶢ≤μŨ(x).

Proof 

Let Ũ be a f.o.s. in X̃, μP̃^r_x≤μŨ(x), then Ũ c is a f. c. s. in X̃ and μP̃^r_x>μ(Ũ(x))^c. since (X̃, T̃) be a f.g.r.s. (by hypothesis), then there exist two f.o.s. à and Ṽ such that μP̃^r_x≤μṼ(x), μ(Ũ(x))^c≤μÃ(x) and min {μṼ(x), μÃ(x)} = 0 μṼ(x)≤μ(Ã(x))^c, where à and Ṽ are f.g.o.s. in X̃, then Ã^c is a f.g.c.s., μ(Ṽ̄(x))ᶢ≤μÃ(x))^c and μÃ(x))^c≤μŨ(x) and thus μ(Ṽ̄(x))ᶢ≤μŨ(x). 

Definition 2.17

A Fuzzy topological space (X̃, T̃) is said to be a fuzzy Urysohn space (f.ury.s.). If (X̃, T̃) is a fuzzy generalized regular space and a fuzzy symmetric space then (X̃, T̃) is a fuzzy Urysohn space.

Theorem 2.18

If (X̃, T̃) is a f.g.r.s. and a f. sym. s. then (X̃, T̃) is a f.ury.s

Proof

Let (X̃, T̃) be a f.g.r.s. and a f. sym. s. and let P̃^r1_x1 and P̃^r2_x2 be any distinct fuzzy points in X̃, then μP̃^r1_x1 is a f.g.r.s. 

Since (X̃, T̃) is a f.g.r.s. and P̃^r2_x2>P̃^r1_x1, then there exist two f.o.s. Ũ and Ṽ such that μP̃^r2_x2≤μṼ(x) and μP̃^r1_x1≤μŨ(x), min {μŨ(x), μṼ(x)} = 0, (by theorem 2.16) so we get (X̃, T̃) is a f.ury.space.

Theorem 2.19

If (X̃, T̃) be a f.g.r.s. and let Ỹ be a f.g.c.s.in X̃ then the fuzzy subspace Ỹ is a f.g.r.s.

Proof

Let (X̃, T̃) be a f.g.r.s. and let à be a f.g.c.s. in Ỹ. let μP̃^r_x≤μÃ(x)^cy, since Ỹ be a f.g.c.s. in X̃ and since (X̃, T̃) is a f.g.r.s. (by hypothesis) then there exist two f.o.s. Ũ and Ṽ in Ỹ such that μP̃^r_x≤μŨ(x) and μÃ(x)≤μṼ(x) and min {μŨ(x), μṼ(x)} = 0. 

So, min {μŨ(x), μỸ(x)},and min {μṼ(x), μỸ(x)} are a membership of a f.o.s. in Ỹ such that μP̃^r_x≤min {μŨ(x), μỸ(x)}, μÃ(x)≤min {μṼ(x), μỸ(x)} and min {min {μŨ(x), μỸ(x)}, min {μṼ(x), μỸ(x)}} = min {min {μŨ(x), μṼ(x), μỸ(x)}} = min {μ∅̃(x), μỸ(x)} = 0, there for Ỹ is a f.g.r.s.

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