𝑊𝜃𝑔-Closed and 𝑊𝛿𝑔-Closed in [𝟎,𝟏]-Topological Spaces

Abstract

We investigate various classes of generalized closed fuzzy sets in [0,1]-topological spaces, namely, 𝑊𝜃𝑔-closed fuzzy sets and 𝑊𝛿𝑔-closed fuzzy sets. Also, we introduce a new separation axiom 𝐹𝑇^∗_3/4 of the [0,1]-topological spaces, and we prove that every 𝐹𝑇^∗_3/4-space is a 𝐹𝑇_3/4-space. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy mappings.

1. Introduction

In 1970, Levine [1] introduced the notion of generalized closed sets in topological spaces as a generalization of closed sets. Since then, many concepts related to generalized closed sets were defined and investigated. In 1997, Balasubramanian and Sundaram [2] introduced the concepts of generalized closed sets in fuzzy setting. Also, they studied various generalizations fuzzy continues mappings.

Recently, El-Shafei and Zakari [3–5] introduced new types of generalized closed fuzzy sets in [0,1]-topological spaces and studied many of their properties. Also, they studied various generalizations fuzzy continues mappings.

In the present paper, we introduce the concepts of 𝑊𝜃𝑔-closed fuzzy sets and 𝑊𝛿𝑔-closed fuzzy sets and study some of their properties. Also, we introduce the concept of 𝐹𝑇^∗_3/4-space. Moreover, we introduce and study the concepts of two new classes of fuzzy mappings, namely, fuzzy 𝑊𝜃𝑔-continuous mappings and fuzzy 𝑊𝛿𝑔-irresolute mappings.

2. Preliminaries

Let 𝑋 be a set and 𝐼 the unit interval. A fuzzy set in 𝑋 is an element of the set of all functions from 𝑋 into 𝐼. The family of all fuzzy sets in 𝑋 is denoted by 𝐼^𝑋. A fuzzy singleton 𝑥_𝑡 is a fuzzy set in 𝑋 defined by 𝑥_𝑡(𝑥)=𝑡, 𝑥_𝑡(𝑦)=0 for all 𝑦≠𝑥, 𝑡∈(0,1]. The set of all fuzzy singletons in 𝑋 is denoted by 𝑆(𝑋). For every 𝑥_𝑡∈𝑆(𝑋) and 𝜇∈𝐼^𝑋, we define 𝑥_𝑡∈𝜇 if and only if 𝑡≤𝜇(𝑥). A fuzzy set 𝜇 is called quasicoincident with a fuzzy set 𝜌, denoted by 𝜇𝑞𝜌, if and only if there exists 𝑥∈𝑋 such that 𝜇(𝑥)+𝜌(𝑥)>1. If 𝜇 is not quasicoincident with 𝜌, then we write 𝜇𝑞𝜌. By cl(𝜇), int(𝜇), 𝜇^𝑐, 𝑁(𝑥_𝑡,𝜏), and 𝑁_𝑄(𝑥_𝑡,𝜏), we mean the fuzzy closure of 𝜇, the fuzzy interior of 𝜇, the complement of 𝜇, the class of all open neighborhoods of 𝑥_𝑡, and the class of all open 𝑄-neighborhoods of 𝑥_𝑡, respectively.

Definition 2.1 (see [6, 7]). A fuzzy subset 𝜇 of a [0,1]-topological space (𝑋,𝜏) is called(i)regular open if and only if 𝜇=int(cl(𝜇)),(ii)preopen if and only if 𝜇≤int(cl(𝜇)).
The complement of a regular open (resp. preopen) fuzzy set is called a regular closed (resp. preclosed).

Definition 2.2 (see [8, 9]). Let (𝑋,𝜏) be a [0,1]-topological space, 𝑥_𝑡∈𝑆(𝑋), and 𝜇∈𝐼^𝑋. Then,(i)the 𝜃-closure of 𝜇, denoted by cl_𝜃(𝜇), is defined by𝑥_𝑡∈cl_𝜃(𝜇) if and only if cl(𝜂)𝑞𝜇 for each 𝜂∈𝑁_𝑄(𝑥_𝑡,𝜏),(ii)the 𝛿-closure of 𝜇 denoted by cl_𝛿(𝜇), is defined by𝑥_𝑡∈cl_𝛿(𝜇) if and only if int(cl(𝜂))q𝜇 for each 𝜂∈𝑁_𝑄(𝑥_𝑡,𝜏),(iii)𝜇 is called 𝜃-closed (resp. 𝛿-closed) if and only if 𝜇=cl_𝜃(𝜇) (resp. 𝜇=cl_𝛿(𝜇)).

Definition 2.3 (see [9]). Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋. Then,(i)the family 𝛾={𝜂_𝑗∶𝑗∈𝐽}⊆𝜏 is called an open 𝑃-cover of 𝜇 if and only if for every 𝑥_𝑡∈𝜇, there exists 𝑗₀∈𝐽 such that 𝑥_𝑡∈𝜂_𝑗0,(ii)𝜇 is called a 𝐶-set if and only if every open 𝑃-cover of 𝜇 has a finite subcover.

Definition 2.4 (see [2–4]). Let (𝑋,𝜏) be a [0,1]-topological space. A fuzzy set 𝜇∈𝐼^𝑋 is called(i)a generalized closed (𝑔-closed, for short) if and only if cl(𝜇)≤𝜂 whenever 𝜇≤𝜂 and 𝜂 is open fuzzy set,(ii)a 𝜃-generalized closed (𝜃𝑔-closed, for short) if and only if cl_𝜃(𝜇)≤𝜂 whenever 𝜇≤𝜂 and 𝜂 is open fuzzy set,(iii)a 𝛿-generalized closed (𝛿𝑔-closed, for short) if and only if cl_𝛿(𝜇)≤𝜂 whenever 𝜇≤𝜂 and 𝜂 is open fuzzy set.

Definition 2.5 (see [2, 4, 6, 10]). A [0,1]-topological space (𝑋,𝜏) is called(i)𝐹𝑅₁ if and only if 𝑥_𝑡𝑞cl(𝑦_𝑟) implies that there exist 𝜂∈𝑁(𝑥_𝑡,𝜏) and 𝜐∈𝑁(𝑦_𝑟,𝜏) such that 𝜂𝑞𝜐,(ii)𝐹𝑅₂ or 𝐹-regular if and only if 𝑥_𝑡𝑞𝜆, 𝜆 is closed fuzzy set implies that there exist 𝜂∈𝑁(𝑥_𝑡,𝜏) and 𝜐∈𝜏, 𝜆≤𝜐 such that 𝜂𝑞𝜐,(iii)𝐹𝑇_1/2 if and only if every 𝑔-closed fuzzy set in 𝑋 is closed,(iv)𝐹𝑇_3/4 if and only if every 𝛿𝑔-closed fuzzy set in 𝑋 is 𝛿-closed,(v)fuzzy weakly Hausdorff (𝐹𝑊𝑇₂, for short) if 𝑥_𝑡𝑞𝑦_𝑟 implies that there exists regular open fuzzy set 𝜂∈𝑁(𝑥_𝑡,𝜏) such that 𝑦_𝑟𝑞𝜂,(vi)fuzzy semiregular if and only if the collection of all regular open fuzzy sets in 𝑋 forms a base for the [0,1]-topology 𝜏,(vii)a fuzzy partition space if and only if every open fuzzy subset is closed.

Definition 2.6 (see [2–4, 11]). A fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is called(i)fuzzy generalized continuous (fuzzy 𝑔-continuous, for short) if and only if 𝑓⁻¹(𝜂) is 𝑔-closed in 𝑋 for any closed fuzzy set 𝜂 in 𝑌,(ii) fuzzy 𝜃-generalized continuous (fuzzy 𝜃𝑔-continuous, for short) if and only if 𝑓⁻¹(𝜂) is 𝜃𝑔-closed in 𝑋 for any closed fuzzy set 𝜂 in 𝑌,(iii) fuzzy 𝛿-generalized continuous (fuzzy 𝛿𝑔-continuous, for short) if and only if 𝑓⁻¹(𝜂) is 𝛿𝑔-closed in 𝑋 for any closed fuzzy set 𝜂 in 𝑌,(iv) fuzzy 𝛿-continuous if the inverse image of every 𝛿-open fuzzy set in 𝑌 is 𝛿-open in 𝑋,(v) fuzzy 𝛿-open (fuzzy 𝛿-open, for short) if and only if 𝑓(𝜂) is 𝛿-open in 𝑌 for any 𝛿-open fuzzy set 𝜂 in 𝑋,(vi)fuzzy 𝛿-closed (fuzzy 𝛿-closed, for short) if and only if 𝑓(𝜂) is 𝛿-closed in 𝑌 for any 𝛿-closed fuzzy set 𝜂 in 𝑋.

Theorem 2.7 (see [3]). A fuzzy subset 𝜇 of an 𝐹𝑅₂-fts (𝑋,𝜏) is 𝜃𝑔-closed if and only if it is 𝑔-closed.

Theorem 2.8 (see [3]). Let (𝑋,𝜏) be a [0,1]-topological space. Then, the following conditions are equivalent:(i)(𝑋,𝜏) is 𝐹𝑅₁-space,(ii)for each 𝐶-set 𝜇∈𝐼^𝑋, cl(𝜇)=cl_𝜃(𝜇),(iii)for each 𝑥_𝑡∈𝑆(𝑋), cl(𝑥_𝑡)=cl_𝜃(𝑥_𝑡).

Theorem 2.9 (see [3, 4]). Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋 be a preopen. Then, 𝜇 is 𝜃𝑔-closed (resp.𝛿𝑔-closed) if and only if it is 𝑔-closed.

Theorem 2.10 (see [3]). Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇,𝜂∈𝐼^𝑋. Then,(i)cl_𝜃(𝜇∨𝜂)=cl_𝜃(𝜇)∨cl_𝜃(𝜂),(ii)cl_𝜃(𝜇∧𝜂)≤cl_𝜃(𝜇)∧cl_𝜃(𝜂).

Theorem 2.11 (see [4]). Let (𝑋,𝜏) be a fuzzy semiregular space and 𝜇∈𝐼^𝑋. Then, (i)𝜇 is 𝛿𝑔-closed if and only if 𝜇 is 𝑔-closed,(ii) If, in addition, (𝑋,𝜏) is 𝐹𝑇_1/2, then 𝜇 is 𝛿𝑔-closed if and only if 𝜇 is closed.

Theorem 2.12 (see [4]). Let (𝑋,𝜏) be an 𝐹𝑅₁-space and 𝜇∈𝐼^𝑋 be a 𝐶-set. Then, 𝜇 is𝛿𝑔-closed if and only if it is 𝑔-closed.

Theorem 2.13 (see [4]). Let (𝑋,𝜏) be a fuzzy partition space and 𝜇∈𝐼^𝑋. Then, 𝜇 is 𝛿𝑔-closed if and only if it is 𝑔-closed.

Theorem 2.14 (see [4]). Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇,𝜂∈𝐼^𝑋. Then,(i)cl_𝛿(𝜇∨𝜂)=cl_𝛿(𝜇)∨cl_𝛿(𝜂),(ii)cl_𝛿(𝜇∧𝜂)≤cl_𝛿(𝜇)∧cl_𝛿(𝜂).

Theorem 2.15 (see [4]). A [0,1]-topological space (𝑋,𝜏) is 𝐹𝑇_3/4-space if for every 𝑥_𝑡∈𝑆(𝑋) either 𝑥_𝑡 is 𝛿-open or 𝑥_𝑡 is closed.

Theorem 2.16 (see [4]). Let (𝑋,𝜏) be a [0,1]-topological space. Then, the following conditions are equivalent: (i)𝑋 is an 𝐹𝑊𝑇₂-space,(ii)𝑥_𝑡=cl_𝛿(𝑥_𝑡) for each 𝑥_𝑡∈𝑆(𝑋).

3. 𝑊𝜃𝑔-Closed Fuzzy Sets

In this section, we introduce the concept of weakly 𝜃-generalized closed fuzzy sets, and we study some of their properties.

Definition 3.1. A fuzzy subset 𝜇 of a [0,1]-topological space (𝑋,𝜏) is said to be weakly 𝜃-generalized closed (𝑊𝜃𝑔-closed, for short) if and only if cl_𝜃(𝜇)≤𝜂 whenever 𝜇≤𝜂 and 𝜂 is 𝜃-open fuzzy set.
The complement of a 𝑊𝜃𝑔-closed fuzzy set is called 𝑊𝜃𝑔-open.

Theorem 3.2. Let (𝑋,𝜏) be a [0,1]-topological space. Then,(i)Every 𝜃-closed fuzzy set is 𝑊𝜃𝑔-closed,(ii)Every 𝜃𝑔-closed fuzzy set is 𝑊𝜃𝑔-closed.

Proof. Obvious.

From the above discussion, we introduce the following diagram.(3.1)

None of these implications is reversible as the following examples show.

Example 3.3. Let 𝑋={𝑥,𝑦} and 𝜏={0_𝑋,𝑦_0.7,1_𝑋}. If 𝜇=𝑥_0.5∨𝑦_0.6, then 𝜇 is 𝑊𝜃𝑔-closed fuzzy set but not 𝜃-closed.

Example 3.4. Let 𝑋={𝑥} and 𝜏={0_𝑋,𝑥_0.6,1_𝑋}. If 𝜇=𝑥_0.5, then 𝜇 is 𝑊𝜃𝑔-closed, since the only 𝜃-open superset of 𝜇 is 1_𝑋. But 𝜇 is not 𝜃𝑔-closed, since 𝜇≤𝑥_0.6 and cl_𝜃(𝜇)=1_𝑋≰𝑥_0.6.

Theorem 3.5. A fuzzy subset 𝜇 of a [0,1]-topological space (𝑋,𝜏) is 𝑊𝜃g-closed if for every 𝑥_𝑡∈𝑆(𝑋) such that 𝑥_𝑡𝑞cl_𝜃(𝜇), one has cl_𝜃(𝑥_𝑡)𝑞𝜇.

Proof. Let 𝜂 be 𝜃-open and 𝜇≤𝜂. If 𝑥_𝑡𝑞cl_𝜃(𝜇), then by assumption, cl_𝜃(𝑥_𝑡)𝑞𝜇. Hence, there exists 𝑦∈𝑋 such that cl_𝜃(𝑥_𝑡)(𝑦)+𝜇(𝑦)>1. Put cl_𝜃(𝑥_𝑡)(𝑦)=𝜀. Then, 𝑦_𝜀∈cl_𝜃(𝑥_𝑡) and 𝑦_𝜀𝑞𝜇. Thus, 𝜌𝑞𝑥_𝑡 for each 𝜌∈𝑁_𝑄(𝑦_𝜀,𝜏_𝜃). Since 𝑦_𝜀𝑞𝜂, then 𝜂𝑞𝑥_𝑡 and so cl_𝜃(𝜇)≤𝜂. Thus, 𝜇 is 𝑊𝜃𝑔-closed.

Theorem 3.6. Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋. Then, 𝜇 is 𝑊𝜃g-closed if there is not any 𝜃-closed fuzzy set 𝜆 such that 𝜆𝑞𝜇 and 𝜆qcl_𝜃(𝜇).

Proof. Suppose that 𝜇 is not 𝑊𝜃𝑔-closed. Then, there exists 𝜃-open fuzzy set 𝜂 such that 𝜇≤𝜂 and cl_𝜃(𝜇)≰𝜂. Put 𝜆=𝜂^𝑐. Then, there exists 𝜃-closed fuzzy set 𝜆 such that 𝜆𝑞𝜇 and 𝜆𝑞cl_𝜃(𝜇). This is a contradiction.

Theorem 3.7. Let (𝑋,𝜏) be an 𝐹𝑅₁-space and 𝜇∈𝐼^𝑋 be a 𝐶-set and 𝑔-closed. Then, 𝜇 is 𝑊𝜃𝑔-closed.

Proof. Suppose that (𝑋,𝜏) is an 𝐹𝑅₁-space and 𝜇 is a 𝐶-set in 𝑋. If 𝜇 is 𝑔-closed, then by Theorem 2.8  𝜇 is 𝜃𝑔-closed and hence 𝑊𝜃𝑔-closed.

Theorem 3.8. Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋 be a preopen and 𝑔-closed. Then, 𝜇 is 𝑊𝜃g-closed.

Proof. It is an immediate consequence of Theorems 2.9 and 3.2.

Theorem 3.9. Let (𝑋,𝜏) be an 𝐹𝑅₂-space and 𝜇∈𝐼^𝑋 be a 𝑔-closed. Then, 𝜇 is 𝑊𝜃g-closed.

Proof. It is an immediate consequence of Theorems 2.7 and 3.2.

Theorem 3.10. A finite union of 𝑊𝜃𝑔-closed fuzzy sets, is always 𝑊𝜃𝑔-closed fuzzy set.

Proof. Suppose that 𝜇,𝜂∈𝐼^𝑋 are 𝑊𝜃𝑔-closed fuzzy sets and let 𝜐∈𝜏_𝜃 such that 𝜇∨𝜂≤𝜐. Since 𝜇 and 𝜂 are 𝑊𝜃𝑔-closed, then we have cl_𝜃(𝜇)∨cl_𝜃(𝜂)≤𝜐 and by Theorem 2.10(i) cl_𝜃(𝜇∨𝜂)≤𝜐. Hence, 𝜇∨𝜂 is 𝑊𝜃𝑔-closed.

4. 𝑊𝛿𝑔-Closed Fuzzy Sets

In this section, we introduce the concept of weakly 𝛿-generalized closed fuzzy sets, and we study some of their properties. Also, we introduce the notion of 𝐹𝑇^∗_3/4-space, and we prove that every 𝐹𝑇^∗_3/4-space is a 𝐹𝑇_3/4-space.

Definition 4.1. A fuzzy subset 𝜇 of [0,1]-topological space (𝑋,𝜏) is said to be weakly 𝛿-generalized closed (𝑊𝛿𝑔-closed, for short) if and only if cl_𝛿(𝜇)≤𝜂 whenever 𝜇≤𝜂 and 𝜂 is 𝛿-open fuzzy set.

The complement of a 𝑊𝛿𝑔-closed fuzzy set is called 𝑊𝛿𝑔-open.

Theorem 4.2. Let (𝑋,𝜏) be a [0,1]-topological space. Then,(i)Every 𝛿-closed fuzzy set is 𝑊𝛿𝑔-closed,(ii)Every 𝛿𝑔-closed fuzzy set is 𝑊𝛿𝑔-closed.

Proof. Obvious.

From the above discussion, we introduce the following diagram.(4.1)

None of these implications is reversible as the following examples show.

Example 4.3. Let 𝑋={𝑥} and 𝜏={0_𝑋,𝑥_0.8,1_𝑋}. If 𝜇=𝑥_0.7, then 𝜇 is 𝑊𝛿𝑔-closed, since the only 𝛿-open superset of 𝜇 is 1_𝑋. But 𝜇 is not 𝛿𝑔-closed, since 𝜇≤𝑥_0.8 and cl_𝛿(𝜇)=1_𝑋≰𝑥_0.8.

Example 4.4. Let 𝑋={𝑥,𝑦} and 𝜏={0_𝑋,𝑦_0..8,1_𝑋}. A fuzzy subset 𝜇=𝑥_0.2∨𝑦_0.3 is 𝛿𝑔-closed and hence 𝑊𝛿𝑔-closed, but it is not 𝛿-closed.

Theorem 4.5. A fuzzy subset 𝜇 of a [0,1]-topological space (𝑋,𝜏) is 𝑊𝛿𝑔-closed if and only if for every 𝑥_𝑡∈𝑆(𝑋) such that 𝑥_𝑡𝑞cl_𝛿(𝜇) one has cl_𝛿(𝑥_𝑡)𝑞𝜇.

Proof. Let 𝑥_𝑡𝑞cl_𝛿(𝜇) and suppose that cl_𝛿(𝑥_𝑡)𝑞𝜇. Since 𝜇 is 𝑊𝛿𝑔-closed, then it is easy to observe that cl_𝛿(𝑥_𝛿)𝑞cl_𝛿(𝜇) which implies that 𝑥_𝑡𝑞cl_𝛿(𝜇). This is a contradiction.
The converse is similar to the proof of Theorem 3.5.

Theorem 4.6. Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋. Then, 𝜇 is 𝑊𝛿𝑔-closed if and only if there is not any 𝛿-closed fuzzy set 𝜆 such that 𝜆𝑞𝜇 and 𝜆qcl_𝛿(𝜇).

Proof. Suppose that there is a 𝛿-closed fuzzy set 𝜆 such that 𝜆𝑞𝜇 and 𝜆𝑞cl_𝛿(𝜇). Then, there exists some 𝑥_𝑡∈𝜆 such that 𝑥_𝑡𝑞cl_𝛿(𝜇). Since 𝜇 is 𝑊𝛿𝑔-closed, then by using Theorem 4.5, cl_𝛿(𝑥_𝑡)𝑞𝜇 and hence cl_𝛿(𝜆)𝑞𝜇. Since 𝜆 is 𝛿-closed, then we have 𝜆𝑞𝜇. This is a contradiction.
The converse is similar to the proof of Theorem 3.6.

Theorem 4.7. Let (𝑋,𝜏) be a fuzzy semiregular space and 𝜇∈𝐼^𝑋. Then,(i)𝜇 is 𝑊𝛿𝑔-closed if and only if it is 𝛿𝑔-closed,(ii)If, in addition, (𝑋,𝜏) is 𝐹𝑇_1/2-space, then 𝜇 is 𝑊𝛿𝑔-closed if and only if it is closed.

Proof. (i) Since (𝑋,𝜏) is semiregular space, then 𝜏=𝜏_𝛿, and so 𝜇 is 𝑊𝛿𝑔-closed if and only if it is 𝛿𝑔-closed.
(ii) From (i), Theorem 2.11, and by 𝐹𝑇_1/2-ness, the result is given.

Theorem 4.8. Let (𝑋,𝜏) be an 𝐹𝑅₁-space and 𝜇∈𝐼^𝑋 be a 𝐶-set and 𝑔-closed. Then, 𝜇 is 𝑊𝛿𝑔-closed.

Proof. Suppose that (𝑋,𝜏) is an 𝐹𝑅₁-space and 𝜇 is a 𝐶-set in 𝑋. If 𝜇 is 𝑔-closed, then by Theorem 2.12  𝜇 is 𝛿𝑔-closed and hence 𝑊𝛿𝑔-closed.

Theorem 4.9. Let (𝑋,𝜏) be a [0,1]-topological space and 𝜇∈𝐼^𝑋 be a preopen and 𝑔-closed. Then, 𝜇 is 𝑊𝛿𝑔-closed.

Proof. It is an immediate consequence of Theorems 2.9 and 4.2.

Theorem 4.10. Every fuzzy subset of a fuzzy partition space (𝑋,𝜏) is 𝑊𝛿𝑔-closed.

Proof. Let (𝑋,𝜏) be a fuzzy partition space, and let 𝜇 be a fuzzy subset of 𝑋. Then, by Theorem 2.13, 𝜇 is 𝛿𝑔-closed and hence, by Theorem 4.2, 𝜇 is 𝑊𝛿𝑔-closed.

Theorem 4.11. A finite union of 𝑊𝛿𝑔-closed fuzzy sets is always 𝑊𝛿𝑔-closed fuzzy set.

Proof. Similar to the proof of Theorem 3.10.

The following example shows that the finite intersection of 𝑊𝛿𝑔-closed fuzzy set may fail to be 𝑊𝛿𝑔-closed fuzzy set.

Example 4.12. Let 𝑋={𝑎,𝑏,𝑐,𝑑,𝑒}. Define 𝜆₁,𝜆₂,𝜆₃,𝜆₄,𝜆₅∶𝑋→[0,1] as follows: 𝜆₁(𝑎)=1,𝜆₁(𝑏)=1,𝜆₁(𝑐)=0,𝜆₁(𝑑)=0,𝜆₁(𝑒)𝜆=0,₂(𝑎)=0,𝜆₂(𝑏)=0,𝜆₂(𝑐)=1,𝜆₂(𝑑)=0,𝜆₂(𝑒)𝜆=0,₃(𝑎)=1,𝜆₃(𝑏)=1,𝜆₃(𝑐)=1,𝜆₃(𝑑)=0,𝜆₃(𝑒)𝜆=0,₄(𝑎)=1,𝜆₄(𝑏)=0,𝜆₄(𝑐)=1,𝜆₄(𝑑)=1,𝜆₄(𝑒)𝜆=0,₅(𝑎)=0,𝜆₅(𝑏)=1,𝜆₅(𝑐)=1,𝜆₅(𝑑)=0,𝜆₅(𝑒)=1. (4.2)
Consider the [0,1]-topology 𝜏={0_𝑋,𝜆₁,𝜆₂,𝜆₃,1_𝑋}. It is clear that 𝜆₄ and 𝜆₅ are 𝑊𝛿𝑔-closed fuzzy sets. But 𝜆₄∧𝜆₅=𝜆₂ is not 𝑊𝛿𝑔-closed.

Definition 4.13. A [0,1]-topological space (𝑋,𝜏) is called 𝐹𝑇^∗_3/4-space if and only if every 𝑊𝛿𝑔-closed fuzzy set is 𝛿-closed.

Theorem 4.14. Every 𝐹𝑇^∗_3/4-space is 𝐹𝑇_3/4-space.

Proof. It is an immediate consequence of Theorem 4.2(ii).

Theorem 4.15. A [0,1]-topological space (𝑋,𝜏) is  F𝑇^∗_3/4-space if for every 𝑥_𝑡∈𝑆(𝑋) either 𝑥_𝑡 is 𝛿-open or 𝛿-closed.

Proof. Let 𝜇∈𝐼^𝑋 be 𝑊𝛿𝑔-closed, and let 𝑥_𝑡𝑞𝜇. We consider the following two cases.
Case 1. 𝑥_𝑡 is 𝛿-open. Then, 𝑥^𝑐_𝑡 is 𝛿-closed. Since 𝑥_𝑡𝑞𝜇, then 𝜇≤𝑥^𝑐_𝑡. But 𝑥^𝑐_𝑡 is 𝛿-closed. Then, cl_𝛿(𝜇)≤𝑥^𝑐_𝑡. This shows that 𝑥_𝑡𝑞cl_𝛿(𝜇).Case 2. 𝑥_𝑡 is 𝛿-closed. Then, 𝑥^𝑐_𝑡 is 𝛿-open. Since 𝑥_𝑡𝑞𝜇, then 𝜇≤𝑥^𝑐_𝑡. But 𝜇 is 𝑊𝛿𝑔-closed. Then, cl_𝛿(𝜇)≤𝑥^𝑐_𝑡 and hence 𝑥_𝑡𝑞cl_𝛿(𝜇).

Corollary 4.16. Every FW𝑇₂-space is 𝐹𝑇^∗_3/4-space.

Proof. This is an immediate consequence of Theorems 2.16 and 4.15.
The converse of Corollary 4.16 need not be true, in general, and as a sample, we give the following example.

Example 4.17. Let 𝑋={𝑎,𝑏,𝑐}. Define 𝜆₁,𝜆₂,𝜆₃∶𝑋→[0,1] as follows: 𝜆₁(𝑎)=1,𝜆₁(𝑏)=0,𝜆₁(𝑐)𝜆=0,₂(𝑎)=0,𝜆₂(𝑏)=1,𝜆₂(𝑐)𝜆=0,₃(𝑎)=1,𝜆₃(𝑏)=1,𝜆₃(𝑐)=0. (4.3) Consider the [0,1]-topology 𝜏={0_𝑋,𝜆₁,𝜆₂,𝜆₃,1_𝑋}. Then, 𝑋 is 𝐹𝑇^∗_3/4-space but not F𝑊𝑇₂-space.

Theorem 4.18. Let (𝑋,𝜏) be a [0,1]-topological space. Then, the following conditions are equivalent:(i)𝑋 is FW𝑇₂-space,(ii)𝑋 is 𝐹𝑇^∗_3/4 and each 𝑥_𝑡∈𝑆(𝑋) is 𝑊𝛿𝑔-closed.

Proof. Obvious.

5. 𝑊𝜃𝑔-Continuous and 𝑊𝛿𝑔-Continuous Mappings

Definition 5.1. A fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is called(i)fuzzy 𝑊𝜃g-continuous if the inverse image of every closed fuzzy set in 𝑌 is 𝑊𝜃g-closed fuzzy set in 𝑋,(ii)fuzzy 𝑊𝛿𝑔-continuous if the inverse image of every closed fuzzy set in 𝑌 is 𝑊𝛿𝑔-closed fuzzy set in 𝑋.

Theorem 5.2. Every fuzzy 𝜃𝑔-continuous (resp.𝛿𝑔-continuous) mapping is fuzzy 𝑊𝜃g-continuous (resp. 𝑊𝛿𝑔-continuous).

Proof. Obvious.

The converse of the above Theorem may not be true, in general, by the following example.

Example 5.3. Let 𝑋=𝑌={𝑥} and consider a [0,1]-topology 𝜏 of Example 4.3, Δ={0_𝑌,𝑥_0.3,1_𝑌}. If 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is the identity fuzzy mapping, then 𝑓 is fuzzy 𝑊𝜃g-continuous but not fuzzy 𝜃𝑔-continuous, since 𝑥_0.7∈Δ^′ and 𝑓⁻¹(𝑥_0.7)=𝑥_0.7≤𝑥_0.8∈𝜏 but cl_𝜃(𝑥_0.7)=1_𝑋≰𝑥_0.8. Also, 𝑓 is fuzzy 𝑊𝛿𝑔-continuous but not fuzzy 𝛿𝑔-continuous.

Theorem 5.4. Let 𝑓∶(𝑋,𝜏)→(𝑌,Δ) be fuzzy mapping and (𝑋,𝜏) be fuzzy semiregular space. Then, the following conditions are equivalent:(i)𝑓 is fuzzy 𝑊𝛿𝑔-continuous,(ii)𝑓 is fuzzy 𝛿𝑔-continuous,(iii)𝑓 is fuzzy 𝑔-continuous.

Proof. It follows directly from Theorems 2.11 and 4.7(i).

Definition 5.5. A fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is called(i)fuzzy 𝑊𝜃g-irresolute if the inverse image of every 𝑊𝜃g-closed fuzzy set in 𝑌 is 𝑊𝜃g-closed fuzzy set in 𝑋,(ii)fuzzy 𝑊𝛿𝑔-irresolute if the inverse image of every 𝑊𝛿𝑔-closed fuzzy set in 𝑌 is 𝑊𝛿𝑔-closed fuzzy set in 𝑋.

Theorem 5.6. Let 𝑓∶(𝑋,𝜏)→(𝑌,Δ) and 𝑔∶(𝑌,Δ)→(Z,Ω) be two fuzzy mappings. Then, (i)𝑔∘𝑓 is fuzzy 𝑊𝜃g-continuous if 𝑔 is fuzzy continuous and 𝑓 is fuzzy 𝑊𝜃g-continuous,(ii)𝑔∘𝑓 is fuzzy 𝑊𝜃g-irresolute if 𝑔 is fuzzy 𝑊𝜃g-irresolute and 𝑓 is fuzzy 𝑊𝜃g-irresolute, (iii)𝑔∘𝑓 is fuzzy 𝑊𝜃g-continuous if 𝑔 is fuzzy 𝑊𝜃g-continuous and 𝑓 is fuzzy 𝑊𝜃g-irresolute.

Proof. Obvious.  

Theorem 5.7. Let 𝑓∶(𝑋,𝜏)→(𝑌,Δ) and 𝑔∶(𝑌,Δ)→(Z,Ω) be two fuzzy mappings. Then, (i)𝑔∘𝑓 is fuzzy 𝑊𝛿𝑔-continuous if 𝑔 is fuzzy continuous and 𝑓 is fuzzy 𝑊𝛿𝑔-continuous,(ii)𝑔∘𝑓 is fuzzy 𝑊𝛿𝑔-irresolute if 𝑔 is fuzzy 𝑊𝛿𝑔-irresolute and 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute, (iii)𝑔∘𝑓 is fuzzy 𝑊𝛿𝑔-continuous if 𝑔 is fuzzy 𝑊𝛿𝑔-continuous and 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute, (iv)Let (𝑌,Δ) be 𝐹𝑇^∗_3/4-space. Then, 𝑔∘𝑓 is fuzzy 𝑊𝛿𝑔-continuous if 𝑔 is fuzzy 𝑊𝛿𝑔-continuous and 𝑓 is fuzzy 𝑊𝛿𝑔-continuous, (v)Let (𝑌,Δ) be a fuzzy semiregular space. Then, 𝑔∘𝑓 is fuzzy 𝑊𝛿𝑔-continuous if 𝑔 is fuzzy 𝑔-continuous and 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute.

Proof. Obvious.    

Definition 5.8. A fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is called fuzzy 𝜃-open if and only if 𝑓(𝜂) is 𝜃-open in 𝑌  for any 𝜃-open fuzzy set 𝜂 in 𝑋.

Theorem 5.9. If a fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is bijective, fuzzy 𝜃-open, and fuzzy 𝑊𝜃g-continuous, then 𝑓 is fuzzy 𝑊𝜃g-irresolute.

Proof. Let 𝜆 be 𝑊𝜃g-closed fuzzy set in 𝑌, and let 𝑓⁻¹(𝜆)≤𝜂, where 𝜂∈𝜏_𝜃. Clearly, 𝜆≤𝑓(𝜂). Since 𝑓(𝜂)∈Δ_𝜃 and 𝜆 is 𝑊𝜃g-closed in 𝑌, then cl_𝜃(𝜆)≤𝑓(𝜂) and thus 𝑓⁻¹(cl_𝜃(𝜆))≤𝜂. Since 𝑓 is fuzzy 𝑊𝜃g-continuous and cl_𝜃(𝜆) is closed in 𝑌, then 𝑓⁻¹(cl_𝜃(𝜆)) is 𝑊𝜃g-closed in 𝑋 and hence cl_𝜃(𝑓⁻¹(cl_𝜃(𝜆)))≤𝜂. Thus, cl_𝜃(𝑓⁻¹(𝜆))≤𝜂 and so 𝑓⁻¹(𝜆) is 𝑊𝜃g-closed in 𝑋. This shows that 𝑓 is fuzzy 𝑊𝜃g-irresolute.

Theorem 5.10. If a fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is bijective, fuzzy 𝛿-open, and fuzzy 𝑊𝛿𝑔-continuous, then 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute.

Proof. Similar to the proof of Theorem 5.9.

Theorem 5.11. If a fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is fuzzy 𝑊𝛿𝑔-irresolute and (𝑋,𝜏) is 𝐹𝑇^∗_3/4-space, then 𝑓 is fuzzy 𝛿-continuous.

Proof. Let 𝜇 be a 𝛿-closed fuzzy set in 𝑌. By using Theorem 4.2, 𝜇 is 𝑊𝛿𝑔-closed. Since 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute, then 𝑓⁻¹(𝜇) is 𝑊𝛿𝑔-closed in 𝑋. Since 𝑋 is 𝐹𝑇^∗_3/4-space, then 𝑓⁻¹(𝜇) is 𝛿-closed in 𝑋. Thus, 𝑓 is fuzzy 𝛿-continuous.

Theorem 5.12. If a mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) is fuzzy 𝛿-continuous and fuzzy 𝛿-closed, and 𝜇 is 𝑊𝛿𝑔-closed fuzzy set in 𝑋, then 𝑓(𝜇)n is 𝑊𝛿𝑔-closed in 𝑋.

Proof. Let 𝜇 be 𝑊𝛿𝑔-closed in 𝑋, and let 𝑓(𝜇)≤𝜂, where 𝜂 is 𝛿-open fuzzy set in 𝑌. Since 𝜇≤𝑓⁻¹(𝜂), 𝜇 is 𝑊𝛿𝑔-closed fuzzy set in 𝑋 and since 𝑓⁻¹(𝜂) is 𝛿-open in 𝑋, then cl_𝛿(𝜇)≤𝑓⁻¹(𝜂). Thus 𝑓(cl_𝛿(𝜇))≤𝜂. Hence, cl_𝛿(𝑓(𝜇))≤cl_𝛿(𝑓(cl_𝛿(𝜇)))=𝑓(cl_𝛿(𝜇))≤𝜂, since 𝑓 is fuzzy 𝛿-closed. Hence, 𝑓(𝜇) is 𝑊𝛿𝑔-closed in 𝑌.

Theorem 5.13. Let (𝑋,𝜏) be an 𝐹𝑇^∗_3/4-space. If a fuzzy mapping 𝑓∶(𝑋,𝜏)→(𝑌,Δ) be surjective, fuzzy 𝑊𝛿𝑔-irresolute, and fuzzy 𝛿-closed, then (𝑌,Δ) is also 𝐹𝑇^∗_3/4-space.

Proof. Let 𝜇 be 𝑊𝛿𝑔-closed fuzzy set in 𝑌. Since 𝑓 is fuzzy 𝑊𝛿𝑔-irresolute, then 𝑓⁻¹(𝜇) is 𝑊𝛿𝑔-closed in 𝑋. Since 𝑋 is 𝐹𝑇^∗_3/4-space, then 𝑓⁻¹(𝜇) is 𝛿-closed in 𝑋. Thus, 𝜇 is 𝛿-closed in 𝑌, since 𝑓 is surjective and fuzzy 𝛿-closed. Hence, (𝑌,Δ) is 𝐹𝑇^∗_3/4-space.