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# Generalized *w*closed sets in biweak structure spaces

*Journal of the Egyptian Mathematical Society*
**volume 28**, Article number: 24 (2020)

## Abstract

As a generalization of the classes of *gw*closed (resp. *gw*open, *sgw*closed) sets in a weak structure space (*X*,*w*), the notions of *ij*-generalized *w*closed (resp. *ij*-generalized *w*open, *ij*-strongly generalized *w*closed) sets in a biweak structure space (*X*,*w*_{1},*w*_{2}) are introduced. In terms of these concepts, new forms of continuous function between biweak spaces are constructed. Additionally, the concepts of *ij*-*w*normal, *ij*-*gw*normal, *ij*-\(wT_{\frac {1}{2}}\), and *ij*-\(w^{\sigma }T_{\frac {1}{2}}\) spaces are studied and several characterizations of them are acquired.

## Introduction

In recent years, many researchers studied bitopological, bigeneralized, biminimal, and biweak spaces due to the richness of their structure and potential for doing a generous area for the generalization of topological results in bitopological environment. The concept of a bitopological space was built by Kelly [1], and thereafter, an abundant number of manuscripts was done to generalize the topological notions to bitopological setting. Fukutake [2] presented the concept of generalized closed sets and in bitopological spaces. The notion has been studied extensively in recent years by many topologists. Csaszar and Makai Jr. proposed the concept of bigeneralized topology [3]. In 2010, Boonpok [4, 5] provided the concept of bigeneralized topological spaces and biminimal structure spaces, respectively. Csaszar [6] defined the concept of weak structure which is weaker than a supra topology, a generalized topology, and a minimal structure and then offered various properties of it. Ekici [7] have investigated further properties and the main rules of the weak structure space. In order to extend many of the important properties of *w*closed sets to a larger family, Zahran et al. [8] characterized the concepts of generalized closed and generalized open sets in weak structures and achieved a number of properties of these concepts. As a generalization of bitopological spaces, bigeneralized topological spaces, and biminimal structure spaces, Puiwong et al. [9] in 2017 defined a new space, which is known as biweak structure. The concept of biweak structure can substitute in many situations, biminimal structures and bigeneralized topology. A new space consists of a nonempty set *X* equipped with two arbitrary weak structures *w*_{1},*w*_{2} on *X*. A triple (*X*,*w*_{1},*w*_{2}) is called a biweak structure space (in short, bi*w*ss).

The interior (resp. closure) of a subset *A* with respect to *w*_{j} are denoted by \(int_{w_{j}}(A)\) (resp. \(cl_{w_{j}}(A)\)), for (*j*=1,2). A subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is called *ij*-*w*closed if \(cl_{w_{i}}(cl_{w_{j}}(A))\)=*A*, where *i*,*j*= 1 or 2 and *i*≠*j*. The complement of an *ij*-*w*closed set is called *ij*-*w*open.

The concepts of generalized closed sets in weak structures [8] and biweak structure spaces [9] motivated us to define a new class of sets which is called generalized *w*closed sets in a biweak structure space which are found to be effective in the study of digital topology. The purpose of this article is introducing the notions of *ij*-generalized *w*closed (written henceforth as *ij*-*gw*closed), *ij*-generalized *w*open (written henceforth as *ij*-*gw*open), and *ij*-strongly generalized *w*closed (*ij*- *σ**g**w*closed, for short) sets in a bi*w*ss (*X*,*w*_{1},*w*_{2}) as a generalization of the concept of *gw*closed, *gw*open, and *sgw*closed sets, respectively, in a weak structure space (*X*,*w*) which presented in [8] and determining some of their behaviors. In terms of *ij*-*gw*closed and *ij*-*gw*open sets, new forms of continuous function between biweak spaces are constructed. Additionally, we try to extend the concepts of separation axioms on weak structures [8] to bi*w*ss and study some of their features. Some considerable results in articles [2, 8, 10] can be treated as particular cases of our outcomes.

## Preliminaries

To prepare this article as self-contained as possible, we recollect the next definitions and results which are due to various references [8, 9, 11].

###
**Definition 1**

*[*8*]* Let *w* be a weak structure on *X*. Then,

A subset

*A*is called generalized*w*closed (*gw*closed, for short) if*c**l*_{w}(*A*)⊆*U*, whenever*A*⊆*U*and*U*is*w*open.The complement of a generalized

*w*closed set is called generalized*w*open (*gw*open, for short), i.e, a subset*A*is*gw*open if and only if*i**n**t*_{w}(*A*)⊇*F*, whenever*A*⊇*F*and*F*is*w*closed.

The family of all *gw*closed (resp. *gw*open) sets in a weak structure *X* will be denoted by *G**W**C*(*X*) (resp. *G**W**O*(*X*)).

###
**Definition 2**

*[*11*]* Let *w* and *w*^{⋆} be weak structures on *X* and *Y*, respectively. A function *f*:(*X*,*w*)→(*Y*,*w*^{⋆}) is called (*w*,*w*^{⋆})-continuous if for *x*∈*X* and *w*^{⋆}open set *V* containing *f*(*x*), there is *w*open set *U* containing *x* s.t. *f*(*U*)⊆*V*.

###
**Theorem 1**

*[*11*]* Let *w* and *w*^{⋆} be weak structures on *X* and *Y*, respectively. For a function *f*:(*X*,*w*)→(*Y*,*w*^{⋆}), the following statements are equivalent:

*f*is (*w*,*w*^{⋆})-continuous,*f*^{−1}(*B*)=*i**n**t*_{w}(*f*^{−1}(*B*)), for every*w*^{⋆}open set*B*in*Y*,\(f(cl_{w}(A)){\subseteq }cl_{w^{\star }}(f(A))\), for every set

*A*in*X*,\(cl_{w}(f^{-1}(B)){\subseteq }(f^{-1}(cl_{w^{\star }}(B))\), for every set

*B*in*Y*,\(f^{-1}(int_{w^{\star }}(B)){\subseteq }int_{w}(f^{-1}(B))\), for every set

*B*in*Y*,*c**l*_{w}(*f*^{−1}(*F*))=*f*^{−1}(*F*), for every*w*^{⋆}closed set*F*in*Y*.

###
**Theorem 2**

*[*9*]* Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss and *A* be a subset of *X*. Then, the following are equivalent:

*A*is*ij*-*w*closed,*A*=\(cl_{w_{i}}(A)\) and*A*=\(cl_{w_{j}}(A)\),*A*=\(cl_{w_{j}}(cl_{w_{i}}(A))\), where*i*,*j*= 1 or 2 and*i*≠*j*.

###
**Proposition 1**

*[*9*]* Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss and *A*⊆*X*. Then, *A* is a *ij*-*w*closed set, if *A* is both *w*_{i}closed and *w*_{j}closed, where *i*,*j*= 1 or 2 and *i*≠*j*.

###
**Proposition 2**

*[*9*]* Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. If *A*_{α} is *ij*-*w*closed for all *α*∈*Λ*≠*∅*, then ∩_{α∈Λ}*A*_{α} is *ij*-*w*closed and the union of two *ij*-*w*closed sets is not a *ij*-*w*closed set, where *i*,*j*= 1 or 2 and *i*≠*j*.

In the rest of this article *i*,*j* will stand for fixed integers in the set {1,2} and *i*≠*j*.

## On *ij*-*gw*closed sets

In this part, a new family of sets called *ij*-generalized *w*closed (briefly, *ij*-*gw*closed) is presented and its properties are investigated.

###
**Definition 3**

A subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is called *ij*-generalized *w*closed (*ij*-*gw*closed, for short) if \(cl_{w_{j}}(A){\subseteq }U\), whenever *A*⊆*U* and *U* is *w*_{i}open. The complement of *ij*-*gw*closed set is called *ij*-*gw*open.

The family of all *ij*-*gw*closed (resp. *ij*-*gw*open) sets in a bi*w*ss (*X*,*w*_{1},*w*_{2}) will be denoted by *ij*- *G**W**C*(*X*) (resp. *ij*- *G**W**O*(*X*)).

###
**Remark 1**

If *A*∈*i**j*- *G**W**C*(*X*)∩*j**i*- *G**W**C*(*X*), then a subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is called pairwise *gw*closed and its complement is pairwise *gw*open.

###
**Example 1**

Let *X*= {1,2,3},*w*_{1}={*∅*,{1},{1,2}}, and *w*_{2}= {*∅*,{3}}. A set {3} is pairwise *gw*closed.

Certainly, the next theorems are obtained:

###
**Theorem 3**

A subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-*gw*open iff \(int_{w_{j}}(A){\supseteq }F\), whenever *A*⊇*F* and *F* is *w*_{i}closed.

###
**Theorem 4**

If *A* is an *ij*-*gw*closed and *w*_{i}open set in (*X*,*w*_{1},*w*_{2}), then *A*=\(cl_{w_{j}}(A)\).

###
**Theorem 5**

Every *w*_{j}closed set in a bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-*gw*closed.

###
*Proof*

Let *A* be a *w*_{j}closed set and *U* be a *w*_{i}open set in *X* s.t. *A*⊆*U*. Then, \(cl_{w_{j}}(A)\)=*A*. It implies that *A*∈*i**j*- *G**W**C*(*X*). □

###
**Corollary 1**

If *A* is a *w*_{j}open set in a bi*w*ss (*X*,*w*_{1},*w*_{2}), then *A*∈*i**j*- *G**W**O*(*X*).

###
**Remark 2**

By the following example, we have a tendency to show that the converse of Theorem *5* is not always true.

###
**Example 2**

In Example *1*, a set {2} is 12-*gw*closed and not *w*_{2}closed.

###
**Proposition 3**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Then,

If

*X*∈*w*_{j}and each*w*_{i}open set is*w*_{j}closed, then,*A*∈*i**j*-*G**W**C*(*X*), for each*A*⊂*X*.*A*∈*i**j*-*G**W**C*(*X*), for each*A*⊂*X*iff \(cl_{w_{j}}{U}={U}\) for each*w*_{i}open set*U*.

###
*Proof*

We prove only (2) and the rest of the proof is simple. Suppose that *A*∈*i**j*- *G**W**C*(*X*), for each *A*⊂*X*. Then, every *w*_{i}open set *U*, *A*∈*i**j*- *G**W**C*(*X*). If *U*⊆*U*, hence \(cl_{w_{j}}(U){\subseteq }U\). Thus, \(cl_{w_{j}}(U)\)=*U*, for each *w*_{i}open set *U*. Conversely, suppose that *A*⊆*U* and *U* be a *w*_{i}open set. Then, \(cl_{w_{j}}(A){\subseteq }cl_{w_{j}}(U)\). From assumption, \(cl_{w_{j}}(A){\subseteq }U\) and so *A*∈*i**j*- *G**W**C*(*X*). □

###
**Remark 3**

In the bi*w*ss (*X*,*w*_{1},*w*_{2}), the converse of the Proposition *3*(1) need not be true in general as shown by the next example.

###
**Example 3**

Let *X*= {1,2,3},*w*_{1}={*∅*,{2},{1,3}}, and *w*_{2}= {*∅*,*X*,{1},{2},{3},{1,2},{2,3}}. One may notice that every subset of *X* is 12-*gw*closed, but *A*={ 2} is a *w*_{1}open set in *X* and it is not *w*_{2}closed.

###
**Remark 4**

In general, 21- *G**W**C*(*X*)≠12- *G**W**C*(*X*) as in Example *3*.

###
**Proposition 4**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. If *w*_{1}⊆*w*_{2}, then 21- *G**W**C*(*X*)⊆12- *G**W**C*(*X*).

###
*Proof*

Straightforward. □

The converse of the Proposition 4 is not true as seen from the next example.

###
**Example 4**

In Example *3*, then 21- *G**W**C*(*X*)⊆12- *G**W**C*(*X*), but *w*_{1}⫅̸*w*_{2}.

Now, one can conclude attitudes relative to the union as well as the intersection of two *ij*-*gw*closed sets in a bi*w*ss (*X*,*w*_{1},*w*_{2}).

###
**Example 5**

Let *X*= {1,2,3,4},*w*_{1}={*∅*,{3},{1,3},{1,3,4},{1,2,4}} and *w*_{2} = {*∅*,{2},{3},{2,3,4}}. Let us consider *A*= {2} and *B*= {3}. Note that *A* and *B* are 21-*gw*closed sets but its union is not 21-*gw*closed.

###
**Example 6**

Let *X*= {1,2,3},*w*_{1}={*∅*,{1},{3}} and *w*_{2}= {*∅*,{1}}. Consider two 21-*gw*closed sets *A*= {1,2} and *B*= {1,3}, then *A*∩*B*= {1} is not 21-*gw*closed.

###
**Theorem 6**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss and \(cl_{w_{j}}(\emptyset)\)= *∅*. Then, the family of all *ij*-*gw*closed sets is a biminimal structure in *X*.

###
*Proof*

Obvious. □

###
**Theorem 7**

Suppose *X*∈*w*_{i}. Then, {*x*} is *w*_{i}closed or *X*∖{*x*}∈*i**j*- *G**W**C*(*X*), for each *x*∈*X*.

###
*Proof*

Suppose that the singleton {*x*} is not *w*_{i}closed for some *x*∈*X*. Then, *X*∖{*x*} is not *w*_{i}open. Since *X* is *w*_{i}open set and *X*∖{*x*}⊆*X*. Hence, *X*∖{*x*}∈*i**j*- *G**W**C*(*X*). □

###
**Theorem 8**

If *A*∈*i**j*- *G**W**C*(*X*), then \(cl_{w_{j}}(A){\setminus }A\) contains no nonempty *w*_{i}closed.

###
*Proof*

For an *ij*-*gw*closed set *A*, let *S* be a nonempty *w*_{i}closed set s.t. \(S{\subseteq }cl_{w_{j}}(A){\setminus }A\). Then, \(S{\subseteq }cl_{w_{j}}(A)\) and *S*⊆*X*∖*A*. Since *X*∖*S* is *w*_{i}open and *A* is *ij*-*gw*closed, then \(cl_{w_{j}}(A){\subseteq }X \setminus S\) or \(S{\subseteq }X{\setminus }cl_{w_{j}}(A)\). Thus, *S*= *∅*. Therefore, \(cl_{w_{j}}(A){\setminus }A\) does not contain nonempty *w*_{i}closed. □

###
**Remark 5**

In general, the converse of Theorem *8* is not true as shown in the next example.

###
**Example 7**

In Example *6*, if *A*= {1}, then \(c_{w_{1}}(A){\setminus }A\)= {2}. So we know that there is no any nonempty *w*_{2}closed contained in \(c_{w_{1}}(A){\setminus }A\). But *A*∉21- *G**W**C*(*X*).

It thus follows from Theorem 8 that

###
**Corollary 2**

If *A*∈*i**j*- *G**W**C*(*X*) and \(cl_{w_{j}}(A){\setminus }A\) is a *w*_{i}closed set, then \(cl_{w_{j}}(A)\)=*A*.

###
**Remark 6**

If *A* is an *ij*-*gw*closed set in a bi*w*ss (*X*,*w*_{1},*w*_{2}) and \(cl_{w_{j}}(A)\)=*A*, then \(cl_{w_{j}}(A){\setminus }A\) need not to be *w*_{i}closed as shown by the following example.

###
**Example 8**

Let *X*= {1,2,3},*w*_{1}={*∅*,{2}}, and *w*_{2}= {*∅*,{1},{3},{1,2}}. If *A*= {2}, one may notice that \(c_{w_{2}}(A)\)=*A* and hence \(c_{w_{2}}(A){\setminus }A\)= *∅*, which is not *w*_{1}closed.

###
**Theorem 9**

If *A*∈*i**j*- *G**W**C*(*X*), then \(cl_{w_{j}}(A){\setminus }A{\in }ij\)- *G**W**O*(*X*).

###
*Proof*

Let *A*∈*i**j*- *G**W**C*(*X*) and *F* be a *w*_{i}closed set s.t. \(F{\subseteq }cl_{w_{j}}(A){\setminus }A\). Then, by Theorem 8, we have *F*= *∅* and hence \(F{\subseteq }int_{w_{j}}(cl_{w_{j}}(A){\setminus }A)\). So by Theorem 3, we have \(cl_{w_{j}}(A){\setminus }A{\in }ij\)- *G**W**O*(*X*). □

###
**Remark 7**

The converse of the Theorem *9* need not to be true in general as shown by the following example.

###
**Example 9**

In Example *6*. If *A*= {1}, one may notice that \(cl_{w_{1}}(A){\setminus }A{\in }21\)- *G**W**O*(*X*), but *A*∉21- *G**W**C*(*X*).

###
**Theorem 10**

If *A*∈*i**j*- *G**W**C*(*X*) and \(A{\subseteq }B{\subseteq }cl_{w_{j}}(A)\), then *B*∈*i**j*- *G**W**C*(*X*).

###
*Proof*

Let *U* be any *w*_{i}open set s.t. *B*⊆*U*. Since *A*⊆*B* and *A*∈*i**j*- *G**W**C*(*X*), then \(cl_{w_{j}}(A){\subseteq }U\). Since \(B{\subseteq }cl_{w_{j}}(A)\), then we have \(cl_{w_{j}}(B){\subseteq }cl_{w_{j}}cl_{w_{j}}(A)\)=\(cl_{w_{j}}(A){\subseteq }U\). Consequently *B*∈*i**j*- *G**W**C*(*X*). □

###
**Corollary 3**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Then,

If

*A*∈*i**j*-*G**W**O*(*X*) and \(int_{w_{j}}(A){\subseteq }B{\subseteq }A\), then,*B*∈*i**j*-*G**W**O*(*X*).\(cl_{w_{j}}(A){\in }ij\)-

*G**W**C*(*X*) if*A*∈*i**j*-*G**W**C*(*X*).\(int_{w_{j}}(A){\in }ij\)-

*G**W**O*(*X*) if*A*∈*i**j*-*G**W**O*(*X*).

In view of Theorems 8 and 10, the next theorem is valid.

###
**Theorem 11**

Let *A* be an *ij*-*gw*closed set with \(A{\subseteq }B{\subseteq }cl_{w_{j}}(A)\), then, \(cl_{w_{j}}(B){\setminus }B\) does not contain nonempty *w*_{i}closed.

###
**Theorem 12**

If *A* is an *ij*-*gw*open set in *X*, then *U*=*X* whenever *U* is *w*_{i}open and \(int_{w_{j}}(A){\cup }(X{\setminus }A){\subseteq }U\).

###
*Proof*

Let *U* be a *w*_{i}open set in *X* and \(int_{w_{j}}(A){\cup }(X{\setminus }A){\subseteq }U\) for any *ij*-*gw*open set *A*. Then, \(X{\setminus }U{\subseteq }(X - int_{w_{j}}(A)){\cap }A\) and so \(X{\setminus }U{\subseteq }cl_{w_{j}}(X{\setminus }A){\setminus }(X{\setminus }A)\). Since *X*∖*A* is *ij*-*gw*closed, then by Theorem 8, we have *X*∖*U*= *∅* and hence *U*=*X*. □

###
**Definition 4**

If \(cl_{w_{j}}(\cup _{\alpha } A_{\alpha })\)=\(\cup _{\alpha } cl_{w_{j}}(A_{\alpha })\), for (*j*=1,2), then a family {*A*_{α}∣*α*∈*Δ*} is called *w*_{j}-locally finite.

###
**Theorem 13**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. If the family {*A*_{α}∣*α*∈*Δ*} is *w*_{j}-locally finite, then the arbitrary union of *ij*-*gw*closed sets *A*_{α},*α*∈*Δ* is an *ij*-*gw*closed set.

###
*Proof*

Direct to prove. □

In the next definition, as an application of *ji*-*gw*open sets, we offer a new type of sets namely *ij*- *σ**g**w*closed sets.

###
**Definition 5**

A subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is called *ij*-strongly generalized *w*closed (briefly, *ij*- *σ**g**w*closed), if \(cl_{w_{j}}(A){\subseteq }U\), whenever *A*⊆*U* and *U* is *ji*-*gw*open. The complement of *ij*- *σ**g**w*closed set is called *ij*- *σ**g**w*open.

The family of all *ij*- *σ**g**w*closed (resp. *ij*- *σ**g**w*open) sets in a bi*w*ss (*X*,*w*_{1},*w*_{2}) will be denoted by *ij*- *σ**G**W**C*(*X*) (resp. *ij*- *σ**G**W**O*(*X*)).

###
**Remark 8**

If *A*∈*i**j*- *σ**G**W**C*(*X*)∩*j**i*- *σ**G**W**C*(*X*), then a subset *A* of a bi*w*ss (*X*,*w*_{1},*w*_{2}) is called pairwise *σ**g**w*closed and its complement is called pairwise *σ**g**w*open.

For brevity the proof of the next proposition is omitted.

###
**Proposition 5**

In a bi*w*ss (*X*,*w*_{1},*w*_{2}), we have the following relation:

*w*_{j}closed set ⇒*ij*- *σ**g**w*closed set ⇒*ij*-*gw*closed set.

###
**Remark 9**

The converse of Proposition *5* is not true as can be seen from the next example.

###
**Example 10**

In Example *6*, one may notice that {4} is 21-*gw*closed set, but it is not 21- *σ**g**w*closed.

###
**Example 11**

In Example *8*. One may notice that, {2} is 12- *σ**g**w*closed set, but it is not *w*_{2}closed.

###
**Theorem 14**

If *A*∈*j**i*- *G**W**O*(*X*)∩*i**j*- *σ**G**W**C*(*X*), then \(cl_{w_{j}}(A)\)=*A*

###
*Proof*

Straightforward. □

###
**Theorem 15**

Let \(cl_{w_{i}}\emptyset \)= *∅*. Then, {*x*}∈*j**i*- *G**W**C*(*X*) or *X*∖{*x*}∈*i**j*- *σ**G**W**C*(*X*), for each *x*∈*X*.

###
*Proof*

Similar to Theorem 7. □

###
**Theorem 16**

If *A*∈*i**j*- *σ**G**W**C*(*X*), then \(cl_{w_{j}}(A){\setminus }A\) contains no nonempty *ji*-*gw*closed.

###
*Proof*

Similar to Theorem 8. □

## Separation axioms in biweak spaces

By using *ij*-*gw*closed, *ij*-*gw*open and *ij*- *σ**g**w*closed sets, we introduce and study the notions of *ij*-\(wT_{\frac {1}{2}}\), *ij*-\(wT_{\frac {1}{2}}^{\sigma }\), *ij*-\(w^{\sigma }T_{\frac {1}{2}}\), *ij*-*w*normal, and *ij*-*gw*normal spaces.

###
**Definition 6**

Let \(cl_{w_{j}}(\emptyset)\)= *∅*. A bi*w*ss (*X*,*w*_{1},*w*_{2}) is called

*ij*-*w**T*_{1}if for each distinct points*x*,*y*∈*X*, there exist a*w*_{i}-open set*U*and*w*_{j}-open set*V*s.t.*x*∈*U*,*y*∉*U*and*y*∈*V*,*x*∉*V*.*ij*-\(wT_{\frac {1}{2}}\) if each*ij*-*gw*closed set*A*of*X*, \(cl_{w_{j}}(A)\)=*A*.*ij*-\(wT_{\frac {1}{2}}^{\sigma }\) if each*ij*-*σ**g**w*closed set*A*of*X*, \(cl_{w_{j}}(A)\)=*A*.

###
**Theorem 17**

A bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*- *w**T*_{1} if every singleton in *X* is *ij*-*w*closed.

###
*Proof*

Let *x*,*y*∈*X* and *x*≠*y*. Then, {*x*},{*y*} are *ij*-*w*closed sets. From Theorem 1, we have \(x \notin cl_{w_{i}}(\{y\})\) and \(y \notin cl_{w_{j}}(\{x\})\). Hence, there exist *w*_{i}-open set *U* containing *x* and *w*_{j}-open set *V* s.t. *x*∈*U*,*y*∉*U*, and *y*∈*V*,*x*∉*V*. Consequently, (*X*,*w*_{1},*w*_{2}) is a *ij*- *w**T*_{1} space. □

In view of Proposition 5, the class of *ij*-\(w T_{\frac {1}{2}}^{\sigma }\) spaces properly contains the class of *ij*-\(w T_{\frac {1}{2}}\) spaces.

###
**Proposition 6**

Every *ij*-\(w T_{\frac {1}{2}}\) space is *ij*-\(w T_{\frac {1}{2}}^{\sigma }\).

The following example supports that the converse of the Proposition 6 is not true in general.

###
**Example 12**

In Example 5, (*X*,*w*_{1},*w*_{2}) is a 21-\(wT^{\sigma }\frac {{~}_{1}}{2}\) space but not 21-\(wT_{\frac {1}{2}}\).

###
**Theorem 18**

Let *X* be a *w*_{i}open set and \(int_{w_{j}}\{x\}\) is *w*_{j}open. A bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-\(wT_{\frac {1}{2}}\) iff {*x*} is *w*_{i}closed or {*x*}=\(int_{w_{j}}\{x\}\) for each *x*∈*X*.

###
*Proof*

Suppose that {*x*} is not *w*_{i}closed for some *x*∈*X*. Then, by using Theorem 7, *X*∖{*x*} is *ij*-*gw*closed. Since (*X*,*w*_{1},*w*_{2}) is *ij*-\(wT_{\frac {1}{2}}\), then {*x*}=\(int_{w_{j}}\{x\}\). On the other hand, let *B* be an *ij*-*gw*closed set. By assumption, {*x*} is *w*_{i}closed or {*x*}=\(int_{w_{j}}\{x\}\) for any \(x{\in }cl_{w_{j}}B\).Case (I): Suppose {*x*} is *w*_{i}closed. If *x*∉*B*, then \(\{x\}{\subseteq }cl_{w_{j}}B {\setminus }B\), which is a contradiction to Theorem 8. Hence *x*∈*B*.Case (II): Suppose {*x*}=\(int_{w_{j}}\{x\}\) and \(x{\in }cl_{w_{j}}B\). Since {*x*}∩*B*≠*∅*, we have *x*∈*B*. Thus, in both cases, we conclude that \(cl_{w_{j}}B\)=*B*. Therefore, (*X*,*w*_{1},*w*_{2}) is *ij*-\(wT_{\frac {1}{2}}\) space. □

###
**Theorem 19**

Suppose \(cl_{w_{i}}\emptyset \)= *∅*. If (*X*,*w*_{1},*w*_{2}) is an *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) space, then {*x*} is *ji*-*gw*closed or {*x*}=\(int_{w_{j}}\{x\}\), for each *x*∈*X*.

###
*Proof*

Follows directly from Theorem 15 and Definition 6. □

###
**Lemma 1**

If {*x*} is *ji*-*gw*closed, then (*X*,*w*_{1},*w*_{2}) is an *ij*-*w*-\(T_{\frac {1}{2}}^{\sigma }\) space, for each *x*∈*X*.

###
*Proof*

Straightforward. □

###
**Definition 7**

A bi*w*ss (*X*,*w*_{1},*w*_{2}) is called

Pairwise \(wT_{\frac {1}{2}}\) if it is both

*ij*-\(wT_{\frac {1}{2}}\) and*ji*-\(wT_{\frac {1}{2}}\).Pairwise \(wT^{\sigma }\frac {{~}_{1}}{2}\) if it is both

*ij*-\(wT_{\frac {1}{2}}^{\sigma }\) and*ji*-\(wT_{\frac {1}{2}}^{\sigma }\).

###
**Proposition 7**

If (*X*,*w*_{1},*w*_{2}) is a pairwise \(wT_{\frac {1}{2}}\) space, then it is pairwise \(wT_{\frac {1}{2}}^{\sigma }\).

###
*Proof*

Uncomplicated. □

###
**Remark 10**

The converse of Proposition *7* is not true as can be seen from the next example.

###
**Example 13**

Let *X*,*w*_{1},*w*_{2} be as in Example *12*. Then, (*X*,*w*_{1},*w*_{2}) is also a 21-\(wT_{\frac {1}{2}}^{\sigma }\) space, and therefore, it is a pairwise \(wT_{\frac {1}{2}}^{\sigma }\) space. But (*X*,*w*_{1},*w*_{2}) is not a pairwise \(wT_{\frac {1}{2}}\) space.

###
**Definition 8**

A bi*w*ss (*X*,*w*_{1},*w*_{2}) is called an *ij*-\(w^{\sigma }T_{\frac {1}{2}}\) if *ij*- *G**W**C*(*X*)=*ij*- *σ**G**W**C*(*X*).

###
**Proposition 8**

Every *ij*-\(wT_{\frac {1}{2}}\) space is *ij*-\(w^{\sigma }T_{\frac {1}{2}}\).

###
*Proof*

Obvious. □

###
**Remark 11**

The converse of Proposition *8* may not be applicable as we see in the next example.

###
**Example 14**

Let *X*= {1,2,3,4}. Define weak structures *w*_{1},*w*_{2} on *X* as follows: *w*_{1}= {*∅*,{1,3},{1,4},{2,3,4}} and *w*_{2}= {*∅*,{2},{1,2},{3,4},{1,3,4}}. Then, (*X*,*w*_{1},*w*_{2}) is an 12-\(w^{\sigma }T_{\frac {1}{2}}\) space but not 12-\(wT_{\frac {1}{2}}\).

###
**Remark 12**

*ij*-\(w^{\sigma }T_{\frac {1}{2}}\) and *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) spaces are independent as may be seen from Example 15 and Example *16*.

###
**Example 15**

Let *w*_{1}= {*∅*,{1},{1,2}},*w*_{2}={*∅*,{3},*X*} be weak structures on *X*= {1,2,3}, then (*X*,*w*_{1},*w*_{2}) is a 12-\(wT^{\sigma }\frac {{~}_{1}}{2}\) space but not 12-\(w^{\sigma }T_{\frac {1}{2}}\).

###
**Example 16**

In Example *14*, (*X*,*w*_{1},*w*_{2}) is an 12-\(w^{\sigma }T_{\frac {1}{2}}\), but it is not 12-*w*-\(T_{\frac {1}{2}}^{\sigma }\).

###
**Theorem 20**

Let \(cl_{w_{j}}(\emptyset)\)= *∅*. A bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-\(wT_{\frac {1}{2}}\) if and only if it is both *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) and *ij*-\(w^{\sigma }T_{\frac {1}{2}}\) space.

###
*Proof*

Suppose that (*X*,*w*_{1},*w*_{2}) is an *ij*-\(wT_{\frac {1}{2}}\) space. Then, by Propositions 6 and 8, (*X*,*w*_{1},*w*_{2}) is both *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) and *ij*-\(w^{\sigma }T_{\frac {1}{2}}\) space. Conversely, suppose that (*X*,*w*_{1},*w*_{2}) is both *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) and *ij*-\(w^{\sigma }T_{\frac {1}{2}}\). Let *A*∈*i**j*- *G**W**C*(*X*). Since (*X*,*w*_{1},*w*_{2}) is an *ij*-\(w^{\sigma }T_{\frac {1}{2}}\) space, *A*∈*i**j*- *σ**G**W**C*(*X*). Since (*X*,*w*_{1},*w*_{2}) is an *ij*-\(wT_{\frac {1}{2}}^{\sigma }\) space, then \(cl_{w_{j}}(A)\)=*A*. Therefore, (*X*,*w*_{1},*w*_{2}) is *ij*-\(wT_{\frac {1}{2}}\). □

###
**Definition 9**

A bi*w*ss (*X*,*w*_{1},*w*_{2}) is called *ij*-*w*normal if for each *w*_{i}closed set *A* and *w*_{j}closed set *B* s.t. *A*∩*B*= *∅*, there are *w*_{j}open set *U* and *w*_{i}open set *V* s.t. *A*⊆*U*,*B*⊆*V*, and *U*∩*V*= *∅*.

###
**Theorem 21**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Consider the following statements:

(

*X*,*w*_{1},*w*_{2}) is*ij*-*w*normal,For each

*w*_{i}closed set*A*and*w*_{j}open set*N*with*A*⊆*N*, there exists*w*_{j}open set*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\),For each

*w*_{i}closed set*A*and each*ij*-*gw*closed set*H*with*A*∩*H*=*∅*, there exist*w*_{j}open set*U*and*w*_{i}open set*V*s.t.*A*⊆*U*,*H*⊆*V*and*U*∩*V*=*∅*,For each

*w*_{i}closed set*A*and*ij*-*gw*open*N*with*A*⊆*N*, there exists*w*_{j}open set*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\).

Then, the implications (1)⇒(2) and (3)⇒(4)⇒(2) are hold.

###
*Proof*

Obvious. □

###
**Theorem 22**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. If \(cl_{w_{i}}(A)\) is *w*_{i}closed for each *w*_{j}open or *ij*-*gw*closed, then the statements in Theorem *21* are equivalent.

###
*Proof*

According to Theorem *21*, we need to prove (2)⇒(1) and (1)⇒(3) only. (2)⇒(1): Let *A* be a *w*_{i}closed set and *B* be a *w*_{j}closed set with *A*∩*B*= *∅*. Then, *X*∖*B* is a *w*_{j}open set with *A*⊆*X*∖*B*. Thus, by (2) there exists *w*_{j} open set *U* s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }X{\setminus }B\). Hence *A*⊆*U* and \(B{\subseteq }X{\setminus }cl_{w_{i}}(U)\). Since \(cl_{w_{i}}(U)\) is *w*_{i}closed for each *w*_{j}open *U*, then \(X{\setminus }cl_{w_{i}}(U)\)=*V* is *w*_{i}open and *U*∩*V*= *∅*. Hence (*X*,*w*_{1},*w*_{2}) is *ij*-*w*normal. (1)⇒(3): Let *A* be a *w*_{i}closed set and *H* be an *ij*-*gw*closed set with *A*∩*H*= *∅*. Then, *H*⊆*X*∖*A*. From Definition 3, we have \(cl_{w_{j}}(H){\subseteq }X{\setminus }A\). Since *H* is *ij*-*gw*closed, then \(cl_{w_{j}}(H)\) is *w*_{j}closed. Since \(A{\cap }cl_{w_{j}}(H)\)= *∅*, then from (1) there exist *w*_{j}open set *U* and *w*_{i}open set *V* s.t. \(A{\subseteq }U, H{\subseteq }cl_{w_{j}}(H){\subseteq }V\) and *U*∩*V*= *∅*. □

###
**Theorem 23**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Consider the following statements:

(

*X*,*w*_{1},*w*_{2}) is*ij*-*w*normal,For each

*w*_{i}closed set*A*and*w*_{j}closed set*B*s.t.*A*∩*B*=*∅*, there exist*ij*-*gw*open*U*and*ji*-*gw*open*V*s.t.*A*⊆*U*,*B*⊆*V*and*U*∩*V*=*∅*,For each

*w*_{i}closed set*A*and*w*_{j}open*N*with*A*⊆*N*, there exists*ij*-*gw*open*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\).

Then, the implication (1)⇒(2)⇒(3) is hold.

###
*Proof*

(1)⇒(2): Let *A* be a *w*_{i}closed set and *B* be a *w*_{j}closed set with *A*∩*B*= *∅*. Since (*X*,*w*_{1},*w*_{2}) is *ij*-*w*normal, then there exist *w*_{j}open set *U* and *w*_{i}open set *V* s.t. *A*⊆*U*,*B*⊆*V* and *U*∩*V*= *∅*. From Corollary 1, there exist *ij*-*gw*open *U* and *ji*-*gw*open *V* s.t. *A*⊆*U*,*B*⊆*V* and *U*∩*V*= *∅*. (2)⇒(3): Let *A* be a *w*_{i}closed set and *N* be a *w*_{j}open set with *A*⊆*N*. Then, *A*∩*X*∖*N*= *∅*. From (2), there exist *ij*-*gw*open *U* and *ji*-*gw*open *V* s.t. *A*⊆*U*,*X*∖*N*⊆*V*, and *U*∩*V*= *∅*. Since *X*∖*V* is *ji*-*gw*closed, *N* is *w*_{j}open, and *X*∖*V*⊆*N*, then from Definition 3, we have \(cl_{w_{i}}(X{\setminus }V){\subseteq }N\). Since *U*⊆*X*∖*V*, hence \(U{\subseteq }cl_{w_{i}}(U){\subseteq }cl_{w_{i}}(X {\setminus } V)\). Consequently, \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\). □

###
**Theorem 24**

Let (*X*,*w*_{1},*w*_{2}) be an *ij*-\(wT_{\frac {1}{2}}\). If \(cl_{w_{i}}(U)\) is *w*_{i}closed for each *ij*-*gw*closed and \(int_{w_{j}}(U)\) is *w*_{j}open for each *ij*-*gw*closed *U*, then the statements in Theorem *23* are equivalent.

###
*Proof*

According to Theorem 23, we need to prove (3)⇒(1). (3)⇒(1): Let *A* be a *w*_{i}closed set and *B* be a *w*_{j}closed set with *A*∩*B*= *∅*. Take *N*= *X*∖*B*, then by using (3) there exists *ij*-*gw*open *U* s.t. \(A \subseteq U \subseteq cl_{w_{i}}(U) \subseteq N\). Since (*X*,*w*_{1},*w*_{2}) is an *ij*-\(wT_{\frac {1}{2}}\) space, then, \(int_{w_{j}}(U)\)=*U*. By assumption *U* is *w*_{j}open. Also, \(X{\setminus }cl_{w_{i}}(U)\) is *w*_{i}open and \(B{\subseteq }X{\setminus }cl_{w_{i}}(U)\). □

###
**Definition 10**

A bi*w*ss (*X*,*w*_{1},*w*_{2}) is called *ij*-*gw*normal if for each *ji*-*gw*closed set *A* and *ij*-*gw*closed set *B* s.t. *A*∩*B*= *∅*, there are *w*_{j}open set *U* and *w*_{i}open set *V* s.t. *A*⊆*U*,*B*⊆*V* and *U*∩*V*= *∅*.

###
**Remark 13**

It is clear that every *ij*-*gw*normal space is *ij*-*w*normal. It can be checked that the converse is not true by the following example.

###
**Theorem 25**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Consider the following statements:

(

*X*,*w*_{1},*w*_{2}) is*ij*-*gw*normal,For each

*ji*-*gw*closed set*A*and*ij*-*gw*open set*N*with*A*⊆*N*, there exists*w*_{j}open set*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\),For each

*ji*-*gw*closed set*A*and*ij*-*gw*closed set*B*s.t.*A*∩*B*=*∅*, there exist*w*_{j}open set*U*s.t.*A*⊆*U*and \(cl_{w_{i}}(U){\cap }B\)=*∅*.

Then, the implication (1)⇒(2)⇒(3) is hold.

###
*Proof*

Obvious. □

###
**Remark 14**

If \(cl_{w_{i}}(U)\) is *w*_{i}closed for each *w*_{i}open set *U*, then the statements in Theorem *25* are equivalent.

###
**Theorem 26**

Let (*X*,*w*_{1},*w*_{2}) be a bi*w*ss. Consider the following statements:

(

*X*,*w*_{1},*w*_{2}) is*ij*-*gw*normal,For each

*ji*-*gw*closed set*A*and*ij*-*gw*closed set*B*s.t.*A*∩*B*=*∅*, there exist*ij*-*σ**g**w*open set*U*,*ji*-*σ**g**w*open set*V*s.t.*A*⊆*U*,*B*⊆*V*and*U*∩*V*=*∅*,For each

*ji*-*gw*closed set*A*and*ij*-*gw*open set*N*with*A*⊆*N*, there exists*ij*-*σ**g**w*open set*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\).

Then, the implication (1)⇒(2)⇒(3) is hold.

###
*Proof*

(1)⇒(2) Follows directly from Proposition 5. (2)⇒(3) Let *A* be a *ji*-*gw*closed set and *N* be an *ij*-*gw*open set with *A*⊆*N*. Take *B*= *X*∖*N*. Then, by assumption, there exist *ij*- *σ**g**w*open set *U*, *ji*- *σ**g**w*open set *V* s.t. *A*⊆*U*,*B*⊆*V* and *U*∩*V*= *∅*. Hence, *U*⊆*X*∖*V*,*X*∖*V*⊆*N*. Since *X*∖*V* is *ji*- *σ**g**w*closed, then \(cl_{w_{i}}(X{\setminus }V){\subseteq }N\) and so \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\). □

The question that comes to our mind, under what conditions can be achieved parity in Theorem 26.

###
**Theorem 27**

Let (*X*,*w*_{1},*w*_{2}) be an *ij*-*w*-\(T_{\frac {1}{2}}^{\sigma }\) space. If \(int_{w_{j}}(U)\) is *w*_{j}open and \(int_{w_{i}}(U)\) is *w*_{i}open for each *ij*- *σ**g**w*open set *U*, then the statements in Theorem *26* are equivalent.

###
*Proof*

Straightforward. □

###
**Corollary 4**

If a bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-*gw*normal, then for each *ji*-*gw*closed set *A* and *ij*- *σ**g**w*open set *N* with *A*⊆*N*, there exists *ij*- *σ**g**w*open set *U* s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\).

###
*Proof*

Obvious from Proposition 5. □

###
**Theorem 28**

If a bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ij*-*gw*normal, then for each *ji*-*gw*closed set *A* and *ij*-*gw*closed set *B* s.t. *A*∩*B*= *∅*, there exist *ij*-*gw*open set *U* and *ji*-*gw*open set *V* s.t. *A*⊆*U*,*B*⊆*V* and *U*∩*V*= *∅*.

###
*Proof*

Clear. □

###
**Theorem 29**

If a bi*w*ss (*X*,*w*_{1},*w*_{2}) is *ji*-*w*-\(T_{\frac {1}{2}}^{\sigma }\) and \(cl_{w_{i}}(\emptyset)\)= *∅*. Consider the following statements:

(

*X*,*w*_{1},*w*_{2}) is*ij*-*gw*normal,For each

*ji*-*gw*closed set*A*and*ij*-*gw*open set*N*with*A*⊆*N*, there exists*ij*-*gw*open set*U*s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\).

Then, the implication (1)⇒(2) is hold.

###
*Proof*

Obvious. □

## Some types of *ij*- (*w*,*w*^{⋆}) continuous functions

In this section, types of continuous functions between biweak spaces are defined and some of their features are established.

###
**Definition 11**

A function \(f : (X, w_{1}, w_{2}){\longrightarrow }(Y, w^{\star }_{1}, w^{\star }_{2})\) is called:

*j*- (*w*,*w*^{⋆})-continuous if for*x*∈*X*and \(w^{\star }_{j}\)open set*V*containing*f*(*x*), there is a*w*_{j}open set*U*containing*x*s.t.*f*(*U*)⊆*V*.*ij*-*g*(*w*,*w*^{⋆})-continuous if for*x*∈*X*and \(w^{\star }_{j}\)open set*V*containing*f*(*x*), there is an*ij*-*gw*open set*U*containing*x*s.t.*f*(*U*)⊆*V*.*ij*-*g*(*w*,*w*^{⋆})closed if for each*w*_{j}closed set*B*,*f*(*B*) is*ji*-*g**w*^{⋆}closed set.

We describe *ij*- *g*(*w*,*w*^{⋆})-continuous function in the following part.

###
**Theorem 30**

Let (*X*,*w*_{1},*w*_{2}) be an *ij*-\(wT_{\frac {1}{2}}\) space. If \(int_{w_{j}}(U)\) is *w*_{j}open for each *ij*-*gw*open set *U*, then, a function \(f:(X, w_{1}, w_{2})\longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) is *ij*- *g*(*w*,*w*^{⋆})-continuous iff *f*^{−1}(*V*)=\(int_{w_{j}}f^{-1}(V)\) for each \(w^{\star }_{j}\)open set *V*.

###
*Proof*

(⇒): Let *V* be a \(w^{\star }_{j}\)open set and *x*∈*f*^{−1}(*V*). Since *f* is *ij*- *g*(*w*,*w*^{⋆})-continuous, then there is an *ij*-*gw*open set *U* containing *x* s.t. *f*(*U*)⊆*V*. Hence, *U*⊆*f*^{−1}(*V*). Since (*X*,*w*_{1},*w*_{2}) is an *ij*-\(wT_{\frac {1}{2}}\) space, then \(int_{w_{j}}(U)\)=*U*. From assumptions, *U* is a *w*_{j}open set s.t. *x*∈*U*⊆*f*^{−1}(*V*) and so \(x \in int_{w_{j}}f^{-1}(V)\). Therefore, *f*^{−1}(*V*)=\(int_{w_{j}}f^{-1}(V)\).(⇐): Let *x*∈*X* and *V* be a \(w^{\star }_{j}\)open set in *Y* with *f*(*x*)∈*V*, then *x*∈*f*^{−1}(*V*). Since *f*^{−1}(*V*)=\(int_{w_{j}}f^{-1}(V)\), then there exists *w*_{j}open set *U* s.t. *x*∈*U*⊆*f*^{−1}(*V*). From Corollary 1, *U* is an *ij*-*gw*open set containing *x* s.t. *f*(*U*)⊆*V*. Consequently, *f* is *ij*- *g*(*w*,*w*^{⋆})-continuous. □

###
**Theorem 31**

For a function \(f : (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\), the following are equivalent:

*f*^{−1}(*V*)=\(int_{w_{j}}(f^{-1}(V))\), for every \(w^{\star }_{j}\)open set*V*in*Y*,\(f(cl_{w_{j}}(A)){\subseteq }cl_{w^{\star }_{j}}(f(A))\), for every set

*A*in*X*,\(cl_{w_{j}}(f^{-1}(V)){\subseteq }(f^{-1}(cl_{w^{\star }_{j}}(V))\), for every set

*V*in*Y*,\(f^{-1}(int_{w^{\star }_{j}}(V)){\subseteq }int_{w_{j}}(f^{-1}(V))\), for every set

*V*in*Y*,\(cl_{w_{j}}(f^{-1}(F))\)=

*f*^{−1}(*F*), for every \(w^{\star }_{j}\)closed set*F*in*Y*.

###
*Proof*

Obvious. □

###
**Theorem 32**

For any function \(f : (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\), every *j*- (*w*,*w*^{⋆})-continuous function is *ij*- *g*(*w*,*w*^{⋆})-continuous.

###
*Proof*

Obvious from Theorem 30. □

###
**Remark 15**

The following example justifies the converse of the Theorem *32* need not to be true in general.

###
**Example 17**

Let *X*= {*a*,*b*,*c*,*d*}, *Y*= {1,2,3}, *w*_{1}=\(\{\emptyset, \{a\}, \{a, d\}\}, w_{2}=\{\emptyset, \{a, b\}, \{c, d\}\}, w^{\star }_{1}\)= {*∅*,{1},{2,3}}, and \(w^{\star }_{2}\)= {*∅*,{2},{1,2}}. If *f* is defined by *f*(*a*)= *f*(*b*)=2, *f*(*c*)=1, *f*(*d*)=3, we have *f* is 12- *g*(*w*,*w*^{⋆})-continuous, but it is not 2- (*w*,*w*^{⋆})-continuous.

###
**Proposition 9**

For any surjection function \(f: (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2}),\) the following are equivalent.

*f*is an*ij*-*g*(*w*,*w*^{⋆})closed function.For any set

*B*in*Y*and each*w*_{i}open*U*s.t.*f*^{−1}(*B*)⊆*U*, there exists*ij*-*g**w*^{⋆}open set*V*of*Y*s.t.*B*⊆*V*and*f*^{−1}(*V*)⊆*U*.

###
*Proof*

(1)⇒(2): Let *B*⊆*Y* and *U* be a *w*_{i}open set s.t. *f*^{−1}(*B*)⊆*U*. Since *f* is an *ij*- *g*(*w*,*w*^{⋆})closed function, then *f*(*U*) is an *ij*- *g**w*^{⋆}open set in *Y*. Take *f*^{−1}(*V*)=*U*. Since *f* is a surjection function and *f*^{−1}(*B*)⊆*U*, then *B*= *f*(*f*^{−1}(*B*))⊆*f*(*U*)=*V*. (2)⇒(1): Let *U* be a *w*_{i}open set, *F*⊆*f*(*U*) s.t. *F* is a \(w^{\star }_{i}\)closed set, then *f*^{−1}(*F*)⊆*U*. This implies that there exists *ij*- *g**w*^{⋆}open set *V* in *Y* s.t. *F*⊆*V* and *f*^{−1}(*V*)⊆*U*. Consequently, \(F{\subseteq }int_{w^{\star }_{j}}(V)\) and so \(F{\subseteq }int_{w^{\star }_{j}}(f(U))\). This implies that *f*(*U*) is *ij*- *g**w*^{⋆}open in *Y*. Therefore, *f* is an *ij*- *g*(*w*,*w*^{⋆})closed function. □

###
**Theorem 33**

Let \((Y, w^{\star }_{1}, w^{\star }_{2})\) be an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space. If \(int_{w^{\star }_{j}}(A)\) is \(w^{\star }_{i}\)open for each *ij*- *g**w*^{⋆}open set *A*. If \(f :(X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) is a surjection *ij*- *g*(*w*,*w*^{⋆})closed and *ij*- *g*(*w*,*w*^{⋆})-continuous function, then *f*^{−1}(*B*) is *ij*-*gw*closed set of *X* for every *ij*- *g**w*^{⋆}closed set *B* of *Y*.

###
*Proof*

Let *B*⊆*Y* be an *ij*- *g**w*^{⋆}closed set. Let *U* be a *w*_{i}open set of *X* s.t. *f*^{−1}(*B*)⊆*U*. Since *f* is a surjection *ij*- *g*(*w*,*w*^{⋆})closed function, then by Proposition 9, there exists *ij*- *g**w*^{⋆}open set *V* of *Y* s.t. *B*⊆*V* and *f*^{−1}(*V*)⊆*U*. Since \((Y, w^{\star }_{1}, w^{\star }_{2})\) is an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space, then \(int_{w^{\star }_{j}}(V)\)=*V*. From assumptions, *V* is a \(w^{\star }_{i}\)open set. Since *B* is *ij*- *g**w*^{⋆}closed, then \(cl_{w^{\star }_{j}}(B) \subseteq V\). Hence, \(f^{-1}(cl_{w^{\star }_{j}}(B)) \subseteq f^{-1}(V) \subseteq U\). By Theorems 30 and 31, \(cl_{w_{j}}f^{-1}(B) \subseteq U\), and hence, *f*^{−1}(*B*) is *ij*-*gw*closed set in *X*. □

###
**Lemma 2**

Let \((Y, w^{\star }_{1}, w^{\star }_{2})\) be an *ji*-\(w^{\star } T_{\frac {1}{2}}\) space. If \(f :(X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) is an *ij*- *g*(*w*,*w*^{⋆})closed function, then \(cl_{w^{\star }_{i}}f(A)\)=\(f(cl_{w_{j}}(A))\), for every *w*_{j}closed set *A* in *X*.

###
*Proof*

Let *A* be a *w*_{j}closed set in *X*, then *A*=\(cl_{w_{j}}(A)\). Since *f* is an *ij*- *g*(*w*,*w*^{⋆})closed function, then *f*(*A*) is *ji*- *g**w*^{⋆}closed set since \((Y, w^{\star }_{1}, w^{\star }_{2})\) is a *ji*-\(w^{\star } T_{\frac {1}{2}}\) space, then \(cl_{w^{\star }_{i}}f(A)\)= *f*(*A*). Hence, \(cl_{w^{\star }_{i}}f(A)\)=\(f(cl_{w_{j}}(A))\). □

###
**Lemma 3**

Let \((Y, w^{\star }_{1}, w^{\star }_{2})\) be an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space. and \(f: (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) be a *ji*- *g*(*w*,*w*^{⋆})closed function. If \(cl_{w_{j}}(A)\) is a *w*_{i}closed set for each set *A* in *X*, then \(cl_{w^{\star }_{j}}f(A) \subseteq f(cl_{w_{j}}(A))\).

###
*Proof*

Suppose \(cl_{w_{j}}(A)\) is a *w*_{i}closed set in *X*. Since *f* is an *ji*- *g*(*w*,*w*^{⋆})closed function, then \(f(cl_{w_{j}}(A))\) is *ij*- *g**w*^{⋆}closed set containing *f*(*A*). Since \((Y, w^{\star }_{1}, w^{\star }_{2})\) is an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space, then \(cl_{w^{\star }_{j}}f(cl_{w_{j}}(A))\)=\(f(cl_{w_{j}}(A))\). Hence, \(cl_{w^{\star }_{j}}f(A){\subseteq }f(cl_{w_{j}}(A))\). □

###
**Theorem 34**

Let \((Y, w^{\star }_{1}, w^{\star }_{2})\) be an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space. If \(int_{w_{i}}f^{-1}(U)\) is *w*_{i}open for each \(w^{\star }_{i}\)open set *U* in *Y* and \(cl_{w_{j}}(A)\) is a *w*_{i}closed set for each set *A* in *X*. If \(f : (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) is a *ji*- *g*(*w*,*w*^{⋆})closed and *ji*- *g*(*w*,*w*^{⋆})-continuous function, then *f*(*A*) is *ij*- *g**w*^{⋆}closed set of *Y* for every *ij*-*gw*closed set *A* of *X*.

###
*Proof*

Follows directly from Theorem 30, Theorem 31, and Lemma 3. □

###
**Theorem 35**

Let \((Y, w^{\star }_{1}, w^{\star }_{2})\) be an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space. If \(i_{w^{\star }_{j}}(A)\) is \(w^{\star }_{j}\)open for each *ij*- *g**w*^{⋆}open set *A* of *Y*. If \(f: (X, w_{1}, w_{2}) \longrightarrow (Y, w^{\star }_{1}, w^{\star }_{2})\) and \(h : (Y, w^{\star }_{1}, w^{\star }_{2}) \longrightarrow (Z, \upsilon _{1}, \upsilon _{2})\) are *ij*- *g*(*w*,*w*^{⋆})-continuous and *ij*- *g*(*w*^{⋆},*υ*)-continuous functions, respectively, then *h*∘*f*:(*X*,*w*_{1},*w*_{2})→(*Z*,*υ*_{1},*υ*_{2}) is *ij*- *g*(*w*,*υ*)-continuous.

###
*Proof*

Let *x*∈*X* and *V* be a *υ*_{j}open set of *Z* containing *h*∘*f*(*x*). Since *h* is *ij*- *g*(*w*^{⋆},*υ*)-continuous, then there is an *ij*- *g**w*^{⋆}open set *U* containing *h*(*x*) s.t. *h*(*U*)⊆*V*. Since \((Y, w^{\star }_{1}, w^{\star }_{2})\) is an *ij*-\(w^{\star } T_{\frac {1}{2}}\) space, hence, \(i_{w^{\star }_{j}}(U)\)=*U*. From assumptions, *U* is a \(w^{\star }_{j}\)open for each *ij*- *g**w*^{⋆}open set *U* of *Y* containing *h*(*x*). Since *f* is an *ij*- *g*(*w*,*w*^{⋆})-continuous function, so there is an *ij*-*gw*open set *G* containing *x* s.t. *f*(*G*)⊆*U*. It follows that there exists an *ij*-*gw*open set *G* containing *x* s.t. *h*∘*f*(*G*)⊆*V*. Consequently, *h*∘*f* is *ij*- *g*(*w*,*υ*)-continuous. □

###
**Theorem 36**

If \(f : (X, w_{1}, w_{2}){\longrightarrow }(Y, w^{\star }_{1}, w^{\star }_{2})\) and \(h: (Y, w^{\star }_{1}, w^{\star }_{2}){\longrightarrow }(Z, \upsilon _{1}, \upsilon _{2})\) are *ij*- *g*(*w*,*w*^{⋆})-continuous and *j*- (*w*^{⋆},*υ*)-continuous respectively, then *h*∘*f*:(*X*,*w*_{1},*w*_{2})→ (*Z*,*υ*_{1},*υ*_{2}) is *ij*- *g*(*w*,*υ*)-continuous.

###
*Proof*

Straightforward. □

## Future work

In the future, we intend to introduce the bisoft weak structure spaces and study the notions *ij*-soft *gw* closed, *ij*-soft *gw*open, and *ij*-soft *σ**g**w*closed sets in it. Also, using these sets, diverse classes of mappings on soft biweak structures can be examined. Further, we suggest studying the properties of some kinds of *ij*-*gw*closed subsets with respect to a biweak structure modified by elements of an ideal or a hereditary class. Accordingly, we construct a kind of continuity depending on the new class of *ij*-*gw*closed subsets. Moreover, one may take research to find the suitable way of defining the biweak structure spaces associated to the digraphs by using *ij*-*gw*closed such that there is a one-to-one correspondence between them. It may also lead to the new properties of separation axioms on these spaces. It will be necessary to perform more research to strengthen a comprehensive framework for the practical applications.

## Availability of data and materials

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

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The authors thank the reviewers for their useful comments that led to the improvement of the original manuscript.

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Abu-Donia, H.M., Hosny, R.A. Generalized *w*closed sets in biweak structure spaces.
*J Egypt Math Soc* **28, **24 (2020). https://doi.org/10.1186/s42787-020-00084-6

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DOI: https://doi.org/10.1186/s42787-020-00084-6

### Keywords

- Biweak structures
*ij*-*gw*closed sets*ij*-*g*(*w*,*w*^{⋆})-continuous functions*ij*-*w*normal spaces

### AMS classification

- 54A05– 54C08– 54C10–54C20