Generalized wclosed sets in biweak structure spaces

Abstract

As a generalization of the classes of gwclosed (resp. gwopen, sgwclosed) sets in a weak structure space (X,w), the notions of ij-generalized wclosed (resp. ij-generalized wopen, ij-strongly generalized wclosed) sets in a biweak structure space (X,w₁,w₂) are introduced. In terms of these concepts, new forms of continuous function between biweak spaces are constructed. Additionally, the concepts of ij-wnormal, ij-gwnormal, 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 wclosed 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 w₁,w₂ on X. A triple (X,w₁,w₂) is called a biweak structure space (in short, biwss).

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 biwss (X,w₁,w₂) is called ij-wclosed if $cl_{w_{i}}(cl_{w_{j}}(A))$=A, where i,j= 1 or 2 and i≠j. The complement of an ij-wclosed set is called ij-wopen.

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 wclosed 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 wclosed (written henceforth as ij-gwclosed), ij-generalized wopen (written henceforth as ij-gwopen), and ij-strongly generalized wclosed (ij- σgwclosed, for short) sets in a biwss (X,w₁,w₂) as a generalization of the concept of gwclosed, gwopen, and sgwclosed sets, respectively, in a weak structure space (X,w) which presented in [8] and determining some of their behaviors. In terms of ij-gwclosed and ij-gwopen 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 biwss 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 wclosed (gwclosed, for short) if cl_w(A)⊆U, whenever A⊆U and U is wopen.

• The complement of a generalized wclosed set is called generalized wopen (gwopen, for short), i.e, a subset A is gwopen if and only if int_w(A)⊇F, whenever A⊇F and F is wclosed.

The family of all gwclosed (resp. gwopen) sets in a weak structure X will be denoted by GWC(X) (resp. GWO(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 wopen 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)= int_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,

• cl_w(f⁻¹(F))= f⁻¹(F), for every w^⋆closed set F in Y.

Theorem 2

[9] Let (X,w₁,w₂) be a biwss and A be a subset of X. Then, the following are equivalent:

• A is ij-wclosed,

• 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,w₁,w₂) be a biwss and A⊆X. Then, A is a ij-wclosed set, if A is both w_iclosed and w_jclosed, where i,j= 1 or 2 and i≠j.

Proposition 2

[9] Let (X,w₁,w₂) be a biwss. If A_α is ij-wclosed for all α∈Λ≠∅, then ∩_α∈ΛA_α is ij-wclosed and the union of two ij-wclosed sets is not a ij-wclosed 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-gwclosed sets

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

Definition 3

A subset A of a biwss (X,w₁,w₂) is called ij-generalized wclosed (ij-gwclosed, for short) if $cl_{w_{j}}(A){\subseteq }U$, whenever A⊆U and U is w_iopen. The complement of ij-gwclosed set is called ij-gwopen.

The family of all ij-gwclosed (resp. ij-gwopen) sets in a biwss (X,w₁,w₂) will be denoted by ij- GWC(X) (resp. ij- GWO(X)).

Remark 1

If A∈ij- GWC(X)∩ji- GWC(X), then a subset A of a biwss (X,w₁,w₂) is called pairwise gwclosed and its complement is pairwise gwopen.

Example 1

Let X= {1,2,3},w₁={∅,{1},{1,2}}, and w₂= {∅,{3}}. A set {3} is pairwise gwclosed.

Certainly, the next theorems are obtained:

Theorem 3

A subset A of a biwss (X,w₁,w₂) is ij-gwopen iff $int_{w_{j}}(A){\supseteq }F$, whenever A⊇F and F is w_iclosed.

Theorem 4

If A is an ij-gwclosed and w_iopen set in (X,w₁,w₂), then A=$cl_{w_{j}}(A)$.

Theorem 5

Every w_jclosed set in a biwss (X,w₁,w₂) is ij-gwclosed.

Proof

Let A be a w_jclosed set and U be a w_iopen set in X s.t. A⊆U. Then, $cl_{w_{j}}(A)$=A. It implies that A∈ij- GWC(X). □

Corollary 1

If A is a w_jopen set in a biwss (X,w₁,w₂), then A∈ij- GWO(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-gwclosed and not w₂closed.

Proposition 3

Let (X,w₁,w₂) be a biwss. Then,

• If X∈w_j and each w_iopen set is w_jclosed, then, A∈ij- GWC(X), for each A⊂X.

• A∈ij- GWC(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∈ij- GWC(X), for each A⊂X. Then, every w_iopen set U, A∈ij- GWC(X). If U⊆U, hence $cl_{w_{j}}(U){\subseteq }U$. Thus, $cl_{w_{j}}(U)$=U, for each w_iopen set U. Conversely, suppose that A⊆U and U be a w_iopen set. Then, $cl_{w_{j}}(A){\subseteq }cl_{w_{j}}(U)$. From assumption, $cl_{w_{j}}(A){\subseteq }U$ and so A∈ij- GWC(X). □

Remark 3

In the biwss (X,w₁,w₂), 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},w₁={∅,{2},{1,3}}, and w₂= {∅,X,{1},{2},{3},{1,2},{2,3}}. One may notice that every subset of X is 12-gwclosed, but A={ 2} is a w₁open set in X and it is not w₂closed.

Remark 4

In general, 21- GWC(X)≠12- GWC(X) as in Example 3.

Proposition 4

Let (X,w₁,w₂) be a biwss. If w₁⊆w₂, then 21- GWC(X)⊆12- GWC(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- GWC(X)⊆12- GWC(X), but w₁⫅̸w₂.

Now, one can conclude attitudes relative to the union as well as the intersection of two ij-gwclosed sets in a biwss (X,w₁,w₂).

Example 5

Let X= {1,2,3,4},w₁={∅,{3},{1,3},{1,3,4},{1,2,4}} and w₂ = {∅,{2},{3},{2,3,4}}. Let us consider A= {2} and B= {3}. Note that A and B are 21-gwclosed sets but its union is not 21-gwclosed.

Example 6

Let X= {1,2,3},w₁={∅,{1},{3}} and w₂= {∅,{1}}. Consider two 21-gwclosed sets A= {1,2} and B= {1,3}, then A∩B= {1} is not 21-gwclosed.

Theorem 6

Let (X,w₁,w₂) be a biwss and $cl_{w_{j}}(\emptyset)$= ∅. Then, the family of all ij-gwclosed sets is a biminimal structure in X.

Proof

Obvious. □

Theorem 7

Suppose X∈w_i. Then, {x} is w_iclosed or X∖{x}∈ij- GWC(X), for each x∈X.

Proof

Suppose that the singleton {x} is not w_iclosed for some x∈X. Then, X∖{x} is not w_iopen. Since X is w_iopen set and X∖{x}⊆X. Hence, X∖{x}∈ij- GWC(X). □

Theorem 8

If A∈ij- GWC(X), then $cl_{w_{j}}(A){\setminus }A$ contains no nonempty w_iclosed.

Proof

For an ij-gwclosed set A, let S be a nonempty w_iclosed 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_iopen and A is ij-gwclosed, 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_iclosed. □

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 w₂closed contained in $c_{w_{1}}(A){\setminus }A$. But A∉21- GWC(X).

It thus follows from Theorem 8 that

Corollary 2

If A∈ij- GWC(X) and $cl_{w_{j}}(A){\setminus }A$ is a w_iclosed set, then $cl_{w_{j}}(A)$=A.

Remark 6

If A is an ij-gwclosed set in a biwss (X,w₁,w₂) and $cl_{w_{j}}(A)$=A, then $cl_{w_{j}}(A){\setminus }A$ need not to be w_iclosed as shown by the following example.

Example 8

Let X= {1,2,3},w₁={∅,{2}}, and w₂= {∅,{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 w₁closed.

Theorem 9

If A∈ij- GWC(X), then $cl_{w_{j}}(A){\setminus }A{\in }ij$- GWO(X).

Proof

Let A∈ij- GWC(X) and F be a w_iclosed 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 $cl_{w_{j}}(A){\setminus }A{\in }ij$- GWO(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 $cl_{w_{1}}(A){\setminus }A{\in }21$- GWO(X), but A∉21- GWC(X).

Theorem 10

If A∈ij- GWC(X) and $A{\subseteq }B{\subseteq }cl_{w_{j}}(A)$, then B∈ij- GWC(X).

Proof

Let U be any w_iopen set s.t. B⊆U. Since A⊆B and A∈ij- GWC(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∈ij- GWC(X). □

Corollary 3

Let (X,w₁,w₂) be a biwss. Then,

• If A∈ij- GWO(X) and $int_{w_{j}}(A){\subseteq }B{\subseteq }A$, then, B∈ij- GWO(X).

• $cl_{w_{j}}(A){\in }ij$- GWC(X) if A∈ij- GWC(X).

• $int_{w_{j}}(A){\in }ij$- GWO(X) if A∈ij- GWO(X).

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

Theorem 11

Let A be an ij-gwclosed set with $A{\subseteq }B{\subseteq }cl_{w_{j}}(A)$, then, $cl_{w_{j}}(B){\setminus }B$ does not contain nonempty w_iclosed.

Theorem 12

If A is an ij-gwopen set in X, then U=X whenever U is w_iopen and $int_{w_{j}}(A){\cup }(X{\setminus }A){\subseteq }U$.

Proof

Let U be a w_iopen set in X and $int_{w_{j}}(A){\cup }(X{\setminus }A){\subseteq }U$ for any ij-gwopen 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-gwclosed, 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,w₁,w₂) be a biwss. If the family {A_α∣α∈Δ} is w_j-locally finite, then the arbitrary union of ij-gwclosed sets A_α,α∈Δ is an ij-gwclosed set.

Proof

Direct to prove. □

In the next definition, as an application of ji-gwopen sets, we offer a new type of sets namely ij- σgwclosed sets.

Definition 5

A subset A of a biwss (X,w₁,w₂) is called ij-strongly generalized wclosed (briefly, ij- σgwclosed), if $cl_{w_{j}}(A){\subseteq }U$, whenever A⊆U and U is ji-gwopen. The complement of ij- σgwclosed set is called ij- σgwopen.

The family of all ij- σgwclosed (resp. ij- σgwopen) sets in a biwss (X,w₁,w₂) will be denoted by ij- σGWC(X) (resp. ij- σGWO(X)).

Remark 8

If A∈ij- σGWC(X)∩ji- σGWC(X), then a subset A of a biwss (X,w₁,w₂) is called pairwise σgwclosed and its complement is called pairwise σgwopen.

For brevity the proof of the next proposition is omitted.

Proposition 5

In a biwss (X,w₁,w₂), we have the following relation:

w_jclosed set ⇒ij- σgwclosed set ⇒ij-gwclosed 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-gwclosed set, but it is not 21- σgwclosed.

Example 11

In Example 8. One may notice that, {2} is 12- σgwclosed set, but it is not w₂closed.

Theorem 14

If A∈ji- GWO(X)∩ij- σGWC(X), then $cl_{w_{j}}(A)$=A

Proof

Straightforward. □

Theorem 15

Let $cl_{w_{i}}\emptyset$= ∅. Then, {x}∈ji- GWC(X) or X∖{x}∈ij- σGWC(X), for each x∈X.

Proof

Similar to Theorem 7. □

Theorem 16

If A∈ij- σGWC(X), then $cl_{w_{j}}(A){\setminus }A$ contains no nonempty ji-gwclosed.

Proof

Similar to Theorem 8. □

Separation axioms in biweak spaces

By using ij-gwclosed, ij-gwopen and ij- σgwclosed 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-wnormal, and ij-gwnormal spaces.

Definition 6

Let $cl_{w_{j}}(\emptyset)$= ∅. A biwss (X,w₁,w₂) is called

• ij- wT₁ 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-gwclosed set A of X, $cl_{w_{j}}(A)$=A.

• ij-$wT_{\frac {1}{2}}^{\sigma }$ if each ij- σgwclosed set A of X, $cl_{w_{j}}(A)$=A.

Theorem 17

A biwss (X,w₁,w₂) is ij- wT₁ if every singleton in X is ij-wclosed.

Proof

Let x,y∈X and x≠y. Then, {x},{y} are ij-wclosed 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,w₁,w₂) is a ij- wT₁ 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,w₁,w₂) is a 21-$wT^{\sigma }\frac {{~}_{1}}{2}$ space but not 21-$wT_{\frac {1}{2}}$.

Theorem 18

Let X be a w_iopen set and $int_{w_{j}}\{x\}$ is w_jopen. A biwss (X,w₁,w₂) is ij-$wT_{\frac {1}{2}}$ iff {x} is w_iclosed or {x}=$int_{w_{j}}\{x\}$ for each x∈X.

Proof

Suppose that {x} is not w_iclosed for some x∈X. Then, by using Theorem 7, X∖{x} is ij-gwclosed. Since (X,w₁,w₂) is ij-$wT_{\frac {1}{2}}$, then {x}=$int_{w_{j}}\{x\}$. On the other hand, let B be an ij-gwclosed set. By assumption, {x} is w_iclosed or {x}=$int_{w_{j}}\{x\}$ for any $x{\in }cl_{w_{j}}B$.Case (I): Suppose {x} is w_iclosed. 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,w₁,w₂) is ij-$wT_{\frac {1}{2}}$ space. □

Theorem 19

Suppose $cl_{w_{i}}\emptyset$= ∅. If (X,w₁,w₂) is an ij-$wT_{\frac {1}{2}}^{\sigma }$ space, then {x} is ji-gwclosed 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-gwclosed, then (X,w₁,w₂) is an ij-w-$T_{\frac {1}{2}}^{\sigma }$ space, for each x∈X.

Proof

Straightforward. □

Definition 7

A biwss (X,w₁,w₂) 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,w₁,w₂) 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,w₁,w₂ be as in Example 12. Then, (X,w₁,w₂) is also a 21-$wT_{\frac {1}{2}}^{\sigma }$ space, and therefore, it is a pairwise $wT_{\frac {1}{2}}^{\sigma }$ space. But (X,w₁,w₂) is not a pairwise $wT_{\frac {1}{2}}$ space.

Definition 8

A biwss (X,w₁,w₂) is called an ij-$w^{\sigma }T_{\frac {1}{2}}$ if ij- GWC(X)=ij- σGWC(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 w₁,w₂ on X as follows: w₁= {∅,{1,3},{1,4},{2,3,4}} and w₂= {∅,{2},{1,2},{3,4},{1,3,4}}. Then, (X,w₁,w₂) 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 w₁= {∅,{1},{1,2}},w₂={∅,{3},X} be weak structures on X= {1,2,3}, then (X,w₁,w₂) is a 12-$wT^{\sigma }\frac {{~}_{1}}{2}$ space but not 12-$w^{\sigma }T_{\frac {1}{2}}$.

Example 16

In Example 14, (X,w₁,w₂) 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 biwss (X,w₁,w₂) 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,w₁,w₂) is an ij-$wT_{\frac {1}{2}}$ space. Then, by Propositions 6 and 8, (X,w₁,w₂) is both ij-$wT_{\frac {1}{2}}^{\sigma }$ and ij-$w^{\sigma }T_{\frac {1}{2}}$ space. Conversely, suppose that (X,w₁,w₂) is both ij-$wT_{\frac {1}{2}}^{\sigma }$ and ij-$w^{\sigma }T_{\frac {1}{2}}$. Let A∈ij- GWC(X). Since (X,w₁,w₂) is an ij-$w^{\sigma }T_{\frac {1}{2}}$ space, A∈ij- σGWC(X). Since (X,w₁,w₂) is an ij-$wT_{\frac {1}{2}}^{\sigma }$ space, then $cl_{w_{j}}(A)$=A. Therefore, (X,w₁,w₂) is ij-$wT_{\frac {1}{2}}$. □

Definition 9

A biwss (X,w₁,w₂) is called ij-wnormal if for each w_iclosed set A and w_jclosed set B s.t. A∩B= ∅, there are w_jopen set U and w_iopen set V s.t. A⊆U,B⊆V, and U∩V= ∅.

Theorem 21

Let (X,w₁,w₂) be a biwss. Consider the following statements:

• (X,w₁,w₂) is ij-wnormal,

• For each w_iclosed set A and w_jopen set N with A⊆N, there exists w_jopen set U s.t. $A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N$,

• For each w_iclosed set A and each ij-gwclosed set H with A∩H= ∅, there exist w_jopen set U and w_iopen set V s.t. A⊆U,H⊆V and U∩V= ∅,

• For each w_iclosed set A and ij-gwopen N with A⊆N, there exists w_jopen 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,w₁,w₂) be a biwss. If $cl_{w_{i}}(A)$ is w_iclosed for each w_jopen or ij-gwclosed, 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_iclosed set and B be a w_jclosed set with A∩B= ∅. Then, X∖B is a w_jopen 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_iclosed for each w_jopen U, then $X{\setminus }cl_{w_{i}}(U)$=V is w_iopen and U∩V= ∅. Hence (X,w₁,w₂) is ij-wnormal. (1)⇒(3): Let A be a w_iclosed set and H be an ij-gwclosed 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-gwclosed, then $cl_{w_{j}}(H)$ is w_jclosed. Since $A{\cap }cl_{w_{j}}(H)$= ∅, then from (1) there exist w_jopen set U and w_iopen set V s.t. $A{\subseteq }U, H{\subseteq }cl_{w_{j}}(H){\subseteq }V$ and U∩V= ∅. □

Theorem 23

Let (X,w₁,w₂) be a biwss. Consider the following statements:

• (X,w₁,w₂) is ij-wnormal,

• For each w_iclosed set A and w_jclosed set B s.t. A∩B= ∅, there exist ij-gwopen U and ji-gwopen V s.t. A⊆U,B⊆V and U∩V= ∅,

• For each w_iclosed set A and w_jopen N with A⊆N, there exists ij-gwopen 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_iclosed set and B be a w_jclosed set with A∩B= ∅. Since (X,w₁,w₂) is ij-wnormal, then there exist w_jopen set U and w_iopen set V s.t. A⊆U,B⊆V and U∩V= ∅. From Corollary 1, there exist ij-gwopen U and ji-gwopen V s.t. A⊆U,B⊆V and U∩V= ∅. (2)⇒(3): Let A be a w_iclosed set and N be a w_jopen set with A⊆N. Then, A∩X∖N= ∅. From (2), there exist ij-gwopen U and ji-gwopen V s.t. A⊆U,X∖N⊆V, and U∩V= ∅. Since X∖V is ji-gwclosed, N is w_jopen, 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,w₁,w₂) be an ij-$wT_{\frac {1}{2}}$. If $cl_{w_{i}}(U)$ is w_iclosed for each ij-gwclosed and $int_{w_{j}}(U)$ is w_jopen for each ij-gwclosed 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_iclosed set and B be a w_jclosed set with A∩B= ∅. Take N= X∖B, then by using (3) there exists ij-gwopen U s.t. $A \subseteq U \subseteq cl_{w_{i}}(U) \subseteq N$. Since (X,w₁,w₂) is an ij-$wT_{\frac {1}{2}}$ space, then, $int_{w_{j}}(U)$=U. By assumption U is w_jopen. Also, $X{\setminus }cl_{w_{i}}(U)$ is w_iopen and $B{\subseteq }X{\setminus }cl_{w_{i}}(U)$. □

Definition 10

A biwss (X,w₁,w₂) is called ij-gwnormal if for each ji-gwclosed set A and ij-gwclosed set B s.t. A∩B= ∅, there are w_jopen set U and w_iopen set V s.t. A⊆U,B⊆V and U∩V= ∅.

Remark 13

It is clear that every ij-gwnormal space is ij-wnormal. It can be checked that the converse is not true by the following example.

Theorem 25

Let (X,w₁,w₂) be a biwss. Consider the following statements:

• (X,w₁,w₂) is ij-gwnormal,

• For each ji-gwclosed set A and ij-gwopen set N with A⊆N, there exists w_jopen set U s.t. $A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N$,

• For each ji-gwclosed set A and ij-gwclosed set B s.t. A∩B= ∅, there exist w_jopen 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_iclosed for each w_iopen set U, then the statements in Theorem 25 are equivalent.

Theorem 26

Let (X,w₁,w₂) be a biwss. Consider the following statements:

• (X,w₁,w₂) is ij-gwnormal,

• For each ji-gwclosed set A and ij-gwclosed set B s.t. A∩B= ∅, there exist ij- σgwopen set U, ji- σgwopen set V s.t. A⊆U,B⊆V and U∩V= ∅,

• For each ji-gwclosed set A and ij-gwopen set N with A⊆N, there exists ij- σgwopen 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-gwclosed set and N be an ij-gwopen set with A⊆N. Take B= X∖N. Then, by assumption, there exist ij- σgwopen set U, ji- σgwopen set V s.t. A⊆U,B⊆V and U∩V= ∅. Hence, U⊆X∖V,X∖V⊆N. Since X∖V is ji- σgwclosed, 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,w₁,w₂) be an ij-w-$T_{\frac {1}{2}}^{\sigma }$ space. If $int_{w_{j}}(U)$ is w_jopen and $int_{w_{i}}(U)$ is w_iopen for each ij- σgwopen set U, then the statements in Theorem 26 are equivalent.

Proof

Straightforward. □

Corollary 4

If a biwss (X,w₁,w₂) is ij-gwnormal, then for each ji-gwclosed set A and ij- σgwopen set N with A⊆N, there exists ij- σgwopen set U s.t. $A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N$.

Proof

Obvious from Proposition 5. □

Theorem 28

If a biwss (X,w₁,w₂) is ij-gwnormal, then for each ji-gwclosed set A and ij-gwclosed set B s.t. A∩B= ∅, there exist ij-gwopen set U and ji-gwopen set V s.t. A⊆U,B⊆V and U∩V= ∅.

Proof

Clear. □

Theorem 29

If a biwss (X,w₁,w₂) is ji-w-$T_{\frac {1}{2}}^{\sigma }$ and $cl_{w_{i}}(\emptyset)$= ∅. Consider the following statements:

• (X,w₁,w₂) is ij-gwnormal,

• For each ji-gwclosed set A and ij-gwopen set N with A⊆N, there exists ij-gwopen 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-gwopen set U containing x s.t. f(U)⊆V.

• ij- g(w,w^⋆)closed if for each w_jclosed set B, f(B) is ji- gw^⋆closed set.

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

Theorem 30

Let (X,w₁,w₂) be an ij-$wT_{\frac {1}{2}}$ space. If $int_{w_{j}}(U)$ is w_jopen for each ij-gwopen 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-gwopen set U containing x s.t. f(U)⊆V. Hence, U⊆f⁻¹(V). Since (X,w₁,w₂) is an ij-$wT_{\frac {1}{2}}$ space, then $int_{w_{j}}(U)$=U. From assumptions, U is a w_jopen 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_jopen set U s.t. x∈U⊆f⁻¹(V). From Corollary 1, U is an ij-gwopen 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}, w₁=$\{\emptyset, \{a\}, \{a, d\}\}, w_{2}=\{\emptyset, \{a, b\}, \{c, d\}\}, w^{\star }_{1}$= {∅,{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_iopen U s.t. f⁻¹(B)⊆U, there exists ij- gw^⋆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_iopen set s.t. f⁻¹(B)⊆U. Since f is an ij- g(w,w^⋆)closed function, then f(U) is an ij- gw^⋆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_iopen 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- gw^⋆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- gw^⋆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- gw^⋆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-gwclosed set of X for every ij- gw^⋆closed set B of Y.

Proof

Let B⊆Y be an ij- gw^⋆closed set. Let U be a w_iopen 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- gw^⋆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- gw^⋆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-gwclosed 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_jclosed set A in X.

Proof

Let A be a w_jclosed set in X, then A=$cl_{w_{j}}(A)$. Since f is an ij- g(w,w^⋆)closed function, then f(A) is ji- gw^⋆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_iclosed 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_iclosed set in X. Since f is an ji- g(w,w^⋆)closed function, then $f(cl_{w_{j}}(A))$ is ij- gw^⋆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_iopen for each $w^{\star }_{i}$open set U in Y and $cl_{w_{j}}(A)$ is a w_iclosed 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- gw^⋆closed set of Y for every ij-gwclosed 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- gw^⋆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,w₁,w₂)→(Z,υ₁,υ₂) is ij- g(w,υ)-continuous.

Proof

Let x∈X and V be a υ_jopen set of Z containing h∘f(x). Since h is ij- g(w^⋆,υ)-continuous, then there is an ij- gw^⋆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- gw^⋆open set U of Y containing h(x). Since f is an ij- g(w,w^⋆)-continuous function, so there is an ij-gwopen set G containing x s.t. f(G)⊆U. It follows that there exists an ij-gwopen 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,w₁,w₂)→ (Z,υ₁,υ₂) 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 gwopen, and ij-soft σgwclosed 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-gwclosed 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-gwclosed subsets. Moreover, one may take research to find the suitable way of defining the biweak structure spaces associated to the digraphs by using ij-gwclosed 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.