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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 wj 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 ij. 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 clw(A)U, whenever AU 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 intw(A)F, whenever AF 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 xX and wopen 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)= intw(f−1(B)), for every wopen 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,

  • clw(f−1(F))= f−1(F), for every wclosed 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 ij.

Proposition 1

[9] Let (X,w1,w2) be a biwss and AX. Then, A is a ij-wclosed set, if A is both wiclosed and wjclosed, where i,j= 1 or 2 and ij.

Proposition 2

[9] Let (X,w1,w2) be a biwss. If Aα is ij-wclosed for all αΛ, then ∩αΛAα is ij-wclosed and the union of two ij-wclosed sets is not a ij-wclosed set, where i,j= 1 or 2 and ij.

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

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 AU and U is wiopen. 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 Aij- 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 AF and F is wiclosed.

Theorem 4

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

Theorem 5

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

Proof

Let A be a wjclosed set and U be a wiopen set in X s.t. AU. Then, \(cl_{w_{j}}(A)\)=A. It implies that Aij- GWC(X). □

Corollary 1

If A is a wjopen set in a biwss (X,w1,w2), then Aij- 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 Xwj and each wiopen set is wjclosed, then, Aij- GWC(X), for each AX.

  • Aij- GWC(X), for each AX iff \(cl_{w_{j}}{U}={U}\) for each wi open set U.

Proof

We prove only (2) and the rest of the proof is simple. Suppose that Aij- GWC(X), for each AX. Then, every wiopen set U, Aij- GWC(X). If UU, hence \(cl_{w_{j}}(U){\subseteq }U\). Thus, \(cl_{w_{j}}(U)\)=U, for each wiopen set U. Conversely, suppose that AU and U be a wiopen set. Then, \(cl_{w_{j}}(A){\subseteq }cl_{w_{j}}(U)\). From assumption, \(cl_{w_{j}}(A){\subseteq }U\) and so Aij- 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 w1w2, 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 w1̸w2.

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 AB= {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 Xwi. Then, {x} is wiclosed or X{x}ij- GWC(X), for each xX.

Proof

Suppose that the singleton {x} is not wiclosed for some xX. Then, X{x} is not wiopen. Since X is wiopen set and X{x}X. Hence, X{x}ij- GWC(X). □

Theorem 8

If Aij- GWC(X), then \(cl_{w_{j}}(A){\setminus }A\) contains no nonempty wiclosed.

Proof

For an ij-gwclosed set A, let S be a nonempty wiclosed set s.t. \(S{\subseteq }cl_{w_{j}}(A){\setminus }A\). Then, \(S{\subseteq }cl_{w_{j}}(A)\) and SXA. Since XS is wiopen 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 wiclosed. □

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 A21- GWC(X).

It thus follows from Theorem 8 that

Corollary 2

If Aij- GWC(X) and \(cl_{w_{j}}(A){\setminus }A\) is a wiclosed 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 wiclosed 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 Aij- GWC(X), then \(cl_{w_{j}}(A){\setminus }A{\in }ij\)- GWO(X).

Proof

Let Aij- GWC(X) and F be a wiclosed 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 A21- GWC(X).

Theorem 10

If Aij- GWC(X) and \(A{\subseteq }B{\subseteq }cl_{w_{j}}(A)\), then Bij- GWC(X).

Proof

Let U be any wiopen set s.t. BU. Since AB and Aij- 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 Bij- GWC(X). □

Corollary 3

Let (X,w1,w2) be a biwss. Then,

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

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

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

Theorem 12

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

Proof

Let U be a wiopen 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 XA is ij-gwclosed, then by Theorem 8, we have XU= 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 {AααΔ} is called wj-locally finite.

Theorem 13

Let (X,w1,w2) be a biwss. If the family {AααΔ} is wj-locally finite, then the arbitrary union of ij-gwclosed sets Aα,αΔ 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 AU 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 Aij- 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:

wjclosed 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 Aji- 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 xX.

Proof

Similar to Theorem 7. □

Theorem 16

If Aij- 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,yX, there exist a wi-open set U and wj-open set V s.t. xU,yU and yV,xV.

  • 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,yX and xy. 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 wi-open set U containing x and wj-open set V s.t. xU,yU, and yV,xV. 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 wiopen set and \(int_{w_{j}}\{x\}\) is wjopen. A biwss (X,w1,w2) is ij-\(wT_{\frac {1}{2}}\) iff {x} is wiclosed or {x}=\(int_{w_{j}}\{x\}\) for each xX.

Proof

Suppose that {x} is not wiclosed for some xX. 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 wiclosed or {x}=\(int_{w_{j}}\{x\}\) for any \(x{\in }cl_{w_{j}}B\).Case (I): Suppose {x} is wiclosed. If xB, then \(\{x\}{\subseteq }cl_{w_{j}}B {\setminus }B\), which is a contradiction to Theorem 8. Hence xB.Case (II): Suppose {x}=\(int_{w_{j}}\{x\}\) and \(x{\in }cl_{w_{j}}B\). Since {x}∩B, we have xB. 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 xX.

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 xX.

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 Aij- GWC(X). Since (X,w1,w2) is an ij-\(w^{\sigma }T_{\frac {1}{2}}\) space, Aij- 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 wiclosed set A and wjclosed set B s.t. AB= , there are wjopen set U and wiopen set V s.t. AU,BV, and UV= .

Theorem 21

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

  • (X,w1,w2) is ij-wnormal,

  • For each wiclosed set A and wjopen set N with AN, there exists wjopen set U s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }N\),

  • For each wiclosed set A and each ij-gwclosed set H with AH= , there exist wjopen set U and wiopen set V s.t. AU,HV and UV= ,

  • For each wiclosed set A and ij-gwopen N with AN, there exists wjopen 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 wiclosed for each wjopen 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 wiclosed set and B be a wjclosed set with AB= . Then, XB is a wjopen set with AXB. Thus, by (2) there exists wj open set U s.t. \(A{\subseteq }U{\subseteq }cl_{w_{i}}(U){\subseteq }X{\setminus }B\). Hence AU and \(B{\subseteq }X{\setminus }cl_{w_{i}}(U)\). Since \(cl_{w_{i}}(U)\) is wiclosed for each wjopen U, then \(X{\setminus }cl_{w_{i}}(U)\)=V is wiopen and UV= . Hence (X,w1,w2) is ij-wnormal. (1)(3): Let A be a wiclosed set and H be an ij-gwclosed set with AH= . Then, HXA. From Definition 3, we have \(cl_{w_{j}}(H){\subseteq }X{\setminus }A\). Since H is ij-gwclosed, then \(cl_{w_{j}}(H)\) is wjclosed. Since \(A{\cap }cl_{w_{j}}(H)\)= , then from (1) there exist wjopen set U and wiopen set V s.t. \(A{\subseteq }U, H{\subseteq }cl_{w_{j}}(H){\subseteq }V\) and UV= . □

Theorem 23

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

  • (X,w1,w2) is ij-wnormal,

  • For each wiclosed set A and wjclosed set B s.t. AB= , there exist ij-gwopen U and ji-gwopen V s.t. AU,BV and UV= ,

  • For each wiclosed set A and wjopen N with AN, 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 wiclosed set and B be a wjclosed set with AB= . Since (X,w1,w2) is ij-wnormal, then there exist wjopen set U and wiopen set V s.t. AU,BV and UV= . From Corollary 1, there exist ij-gwopen U and ji-gwopen V s.t. AU,BV and UV= . (2)(3): Let A be a wiclosed set and N be a wjopen set with AN. Then, AXN= . From (2), there exist ij-gwopen U and ji-gwopen V s.t. AU,XNV, and UV= . Since XV is ji-gwclosed, N is wjopen, and XVN, then from Definition 3, we have \(cl_{w_{i}}(X{\setminus }V){\subseteq }N\). Since UXV, 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 wiclosed for each ij-gwclosed and \(int_{w_{j}}(U)\) is wjopen 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 wiclosed set and B be a wjclosed set with AB= . Take N= XB, 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 wjopen. Also, \(X{\setminus }cl_{w_{i}}(U)\) is wiopen 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. AB= , there are wjopen set U and wiopen set V s.t. AU,BV and UV= .

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 AN, there exists wjopen 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. AB= , there exist wjopen set U s.t. AU 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 wiclosed for each wiopen 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. AB= , there exist ij- σgwopen set U, ji- σgwopen set V s.t. AU,BV and UV= ,

  • For each ji-gwclosed set A and ij-gwopen set N with AN, 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 AN. Take B= XN. Then, by assumption, there exist ij- σgwopen set U, ji- σgwopen set V s.t. AU,BV and UV= . Hence, UXV,XVN. Since XV 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 wjopen and \(int_{w_{i}}(U)\) is wiopen 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 AN, 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. AB= , there exist ij-gwopen set U and ji-gwopen set V s.t. AU,BV and UV= .

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 AN, 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 xX and \(w^{\star }_{j}\)open set V containing f(x), there is a wj open set U containing x s.t. f(U)V.

  • ij- g(w,w)-continuous if for xX 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 wjclosed set B, f(B) is ji- gwclosed 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 wjopen 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 xf−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, Uf−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 wjopen set s.t. xUf−1(V) and so \(x \in int_{w_{j}}f^{-1}(V)\). Therefore, f−1(V)=\(int_{w_{j}}f^{-1}(V)\).(): Let xX and V be a \(w^{\star }_{j}\)open set in Y with f(x)V, then xf−1(V). Since f−1(V)=\(int_{w_{j}}f^{-1}(V)\), then there exists wjopen set U s.t. xUf−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 wiopen U s.t. f−1(B)U, there exists ij- gwopen set V of Y s.t. BV and f−1(V)U.

Proof

(1)(2): Let BY and U be a wiopen set s.t. f−1(B)U. Since f is an ij- g(w,w)closed function, then f(U) is an ij- gwopen 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 wiopen set, Ff(U) s.t. F is a \(w^{\star }_{i}\)closed set, then f−1(F)U. This implies that there exists ij- gwopen set V in Y s.t. FV 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- gwopen 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- gwopen 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- gwclosed set B of Y.

Proof

Let BY be an ij- gwclosed set. Let U be a wiopen 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- gwopen set V of Y s.t. BV 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- gwclosed, 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 wjclosed set A in X.

Proof

Let A be a wjclosed set in X, then A=\(cl_{w_{j}}(A)\). Since f is an ij- g(w,w)closed function, then f(A) is ji- gwclosed 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 wiclosed 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 wiclosed set in X. Since f is an ji- g(w,w)closed function, then \(f(cl_{w_{j}}(A))\) is ij- gwclosed 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 wiopen for each \(w^{\star }_{i}\)open set U in Y and \(cl_{w_{j}}(A)\) is a wiclosed 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- gwclosed 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- gwopen 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 hf:(X,w1,w2)→(Z,υ1,υ2) is ij- g(w,υ)-continuous.

Proof

Let xX and V be a υjopen set of Z containing hf(x). Since h is ij- g(w,υ)-continuous, then there is an ij- gwopen 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- gwopen 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. hf(G)V. Consequently, hf 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 hf:(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.

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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Abu-Donia, H.M., Hosny, R.A. Generalized wclosed 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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Keywords

  • Biweak structures
  • ij-gwclosed sets
  • ij-g (w,w)-continuous functions
  • ij-wnormal spaces

AMS classification

  • 54A05– 54C08– 54C10–54C20