- Original Research
- Open Access
- Published:
Generalized wclosed 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 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 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 clw(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 intw(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)= intw(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,
clw(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 wiclosed and wjclosed, where i,j= 1 or 2 and i≠j.
Proposition 2
[9] Let (X,w1,w2) be a biwss. If Aα is ij-wclosed for all α∈Λ≠∅, then ∩α∈ΛAα 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 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 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 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. A⊆U. Then, \(cl_{w_{j}}(A)\)=A. It implies that A∈ij- GWC(X). □
Corollary 1
If A is a wjopen 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∈wj and each wiopen set is wjclosed, 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 wi 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 wiopen set U, A∈ij- GWC(X). If U⊆U, hence \(cl_{w_{j}}(U){\subseteq }U\). Thus, \(cl_{w_{j}}(U)\)=U, for each wiopen set U. Conversely, suppose that A⊆U 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 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∈wi. Then, {x} is wiclosed or X∖{x}∈ij- GWC(X), for each x∈X.
Proof
Suppose that the singleton {x} is not wiclosed for some x∈X. Then, X∖{x} is not wiopen. Since X is wiopen 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 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 S⊆X∖A. Since X∖S 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 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 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 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 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 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 wiopen 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 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 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 {Aα∣α∈Δ} is called wj-locally finite.
Theorem 13
Let (X,w1,w2) be a biwss. If the family {Aα∣α∈Δ} is wj-locally finite, then the arbitrary union of ij-gwclosed sets Aα,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:
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 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 wi-open set U and wj-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 wi-open set U containing x and wj-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 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 x∈X.
Proof
Suppose that {x} is not wiclosed 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 wiclosed or {x}=\(int_{w_{j}}\{x\}\) for any \(x{\in }cl_{w_{j}}B\).Case (I): Suppose {x} is wiclosed. 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 wiclosed set A and wjclosed set B s.t. A∩B= ∅, there are wjopen set U and wiopen 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 wiclosed set A and wjopen set N with A⊆N, 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 A∩H= ∅, there exist wjopen set U and wiopen set V s.t. A⊆U,H⊆V and U∩V= ∅,
For each wiclosed set A and ij-gwopen N with A⊆N, 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 A∩B= ∅. Then, X∖B is a wjopen set with A⊆X∖B. Thus, by (2) there exists wj 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 wiclosed for each wjopen U, then \(X{\setminus }cl_{w_{i}}(U)\)=V is wiopen and U∩V= ∅. Hence (X,w1,w2) is ij-wnormal. (1)⇒(3): Let A be a wiclosed 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 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 U∩V= ∅. □
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. A∩B= ∅, there exist ij-gwopen U and ji-gwopen V s.t. A⊆U,B⊆V and U∩V= ∅,
For each wiclosed set A and wjopen 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 wiclosed set and B be a wjclosed set with A∩B= ∅. Since (X,w1,w2) is ij-wnormal, then there exist wjopen set U and wiopen 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 wiclosed set and N be a wjopen 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 wjopen, 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 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 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 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. A∩B= ∅, there are wjopen set U and wiopen 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 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. A∩B= ∅, there exist wjopen 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 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. 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 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 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 wj 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 wjclosed 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 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 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 wjopen 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 wjopen 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 wiopen 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 wiopen 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 wiopen 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 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- 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 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- 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 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- 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 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- 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 υjopen 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.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
- 1
Kelly, J.: Bitopological spaces. Proc. London. Math. Soc. 13, 71–89 (1963).
- 2
Fukutake, T.: On generalized closed sets in bitopological spaces. Bull. Fukuoka Univ. Ed. III. 35, 19–28 (1985).
- 3
Csaszar, A., Makai Jr., E.: Further remarks on δ- and θ-modifications. Acta Math. Hungar. 123, 223–228 (2009).
- 4
Boonpok, C.: Weakly open functions on bigeneralized topological spaces. Int. J. Math. Anal. 4, 891–897 (2010).
- 5
Boonpok, C.: Biminimal structure spaces. Int. Math. Forum. 5, 703–707 (2010).
- 6
Csaszar, A.: Weak structures. Acta Math. Hungar. 131(1-2), 193–195 (2011).
- 7
Ekici, E: On weak structures due to Csaszar. Acta Math. Hungar. 134(4), 565–570 (2012).
- 8
Zahran, A, Mousa, A, Ghareeb, A: generalized closed sets and some separation axioms on weak structure. Hacettepe J. Math. Stat.44(3), 669–677 (2015).
- 9
Puiwong, J, Viriyapong, C, Khampakdee, J: Weak separation axioms in biweak structure spaces. Burapha Sci. J. 22(2), 110–117 (2017).
- 10
Al-Omari, A, Noiri, T: a unified theory of generalized closed sets in weak structure. Acta Math. Hungar. 135(1–2), 174–183 (2012).
- 11
Al-Omari, A, Noiri, T: weak continuity between WSS and GTS due to Csaszar. Malays. J. Math. Sci. 7(2), 297–313 (2013).
Acknowledgements
The authors thank the reviewers for their useful comments that led to the improvement of the original manuscript.
Funding
Not applicable.
Author information
Affiliations
Contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
Keywords
- Biweak structures
- ij-gwclosed sets
- ij-g (w,w⋆)-continuous functions
- ij-wnormal spaces
AMS classification
- 54A05– 54C08– 54C10–54C20
