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. □