A novel class of bipolar soft separation axioms concerning crisp points

Definition 2.1

[7] Let Σ = { ℓ 1 , ℓ 2 , … , ℓ n } be a set of parameters. The NOT set of Σ denoted by ¬ Σ is defined by ¬ Σ = { ¬ ℓ 1 , ¬ ℓ 2 , … , ¬ ℓ n } , where ¬ ℓ i = not ℓ i for i = 1 , 2 , … , n .

Definition 2.2

[32] Let the mappings g and g ^ be defined by g : Σ → 2 ℵ and g ^ : ¬ Σ → 2 ℵ with g ( ℓ ) ∩ g ^ ( ¬ ℓ ) = ∅ for all ℓ ∈ Σ . Then, ( g , g ^ , Σ ) is said to be a BS set over ℵ .

Definition 2.3

[32] A BS set ( g 1 , g ^ 1 , Λ ) is a BS subset of BS set ( g 2 , g ^ 2 , B ) , symbolized by ( g 1 , g ^ 1 , Λ ) ⊑ ˜ ˜ ( g 2 , g ^ 2 , B ) , if, (i) Λ ⊆ B and (ii) for all ℓ ∈ Λ , g 1 ( ℓ ) ⊆ g 2 ( ℓ ) and g ^ 2 ( ¬ ℓ ) ⊆ g ^ 1 ( ¬ ℓ ) .

Definition 2.4

[32] Two BS sets ( g 1 , g ^ 1 , Λ ) and ( g 2 , g ^ 2 , B ) are said to be BS equal if ( g 1 , g ^ 1 , Λ ) ⊑ ˜ ˜ ( g 2 , g ^ 2 , B ) and ( g 2 , g ^ 2 , Λ ) ⊑ ˜ ˜ ( g 1 , g ^ 1 , B ) .

Definition 2.5

[32] A BS set ( g , g ^ , Λ ) c is the complement of a BS set ( g , g ^ , Λ ) and is defined by ( g c , g ^ c , Λ ) , where g c and g ^ c are mappings given by g c ( ℓ ) = g ^ ( ¬ ℓ ) and g ^ c ( ¬ ℓ ) = g ( ℓ ) for all ℓ ∈ Λ .

Definition 2.6

[32] A relative null BS set of a BS set ( g , g ^ , Λ ) , denoted by ( Φ ^ , ℵ ^ , Λ ) , is defined as Φ ^ ( ℓ ) = ∅ and ℵ ^ ( ¬ ℓ ) = ℵ for all ℓ ∈ Λ .

Definition 2.7

[32] A relative whole BS set of a BS set ( g , g ^ , Λ ) , denoted by ( ℵ ^ , Φ ^ , Λ ) , is defined as ℵ ^ ( ℓ ) = ℵ and Φ ^ ( ¬ ℓ ) = ∅ for all ℓ ∈ Λ .

Definition 2.8

[32] The union of two BS sets ( g 1 , g ^ 1 , Λ ) and ( g 2 , g ^ 2 , B ) is a BS set ( f , f ^ , Γ ) , symbolized by ( g 1 , g ^ 1 , Λ ) ⊔ ˜ ˜ ( g 2 , g ^ 2 , B ) = ( f , f ^ , Γ ) , where Γ = Λ ∩ B and for all ℓ ∈ Γ , f ( ℓ ) = g 1 ( ℓ ) ∪ g 2 ( ℓ ) and f ^ ( ¬ ℓ ) = g ^ 1 ( ¬ ℓ ) ∩ g ^ 2 ( ¬ ℓ ) .

Definition 2.9

[32] The intersection of two BS sets ( g 1 , g ^ 1 , Λ ) and ( g 2 , g ^ 2 , B ) is a BS set ( f , f ^ , Γ ) , symbolized by ( g 1 , g ^ 1 , Λ ) ⊓ ˜ ˜ ( g 2 , g ^ 2 , B ) = ( f , f ^ , Γ ) , where Γ = Λ ∩ B and for all ℓ ∈ Γ , f ( ℓ ) = g 1 ( ℓ ) ∩ g 2 ( ℓ ) and f ^ ( ¬ ℓ ) = g ^ 1 ( ¬ ℓ ) ∪ g ^ 2 ( ¬ ℓ ) .

Definition 2.10

[38] The difference of two BS sets ( g 1 , g ^ 1 , Λ ) and ( g 2 , g ^ 2 , B ) is a BS set ( f , f ^ , Γ ) where Γ = Λ ∩ B and ( g 1 , g ^ 1 , Λ ) ⧹ ( g 2 , g ^ 2 , B ) = ( g 1 , g ^ 1 , Λ ) ⊓ ˜ ˜ ( g 2 , g ^ 2 , B ) c = ( f , f ^ , Γ ) .

Definition 2.11

[33] Let ( g , g ^ , Σ ) be a BS set over ℵ and η ∈ ℵ . Then, η ∈ ( g , g ^ , Σ ) if η ∈ g ( ℓ ) for all ℓ ∈ Σ . For any η ∈ ℵ , η ∉ ( g , g ^ , Σ ) if η ∉ g ( ℓ ) for some ℓ ∈ Σ .

Definition 2.12

[33] Two BS sets ( g 1 , g ^ 1 , Σ ) and ( g 2 , g ^ 2 , Σ ) are said to be disjoint BS sets if for all ℓ ∈ Σ , g 1 ( ℓ ) ∩ g 2 ( ℓ ) = ∅ . We symbolized by ( g 1 , g ^ 1 , Σ ) ⊓ ˜ ˜ ( g 2 , g ^ 2 , Σ ) = ( Φ ^ , g ^ , Σ ) , where Φ ^ ( ℓ ) = ∅ for all ℓ ∈ Σ and g ^ ( ¬ ℓ ) ⊆ ℵ for all ¬ ℓ ∈ ¬ Σ .

Definition 2.13

[33] Let ( g , g ^ , Σ ) be a BS set over ℵ and ϒ ⊆ ℵ . The sub BS set of ( g , g ^ , Σ ) over ϒ , denoted by ( g ϒ , g ^ ϒ , Σ ) , is defined as g ϒ ( ℓ ) = ϒ ∩ g ( ℓ ) and g ^ ϒ ( ¬ ℓ ) = ϒ ∩ g ^ ( ¬ ℓ ) for each ℓ ∈ Σ .

Definition 2.14

[33] Let η ∈ ℵ , then ( g η , g ^ η , Σ ) denotes the BS set over ℵ defined by g η ( ℓ ) = { η } and g ^ η ( ¬ ℓ ) = ℵ ⧹ { η } for all ℓ ∈ Σ .

Definition 2.15

[33] Let T ℬ S be the collection of BS sets over ℵ , then T ℬ S is said to be a bipolar soft topology (BST) on ℵ if

1. ( Φ ^ , ℵ ^ , Σ ) , ( ℵ ^ , Φ ^ , Σ ) belong to T ℬ S ;

2. the intersection of any two BS sets in T ℬ S belongs to T ℬ S ;

3. the union of any number of BS sets in T ℬ S belongs to T ℬ S .

We termed ( ℵ , T ℬ S , Σ , ¬ Σ ) a BSTS over ℵ . The members of T ℬ S are said to be BS open sets in ℵ , while its complements are said to be BS closed sets.

Definition 2.16

[33] Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS over ℵ and ϒ ⊆ ℵ . Then, T ℬ S ϒ = { ( g ϒ , g ^ ϒ , Σ ) ∣ ( g , g ^ , Σ ) ∈ ˜ ˜ T ℬ S } is said to be a relative BST on ϒ and ( ϒ , T ℬ S ϒ , Σ , ¬ Σ ) is called a BS subspace of ( ℵ , T ℬ S , Σ , ¬ Σ )

Proposition 2.17

[34] Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS. Then, the following collections define the soft topology on ℵ .

1. T S = { ( g , Σ ) ∣ ( g , g ^ , Σ ) ∈ ˜ ˜ T ℬ S } .

2. ¬ T S = { ( g ^ , ¬ Σ ) ∣ ( g , g ^ , Σ ) ∈ ˜ ˜ T ℬ S } (provided that ℵ is finite).

Proposition 2.18

[34,33] Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS. Then, the following collections define topology on ℵ .

1. T ℓ = { g ( ℓ ) ∣ ( g , g ^ , Σ ) ∈ ˜ ˜ T ℬ S } for each ℓ ∈ Σ .

2. T ¬ ℓ = { g ^ ( ¬ ℓ ) ∣ ( g , g ^ , Σ ) ∈ ˜ ˜ T ℬ S } for each ¬ ℓ ∈ ¬ Σ . (provided that ℵ is finite).

Proposition 2.19

[33] Let ( ℵ , T S , Σ ) be an STS. Then, the collection T ℬ S defines a BST over ℵ if it consists of BS sets ( g , g ^ , Σ ) with ( g , Σ ) ∈ ˜ T S and g ^ ( ¬ ℓ ) = ℵ ⧹ g ( ℓ ) for all ℓ ∈ Σ .

Definition 2.20

[14] An STS ( ℵ , T S , Σ ) is said to be:

1. a soft T ˜ 0 -space if for every η ≠ κ ∈ ℵ , there is a soft open set ( g , Σ ) such that η ∈ ( g , Σ ) but κ ∉ ( g , Σ ) or κ ∈ ( g , Σ ) but η ∉ ( g , Σ ) .

2. a soft T ˜ 1 -space if for every η ≠ κ ∈ ℵ , there are soft open sets ( g 1 , Σ ) and ( g 2 , Σ ) such that η ∈ ( g 1 , Σ ) but κ ∉ ( g 1 , Σ ) and κ ∈ ( g 2 , Σ ) but η ∉ ( g 2 , Σ ) .

3. a soft T ˜ 2 -space if for every η ≠ κ ∈ ℵ , there are soft open sets ( g 1 , Σ ) and ( g 2 , Σ ) such that η ∈ ( g 1 , Σ ) but κ ∈ ( g 2 , Σ ) and ( g 1 , Σ ) ⊓ ˜ ( g 2 , Σ ) = ( Φ ^ , Σ ) .

Corollary 2.21

Let ( g , g ^ , Σ ) be a BS set over ℵ and η ∈ ℵ . Then,

1. η ∈ ( g , g ^ , Σ ) if and only if ( g η , g ^ η , Σ ) ⊑ ˜ ˜ ( g , g ^ , Σ ) .

2. If ( g η , g ^ η , Σ ) ⊓ ˜ ˜ ( g , g ^ , Σ ) = ( Φ ^ , g ^ , Σ ) , then η ∉ ( g , g ^ , Σ ) .

Proof

Straightforward.□

Remark 2.22

The converse of Corollary 2.21 (ii) is not true.

Example 2.23

Suppose that ℵ = { η 1 , η 2 } , Σ = { ℓ 1 , ℓ 2 , ℓ 3 } , and ( g , g ^ , Σ ) = { ( ℓ 1 , { η 1 } , { η 2 } ) , ( ℓ 2 , { η 1 } , { η 2 } ) , ( ℓ 3 , ℵ , ∅ ) } . Then, η 2 ∉ ( g , g ^ , Σ ) but ( g η 2 , g ^ η 2 , Σ ) ⊓ ˜ ˜ ( g , g ^ , Σ ) ≠ ( Φ ^ , g ^ , Σ ) .

Proposition 2.24

Let ( ϒ , T ℬ S ϒ , Σ , ¬ Σ ) be a BS subspace of ( ℵ , T ℬ S , Σ , ¬ Σ ) . Then,

1. ( g ϒ , g ^ ϒ , Σ ) is a BS open set in ϒ if and only if g ϒ ( ℓ ) = ϒ ∩ g ( ℓ ) and g ^ ϒ ( ¬ ℓ ) = ϒ ∩ g ^ ( ¬ ℓ ) for some BS open set ( g , g ^ , Σ ) in ℵ .

2. ( f ϒ , f ^ ϒ , Σ ) is a BS closed set in ϒ if and only if f ϒ ( ℓ ) = ϒ ∩ f ( ℓ ) and f ^ ϒ ( ¬ ℓ ) = ϒ ∩ f ^ ( ¬ ℓ ) for some BS closed set ( f , f ^ , Σ ) in ℵ .

Proof

1. Follows from Definition 2.16.

2. ( f ϒ , f ^ ϒ , Σ ) is a BS closed set in ϒ iff ( f ϒ , f ^ ϒ , Σ ) c is a BS open set in ϒ iff f ϒ c ( ℓ ) = ϒ ∩ g ( ℓ ) and f ^ ϒ c ( ¬ ℓ ) = ϒ ∩ g ^ ( ¬ ℓ ) for some BS open set ( g , g ^ , Σ ) in ℵ iff ϒ ⧹ f ϒ c ( ℓ ) = ϒ ⧹ [ ϒ ∩ g ( ℓ ) ] and ϒ ⧹ f ^ ϒ c ( ¬ ℓ ) = ϒ ⧹ [ ϒ ∩ g ^ ( ¬ ℓ ) ] iff f ϒ ( ℓ ) = ϒ ∩ g c ( ℓ ) and f ^ ϒ ( ¬ ℓ ) = ϒ ∩ g ^ c ( ¬ ℓ ) iff f ϒ ( ℓ ) = ϒ ∩ f ( ℓ ) and f ^ ϒ ( ¬ ℓ ) = ϒ ∩ f ^ ( ¬ ℓ ) , where ( f , f ^ , Σ ) = ( g , g ^ , Σ ) c is a BS closed set in ℵ since ( g , g ^ , Σ ) is a BS open set in ℵ .□

Definition 3.1

A BSTS ( ℵ , T ℬ S , Σ , ¬ Σ ) is said to be:

1. a BS T ˜ ˜ 0 -space if for every η ≠ κ ∈ ℵ , there is a BS open set ( g , g ^ , Σ ) with η ∈ ( g , g ^ , Σ ) but κ ∉ ( g , g ^ , Σ ) or κ ∈ ( g , g ^ , Σ ) but η ∉ ( g , g ^ , Σ ) .

2. a BS T ˜ ˜ 1 -space if for every η ≠ κ ∈ ℵ , there are BS open sets ( g 1 , g ^ 1 , Σ ) and ( g 2 , g ^ 2 , Σ ) with η ∈ ( g 1 , g ^ 1 , Σ ) but κ ∉ ( g 1 , g ^ 1 , Σ ) and κ ∈ ( g 2 , g ^ 2 , Σ ) but η ∉ ( g 2 , g ^ 2 , Σ ) .

3. a BS T ˜ ˜ 2 -space if for every η ≠ κ ∈ ℵ , there are BS open sets ( g 1 , g ^ 1 , Σ ) and ( g 2 , g ^ 2 , Σ ) with η ∈ ( g 1 , g ^ 1 , Σ ) and κ ∈ ( g 2 , g ^ 2 , Σ ) and ( g 1 , g ^ 1 , Σ ) ⊓ ˜ ˜ ( g 2 , g ^ 2 , Σ ) = ( Φ ^ , g ^ , Σ ) .

Proposition 3.2

Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS and η ≠ κ ∈ ℵ . If there is a BS open set ( g , g ^ , Σ ) with η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c or κ ∈ ( g , g ^ , Σ ) and η ∈ ( g , g ^ , Σ ) c , then ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space.

Proof

Let η ≠ κ ∈ ℵ and ( g , g ^ , Σ ) be a BS open set with η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c . If κ ∈ ( g , g ^ , Σ ) c , then κ ∈ g c ( ℓ ) for all ℓ ∈ Σ . This means that κ ∉ g ( ℓ ) for all ℓ ∈ Σ . Hence, κ ∉ ( g , g ^ , Σ ) . Similarly, we may verify κ ∈ ( g , g ^ , Σ ) and η ∉ ( g , g ^ , Σ ) . Therefore, ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space.□

Proposition 3.3

If ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BS T ˜ ˜ 0 -space, then ( ℵ , T S , Σ ) is a soft T ˜ 0 -space.

Proof

Let η ≠ κ ∈ ℵ . Since ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space. Then, there is a BS open set ( g , g ^ , Σ ) with η ∈ ( g , g ^ , Σ ) but κ ∉ ( g , g ^ , Σ ) or κ ∈ ( g , g ^ , Σ ) but η ∉ ( g , g ^ , Σ ) . Say, η ∈ ( g , g ^ , Σ ) but κ ∉ ( g , g ^ , Σ ) . This means that η ∈ g ( ℓ ) for all ℓ ∈ Σ and κ ∉ g ( ℓ ) for some ℓ ∈ Σ . Hence, η ∈ ( g , Σ ) and κ ∉ ( g , Σ ) . Similarly, we may verify κ ∈ ( g , Σ ) and η ∉ ( g , Σ ) . Therefore, ( ℵ , T S , Σ ) is a soft T ˜ 0 -space.□

Remark 3.4

If ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BS T ˜ ˜ 0 -space. Then, ( ℵ , T ℓ ) need not be T 0 -space for each ℓ ∈ Σ .

Example 3.5

Let ℵ = { η 1 , η 2 } and Σ = { ℓ 1 , ℓ 2 } . Let T ℬ S = { ( Φ ^ , ℵ ^ , Σ ) , ( ℵ ^ , Φ ^ , Σ ) , ( g , g ^ , Σ ) } , where ( g , g ^ , Σ ) = { ( ℓ 1 , { η 1 } , { η 2 } ) , ( ℓ 2 , ℵ , ∅ ) } . Then, ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space. Also, we have

T ℓ 1 = { ∅ , ℵ , { η 1 } } , and

T ℓ 2 = { ∅ , ℵ } .

Then, T ℓ 2 is not T 0 -space.

Proposition 3.6

Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS and η ≠ κ ∈ ℵ . If there is a BS open set ( g , g ^ , Σ ) with η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c or κ ∈ ( g , g ^ , Σ ) and η ∈ ( g , g ^ , Σ ) c , then ( ℵ , T S , Σ ) is a soft T ˜ 0 -space and ( ℵ , T ℓ ) is a T 0 -space for each ℓ ∈ Σ .

Proof

By Propositions 3.2 and 3.3, it is obvious that ( ℵ , T S , Σ ) is a soft T ˜ 0 -space. Now, for any ℓ ∈ Σ , ( ℵ , T ℓ ) is a topological space and since η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c or κ ∈ ( g , g ^ , Σ ) and η ∈ ( g , g ^ , Σ ) c , then η ∈ g ( ℓ ) but κ ∉ g ( ℓ ) or κ ∈ g ( ℓ ) but η ∉ g ( ℓ ) . Hence, ( ℵ , T ℓ ) is a T 0 -space.□

Proposition 3.7

Let ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS and η ≠ κ ∈ ℵ and let ℵ be a finite set. If there is a BS open set ( g , g ^ , Σ ) with η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c or κ ∈ ( g , g ^ , Σ ) and η ∈ ( g , g ^ , Σ ) c , then ( ℵ , ¬ T S , ¬ Σ ) is a soft T ˜ 0 -space and ( ℵ , T ¬ ℓ ) is a T 0 -space for all ¬ ℓ ∈ ¬ Σ .

Proof

Let η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c . This means that η ∈ g ( ℓ ) and κ ∈ g c ( ℓ ) for all ℓ ∈ Σ . Then, η ∉ g ^ ( ¬ ℓ ) and κ ∈ g ^ ( ¬ ℓ ) for all ℓ ∈ Σ . So, κ ∈ ( g ^ , ¬ Σ ) and η ∉ ( g ^ , ¬ Σ ) . Similarly, we may verify η ∈ ( g ^ , ¬ Σ ) and κ ∉ ( g ^ , ¬ Σ ) . Hence, ( ℵ , ¬ T S , ¬ Σ ) is a soft T ˜ 0 -space. Now, for any ¬ ℓ ∈ ¬ Σ , ( ℵ , T ¬ ℓ ) is a topological space, and since η ∈ ( g , g ^ , Σ ) and κ ∈ ( g , g ^ , Σ ) c or κ ∈ ( g , g ^ , Σ ) and η ∈ ( g , g ^ , Σ ) c , then η ∉ g ^ ( ¬ ℓ ) and κ ∈ g ^ ( ¬ ℓ ) or κ ∉ g ^ ( ¬ ℓ ) and η ∈ g ^ ( ¬ ℓ ) . Hence, ( ℵ , T ¬ ℓ ) is T 0 -space.□

Proposition 3.8

Let ( ℵ , T S , Σ ) be an STS and ( ℵ , T ℬ S , Σ , ¬ Σ ) be a BSTS constructed from ( ℵ , T S , Σ ) as in Proposition 2.19. If ( ℵ , T S , Σ ) be a soft T ˜ 0 -space, then ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space.

Proof

Let η ≠ κ ∈ ℵ . Since ( ℵ , T S , Σ ) is a soft T ˜ 0 -space, then there is a soft open set ( g , Σ ) with η ∈ ( g , Σ ) but κ ∉ ( g , Σ ) or κ ∈ ( g , Σ ) but η ∉ ( g , Σ ) . Say, η ∈ ( g , Σ ) and κ ∉ ( g , Σ ) . This means that η ∈ g ( ℓ ) for all ℓ ∈ Σ and κ ∉ g ( ℓ ) for some ℓ ∈ Σ . Since g ^ ( ¬ ℓ ) = ℵ ⧹ g ( ℓ ) for all ℓ ∈ Σ , then η ∉ g ^ ( ¬ ℓ ) for all ℓ ∈ Σ and κ ∈ g ^ ( ¬ ℓ ) for some ℓ ∈ Σ . Therefore, η ∈ ( g , g ^ , Σ ) and κ ∉ ( g , g ^ , Σ ) . Similarly, we may verify κ ∈ ( g , g ^ , Σ ) but η ∉ ( g , g ^ , Σ ) . Hence, ( ℵ , T ℬ S , Σ , ¬ Σ ) is a BS T ˜ ˜ 0 -space.□