Regularity and normality in hereditary bi m-spaces

Definition 2.1

A subfamily m of the power set P ( X ) of a nonempty set X is called a minimal structure (briefly m-structure) [15] on X if ∅ ∈ m and X ∈ m .

Definition 2.2

Let ( X , m ) be an m-space and A be a subset of X . The m-closure mCl( A ) (resp. m-interior mInt( A )) of A [15] is defined by mCl( A ) = ∩ { F ⊂ X : A ⊂ F , X ⧹ F ∈ m } (resp. mInt( A ) = ∪ { U ⊂ X : U ⊂ A , U ∈ m } ).

Lemma 2.3

[15]. Let X be a nonempty set and m be a minimal structure on X. For subsets A and B of X, the following properties hold:

1. A ⊂ m Cl ( A ) and m Cl ( A ) = A if A is m-closed;

2. m Cl ( ∅ ) = ∅ , m Cl ( X ) = X ;

3. If A ⊂ B , then m Cl ( A ) ⊂ m Cl ( B ) ;

4. m Cl ( A ) ∪ m Cl ( B ) ⊂ m Cl ( A ∪ B ) ;

5. m Cl ( m Cl ( A ) ) = m Cl ( A ) ;

6. m Cl ( X − A ) = X − m Int ( A ) .

Definition 2.4

A minimal structure m on a set X is said to have property ℬ [15] if the union of any collection of elements of m is an element of m.

Lemma 2.5

[1] Let ( X , m ) be an m-space and A be a subset of X.

1. x ∈ m Cl ( A ) if and only if U ∩ A ≠ ∅ for every U ∈ m ( x ) .

2. Let m have property ℬ . Then the following properties hold:

   1. A is m-closed (resp. m-open) if and only if m Cl ( A ) = A (resp. m Int ( A ) = A );

   2. m Cl ( A ) is m-closed and mInt A ) is m-open.

Definition 2.6

A nonempty subfamily ℋ of P ( X ) is called a hereditary class on X [16] if it satisfies the following property: A ∈ ℋ and B ⊂ A imply B ∈ ℋ . A hereditary class ℋ is called an ideal if it satisfies the additional condition: A ∈ ℋ and B ∈ ℋ imply A ∪ B ∈ ℋ .

Definition 2.7

[3] Let ( X , m , ℋ ) be a hereditary m-space. For a subset A of X , the minimal local function A m H ⋆ ( ℋ , m) of A is defined as follows:

Remark 2.8

[3] Let ( X , m , ℋ ) be a hereditary m-space and A be a subset of X . If ℋ = { ∅ } (resp. P ( X ) ), then A m H ⋆ = m Cl ( A ) (resp. A m H ⋆ = ∅ ).

Lemma 2.9

[3] Let ( X , m , ℋ ) be a hereditary m-space. For subsets A and B of X, the following properties hold:

1. If A ⊂ B , then A m H ⋆ ⊂ B m H ⋆ ;

2. A m H ⋆ = m Cl ( A m H ⋆ ) ⊂ m Cl ( A ) ;

3. A m H ⋆ ∪ B m H ⋆ ⊂ ( A ∪ B ) m H ⋆ ;

4. ( A m H ⋆ ) m H ⋆ ⊂ ( A ∪ A m H ⋆ ) m H ⋆ = A m H ⋆ ;

5. If A ∈ ℋ , then A m H ⋆ = ∅ .

Definition 2.10

[3] Let ( X , m , ℋ ) be a hereditary m-space and A be a subset of X . The minimal ⋆ -closure mCl H ⋆ ( A ) of A is defined as mCl H ⋆ ( A ) = A ∪ A m H ⋆ . The m H ⋆ -structure is defined as follows: m H ⋆ = { U ⊂ X : m Cl H ⋆ ( X ⧹ U ) = X ⧹ U } . Each member of m H ⋆ is said to be m H ⋆ - o p e n and the complement of an m H ⋆ -open set is said to be m H ⋆ - c l o s e d .

Lemma 2.11

[3] Let ( X , m , ℋ ) be a hereditary m-space and A, B be two subsets of X. Then, the following properties hold:

1. A ⊂ m Cl H ⋆ ( A ) ;

2. m Cl H ⋆ ( ∅ ) = ∅ and mCl H ⋆ ( X ) = X ;

3. If A ⊂ B , then m Cl H ⋆ ( A ) ⊂ m Cl H ⋆ ( B ) ;

4. m Cl H ⋆ ( A ) ∪ m Cl H ⋆ ( B ) ⊂ m Cl H ⋆ ( A ∪ B ) ;

5. m Cl H ⋆ ( m Cl H ⋆ ( A ) ) = m Cl H ⋆ ( A ) , that is, m Cl H ⋆ ( A ) is m H ⋆ -closed.

Lemma 2.12

[3] For a hereditary m-space ( X , m , ℋ ) , the following properties hold:

1. m H ⋆ is an m-structure on X such that m H ⋆ has property ℬ and m ⊆ m H ⋆ ;

2. m Cl H ⋆ ( A ) = m H ⋆ Cl ( A ) for every subset A of X ;

3. β ( m , ℋ ) = { U ⧹ H : U ∈ m , H ∈ ℋ } is a basis for m H ⋆ such that m ⊂ β ( m , ℋ ) .

Lemma 2.13

Let ( X , m , ℋ ) be a hereditary m-space. If ℋ = { ∅ } , then m H ⋆ = { U ⊂ X : m Int ( U ) = U } .

Proof

By Remark 2.8, we have mCl H ⋆ ( A ) = A m H ⋆ ∪ A = m Cl ( A ) ∪ A = m Cl ( A ) for every subset A of X and hence m H ⋆ = { U ⊂ X : m Cl H ⋆ ( X − U ) = X − U } = { U ⊂ X : m Cl ( X − U ) = X − U } = { U ⊂ X : m Int ( U ) = U } .□

Lemma 2.14

Let ( X , m ) be a minimal space. Then the following hold:

1. m 0 is a minimal structure such that m 0 has property ℬ and m ⊂ m 0 ;

2. If m has property ℬ , then m 0 = m .

Example 2.15

Let X = { a , b , c , d } , m = { X , ∅ , { a } , { c } , { a , b } } and A = { a , c } . Then the minimal structure m does not have property ℬ and A ∈ m 0 but A ∉ m .

Definition 3.1

Let ( X , m , n , ℋ ) be a hereditary bi m-space. A subset A of X is said to be m n - ℋ g - c l o s e d (resp. m n g -closed [4]) if A n H ⋆ ⊂ U (resp. n Cl ( A ) ⊂ U ) whenever A ⊂ U and U ∈ m . A subset A of X is said to be m n - ℋ g -open (resp. m n g -open [4]) if the complement of A is m n - ℋ g -closed (resp. m n g -closed).

Proposition 3.2

For a subset of a hereditary m n -space ( X , m , n , ℋ ) , the following diagram holds:

DIAGRAM I

Lemma 3.3

Let A be a subset of a hereditary bi m-space ( X , m , n , ℋ ) .

1. A is m n - ℋ g -closed if and only if n Cl ℋ ∗ ( A ) ⊆ U whenever A ⊆ U and U ∈ m .

2. A is m n - ℋ g -open [5] if and only if F ⊆ n Int H ⋆ ( A ) whenever F ⊆ A and F is m-closed.

Definition 3.4

A subset N of a hereditary bi m-space ( X , m , n , ℋ ) is called an m n - ℋ g -neighborhood (resp. n ℋ ∗ -neighborhood) of a subset B of X , if there exists an m n - ℋ g -open (resp. n ℋ ∗ -open) set U such that B ⊆ U ⊆ N .

Definition 3.5

A hereditary bi m-space ( X , m , n , ℋ ) , is said to be ( m , n ) - ℋ g - r e g u l a r if for each m-closed set A and each point x ∉ A , there exist disjoint m n - ℋ g -open sets U , V such that x ∈ U and A ⊆ V .

Theorem 3.6

For a hereditary bi m-space ( X , m , n , ℋ ) , the following properties are equivalent:

1. X is ( m , n ) - ℋ g -regular.

2. For any m-open set U containing x ∈ X , there exists an m n - ℋ g -open set V such that x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ U .

3. For any m-closed set A , the intersection of all m n - ℋ g -closed neighborhoods of A is A .

4. For any set A and any m-open set B such that A ∩ B ≠ ∅ , there exists an m n - ℋ g -open set U such that A ∩ U ≠ ∅ and n Cl ℋ ∗ ( U ) ⊆ B .

5. For any nonempty set A and any m-closed set B such that A ∩ B = ∅ , there exist disjoint m n - ℋ g -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Proof

(1) ⇒ (2): Let U be an m-open set such that x ∈ U . Then X − U is an m-closed set not containing x . By hypothesis, there exist disjoint m n - ℋ g -open sets V , W such that x ∈ V and X − U ⊆ W . By Lemma 3.3, X − U ⊆ n Int ℋ ∗ ( W ) and so X − n Int ℋ ∗ ( W ) ⊆ U . Now V ∩ W = ∅ implies that V ∩ n Int ℋ ∗ ( W ) = ∅ and hence n Cl ℋ ∗ ( V ) ⊆ X − n Int ℋ ∗ ( W ) . Thus, x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ U .

(2) ⇒ (3): Let A be an m-closed set and x ∉ A . Then X − A is an m-open set containing x . By hypothesis, there exists an m n - ℋ g -open set V such that x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ X − A . Thus, A ⊆ X − n Cl ℋ ∗ ( V ) ⊆ X − V . Since X − n Cl ℋ ∗ ( V ) is n ℋ ⋆ -open and hence m n - ℋ g -open, then X − V is an m n - ℋ g -closed neighborhood of A and x ∉ X − V . This shows that A is the intersection of all the m n - ℋ g -closed neighborhoods of A .

(3) ⇒ (4): Let A be any set and B be any m-open set such that A ∩ B ≠ ∅ . Let x ∈ A ∩ B . Then, X − B is m-closed and x ∉ X − B . By hypothesis, there exists an m n - ℋ g -closed neighborhood V of X − B such that x ∉ V . Let X − B ⊆ G ⊆ V , where G is m n - ℋ g -open. Then U = X − V is an m n - ℋ g -open set such that x ∈ U and A ∩ U ≠ ∅ .

Furthermore, X − G is m n - ℋ g -closed, X − G ⊆ B and B is m-open, which implies that n Cl ℋ ∗ ( U ) = n Cl ℋ ∗ ( X − V ) ⊆ n Cl ℋ ∗ ( X − G ) ⊆ B .

(4) ⇒ (5): Let A be any nonempty set and B be any m-closed set such that A ∩ B = ∅ . Then X − B is an m-open set and A ∩ ( X − B ) ≠ ∅ . By hypothesis, there exists an m n - ℋ g -open set U such that A ∩ U ≠ ∅ and U ⊆ n Cl H ∗ ( U ) ⊆ X − B . Let V = X − n Cl ℋ ∗ ( U ) . Then U and V are m n - ℋ g -open sets such that B ⊆ X − n Cl ℋ ∗ ( U ) = V .

(5) ⇒ (1): Let B be an m-closed set and x ∉ B . Put A = { x } . Then, there exists disjoint m n - ℋ g -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V , hence x ∈ U . Thus, X is m n ℋ g -regular.□

Theorem 3.7

If every m-open subset of a hereditary bi m-space ( X , m , n , ℋ ) is n -closed, then X is m n ℋ g -regular.

Proof

Suppose every m-open subset of X is n -closed. Let A ⊆ X and U be any m-open set containing A . Then U is n ℋ ∗ -closed and n ℋ ∗ C l ( A ) ⊆ n ℋ ∗ C l ( U ) = U . Hence, every subset of X is m n - ℋ g -closed and every subset of X is m n - ℋ g -open. If B is an m-closed set not containing x , then { x } and B are the required disjoint m n - ℋ g -open sets containing x and B , respectively. Therefore, X is m n ℋ g -regular.□

Corollary 3.8

[20] An ideal topological space ( X , τ , ℐ ) is ℐ g -regular if and only if for any open set V containing any x ∈ X , there exists an ℐ g -open set U such that x ∈ U ⊆ Cl ⋆ ( U ) ⊆ V .

Definition 3.9

Let ( X , m ) be an m-space and A be a subset of X .

1. A is said to be m g - c l o s e d [21] if m Cl ( A ) ⊆ U whenever A ⊆ U and U ∈ m . The complement of an m g -closed set is said to be m g -open.

2. X is m g - r e g u l a r if for each m-closed set A and each point x ∉ A , there exist disjoint m g -open sets U , V such that x ∈ U and A ⊆ V .

Corollary 3.10

For an m-space ( X , m ) , the following properties are equivalent:

1. X is m g -regular.

2. For any m-open set U containing x ∈ X , there exists an m g -open set V such that x ∈ V ⊆ m C l ( V ) ⊆ U .

3. For any m-closed set A , the intersection of all m g -closed neighborhoods of A is A .

4. For any set A and any m-open set B such that A ∩ B ≠ ∅ , there exists an m g -open set U such that A ∩ U ≠ ∅ and m C l ( U ) ⊆ B .

5. For any nonempty set A and any m-closed set B such that A ∩ B = ∅ , there exist disjoint m g -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Corollary 3.11

For a topological space ( X , τ ) , the following properties are equivalent:

1. X is g -regular.

2. For any open set U containing x ∈ X , there exists a g-open set V such that x ∈ V ⊆ C l ( V ) ⊆ U .

3. For any closed set A , the intersection of all g-closed neighborhoods of A is A .

4. For any set A and any open set B such that A ∩ B ≠ ∅ , there exists a g -open set U such that A ∩ U ≠ ∅ and C l ( U ) ⊆ B .

5. For any nonempty set A and any closed set B such that A ∩ B = ∅ , there exist disjoint g -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Definition 3.12

A hereditary bi m-space ( X , m , n , ℋ ) is said to be ( m , n ) - ℋ g ∗ - r e g u l a r if for each m n - ℋ g -closed set A and each point x ∉ A , there exist disjoint n ℋ ∗ -open sets U , V such that x ∈ U and A ⊆ V .

Theorem 3.13

For a hereditary bi m-space ( X , m , n , ℋ ) , the following properties are equivalent:

1. X is ( m , n ) - ℋ g ∗ -regular.

2. For any m n - ℋ g -open set U containing x ∈ X , there exists an n ℋ ∗ -open set V such that x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ U .

3. For any m n - ℋ g -closed set A , the intersection of all n ℋ ∗ -closed neighborhoods of A is A .

4. For any set A and any m n - ℋ g -open set B such that A ∩ B ≠ ∅ , there exists an n ℋ ∗ -open set U such that A ∩ U ≠ ∅ and n Cl ℋ ∗ ( U ) ⊆ B .

5. For any nonempty set A and any m n - ℋ g -closed set B such that A ∩ B = ∅ , there exist disjoint n ℋ ∗ -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Proof

(1) ⇒ (2): Let U be an m n - ℋ g -open set such that x ∈ U . Then X − U is an m n - ℋ g -closed set not containing x . By hypothesis, there exist disjoint n ℋ ∗ -open sets V , W such that x ∈ V and X − U ⊆ W . Hence, X − W ⊆ U , V ∩ W = ∅ and so V ⊆ X − W . Thus, x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ n Cl ℋ ∗ ( X − W ) = X − W ⊆ U .

(2) ⇒ (3): Let A be any m n - ℋ g -closed set and x ∉ A . Then X − A is an m n - ℋ g -open set containing x . By hypothesis, there exists an n ℋ ∗ -open set V such that x ∈ V ⊆ n Cl ℋ ∗ ( V ) ⊆ X − A . Thus, A ⊆ X − n Cl ℋ ∗ ( V ) ⊆ X − V . Since X − n Cl ℋ ∗ ( V ) is n H ∗ -open, then X − V is an n ℋ ∗ -closed neighborhood of A and x ∉ X − V . This shows that A is the intersection of all the n ℋ ∗ -closed neighborhoods of A .

(3) ⇒ (4): Let A be any set and B be any m n - ℋ g -open set such that A ∩ B ≠ ∅ . Let x ∈ A ∩ B . Then, X − B is m n - ℋ g -closed and x ∉ X − B . By hypothesis, there exists an n ℋ ∗ -closed neighborhood V of X − B such that x ∉ V . Let X − B ⊆ G ⊆ V , where G is n H ∗ -open. Then U = X − V is an n H ∗ -open set such that x ∈ U and A ∩ U ≠ ∅ . Furthermore, X − G is n ℋ ∗ -closed and n Cl ℋ ∗ ( U ) = n Cl ℋ ∗ ( X − V ) ⊆ n Cl ℋ ∗ ( X − G ) ⊆ B .

(4) ⇒ (5): Let A be any nonempty set and B be any m n - ℋ g -closed set such that A ∩ B = ∅ . Then X − B is an m n - ℋ g -open set and A ∩ ( X − B ) ≠ ∅ . By hypothesis, there exists an n ℋ ∗ -open set U such that A ∩ U ≠ ∅ and U ⊆ n Cl ℋ ∗ ( U ) ⊆ X − B . Let V = X − n Cl ℋ ∗ ( U ) . Then U and V are disjoint n ℋ ⋆ -open sets such that B ⊆ X − n Cl ℋ ∗ ( U ) = V .

(5) ⇒ (1): Let B be an m n - ℋ g -closed set and x ∉ B . Put A = { x } . Then, there exist disjoint n ℋ ∗ -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V , hence x ∈ U . Thus, X is ( m , n ) - ℋ g ∗ –regular.□

Corollary 3.14

For an m-space ( X , m ) , the following properties are equivalent:

1. X is m 0 -regular.

2. For any m g -open set U containing x ∈ X , there exists an m 0 -open set V such that x ∈ V ⊆ m C l ( V ) ⊆ U .

3. For any m g -closed set A , the intersection of all m 0 -closed neighborhoods of A is A .

4. For any set A and any m g -open set B such that A ∩ B ≠ ∅ , there exists an m 0 -open set U such that A ∩ U ≠ ∅ and m C l ( U ) ⊆ B .

5. For any nonempty set A and any m g -closed set B such that A ∩ B = ∅ , there exist disjoint m 0 -open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Corollary 3.15

For a topological space ( X , τ ) , the following properties are equivalent:

1. X is g -regular (in the sense of Munshi).

2. For any g -open set U containing x ∈ X , there exists an open set V such that x ∈ V ⊆ C l ( V ) ⊆ U .

3. For any g -closed set A , the intersection of all closed neighborhoods of A is A.

4. For any set A and any g -open set B such that A ∩ B ≠ ∅ , there exists an open set U such that A ∩ U ≠ ∅ and C l ( U ) ⊆ B .

5. For any nonempty set A and any g -closed set B such that A ∩ B = ∅ , there exist disjoint open sets U , V such that A ∩ U ≠ ∅ and B ⊆ V .

Definition 4.1

A hereditary bi m-space ( X , m , n , ℋ ) is said to be ( m , n ) - ℋ g -normal if for every pair of disjoint m-closed sets A and B in X , there exist disjoint n ℋ ∗ -open sets U and V such that A ⊆ U and B ⊆ V .

Theorem 4.2

For a hereditary bi m-space ( X , m , n , ℋ ) , the following properties are equivalent:

1. X is ( m , n ) - ℋ g -normal.

2. For any disjoint m-closed sets A and B , there exist disjoint m n - ℋ g -open sets U , V such that A ⊆ U and B ⊆ V .

3. For any m-closed set A and any m-open set U containing A , there exists an m n - ℋ g -open set G such that A ⊆ G ⊆ n Cl ℋ ∗ ( G ) ⊆ U .

4. For any m-closed set A and any m-open set U containing A , there exists an n ℋ ∗ -open set G such that A ⊆ G ⊆ n Cl ℋ ∗ ( G ) ⊆ U .

5. For any disjoint m-closed sets A and B , there exists an m n - ℋ g -open set V such that A ⊆ V and n Cl ℋ ∗ ( V ) ∩ B = ∅ .

6. For any disjoint m-closed sets A and B , there exists an n ℋ ∗ -open set G such that A ⊆ G and n Cl ℋ ∗ ( G ) ∩ B = ∅ .

Proof

(1) ⇒ (2): The proof follows from the fact that every n H ∗ -open set is m n - ℋ g -open.

(2) ⇒ (3): Let A be any m-closed set and U any m-open set such that A ⊆ U . Then, A and X − U are disjoint m-closed sets. By hypothesis, there exist disjoint m n - ℋ g -open sets G , V such that A ⊆ G and X − U ⊆ V . Thus, G ⊆ X − V ⊆ U and since X − V is an m n - ℋ g -closed set, then A ⊆ G ⊆ n Cl ℋ ∗ ( G ) ⊆ n Cl ℋ ∗ ( X − V ) ⊆ U .

(3) ⇒ (4): Let A be any m-closed set and U be any m-open set such that A ⊆ U . By hypothesis, there exists an m n - ℋ g -open set V such that A ⊆ V ⊆ n Cl ℋ ∗ ( V ) ⊆ U . By Lemma 3.3, A ⊆ n Int ℋ ∗ ( V ) . Let G = n Int ℋ ∗ ( V ) . Then G is an n ℋ ∗ -open set and we have A ⊆ G ⊆ n Cl ℋ ∗ ( G ) ⊆ n Cl ℋ ∗ ( V ) ⊆ U .

(4) ⇒ (5): Let A and B be any disjoint m-closed sets. Then, X − B is an m-open set such that A ⊆ X − B and by hypothesis, there exists an n ℋ ∗ -open set V such that A ⊆ V ⊆ n Cl ℋ ∗ ( V ) ⊆ X − B . Thus, V is m n - ℋ g -open, A ⊆ V and n Cl ℋ ∗ ( V ) ∩ B = ∅ .