On the topological covering properties by means of generalized open sets

Definition 1

[20] A subset A of a topological space ( X , τ ) is called g -closed if A ⊆ G ∈ τ implies that A ¯ ⊂ G . A subset B of X is called g -open if B c is g -closed.

Definition 2

[12] A topological space X is GO-compact if for every cover of X by g -open sets of X has finite subcover.

Definition 3

[17]  S f i n ( A , ℬ ) denotes the following selection principle: for each sequence { A n : n ∈ N } of elements of A , there is a sequence { B n : n ∈ N } of finite sets such that for each n ∈ N , B n ⊆ A n , and ⋃ n ∈ N B n ∈ ℬ , where A and ℬ are families of subsets of a space X or collection of families of subsets of a space X .

Proposition 1

In a topological space ( X , τ ) , if A ⊆ U ∈ τ implies that A ¯ ⊆ U then A c ⊇ V , where V is a closed set, implies that int ( A c ) ⊇ V .

Proof

Let A c ⊇ V , where V be any closed set.

1. A ⊆ V c = U (open)

2. A ⊆ U

3. A ¯ ⊆ U = V c [by the given condition.]

4. A ¯ c ⊇ V

5. int ( A ¯ c ) ⊇ V .

Let there exists a open set B such that A ¯ c ⊆ B ⊆ A c . Therefore, A ¯ ⊇ B c ⊇ A , which implies that B c is a closed set smaller than A ¯ containing A . But A ¯ is the smallest closed set containing A . Therefore, A ¯ = B c . Hence, B = A ¯ c . So we can say that A ¯ c is the smallest open set inside A c .

∴ int ( A c ) = A ¯ c .

⇒ int ( A c ) ⊇ V .□

Definition 4

A subset A of a topological space X is called g -open set, if V ⊆ int ( A ) whenever V ⊆ A and for all closed set V .

Example 3.1

Let X = { a , b , c , d , e } and τ = { ∅ , { a } , { e } , { a , e } , X } .

Then ( X , τ ) is a topological space and C = { ∅ , { b , c , d , e } , { a , b , e , d } , { b , e , d } , X } is the collection of all closed subsets of X .

We take A = { a , e , d } , then int ( A ) = { a , e } . Only closed set inside A is ∅ and ∅ ⊆ int ( A ) . Therefore, A is g -open set. Let B = A c = { b , c } then B ¯ = { b , c , d } . Only open set containing B is X and B ¯ ⊆ X . Therefore, B is g -closed set. Now for B = { b , c } , int ( B ) = ∅ . Only closed set inside B is ∅ and ∅ ⊆ int ( B ) . B is both g -open and g -closed. Hence, B is g -clopen set.

Definition 5

Let G O be the collection of all g -open covers of a space X . A topological space will be said to GO-Menger space if it satisfies the selection principle S f i n ( G O , G O ) .

Definition 6

A topological space ( X , τ ) is called a GO-Lindelöf space if for every cover of X by g -open sets of X has countable subcover.

Proposition 2

1. Every GO-compact space is compact space.

2. Every GO-Lindelöf space is Lindelöf space.

3. Every GO-Menger space is Menger space.

Proof

Since every open cover is a g -open cover. Therefore, the aforementioned propositions follow directly.□

Theorem 3.2

1. Every GO-compact space is GO-Menger space.

2. Every GO-Menger space is GO-Lindelöf space.

Proof

(1) Let ( X , τ ) be GO-compact space and { A n : n ∈ N } be a sequence such that A n ∈ G O , where G O is the collection of all g -open covers of X . Since ( X , τ ) is a GO-compact space, then for each n ∈ N , A n has a finite subcover ℬ n . Therefore, there exist a sequence { ℬ n : n ∈ N } of g -open covers, where ℬ n ⊆ A n is finite with ⋃ ℬ n = X , ∀ n ∈ N . Therefore, ⋃ n ∈ N ℬ n is a g -open cover of X , i.e., ⋃ n ∈ N ℬ n ∈ G O . Hence, ( X , τ ) is GO-Menger space.

(2) Let ( X , τ ) be a GO-Menger space and U be an arbitrary g -open cover. Therefore, { A n = U : n ∈ N } is a sequence of g -open covers of X . But ( X , τ ) is GO-Menger space, then for every n ∈ N , there exists a finite subset ℬ n ⊆ A n = U such that ⋃ n ∈ N ℬ n ∈ G O . But countable union of finite sets is countable. Therefore, U ′ = ⋃ n ∈ N B n is a countable g -open subcover of U .

Hence, ( X , τ ) is GO-Lindelöf space.□

Example 3.3

There exists a compact space, which is not GO-compact space.

Let X = N with the topology τ = { ∅ , X , { 2 , 4 , 6 , … } , { 1 , 3 , 5 , … } } . So, ( X , τ ) is trivially compact. Now consider the cover V = { V n = { n } : n ∈ N } . For each n ∈ N , there does not exist any closed subset of { n } other than ∅ . Therefore, each { n } is g -open set but not open. Thus, V is a g -open cover of X . But V is an infinite collection of disjoint g -open sets. So it cannot have a finite subcover. So, ( X , τ ) is not GO-compact.

Example 3.4

There exists a Menger space, which is not GO-Menger space.

Let X = R + ⋃ { 0 } . Then ℬ = { [ 0 , n ) : n ∈ N } ⋃ { ∅ } is a base for suitable topology τ (say). Now consider an arbitrary sequence { A n : n ∈ N } of open covers of X . So there exist a U n ∈ A n such that [ 0 , n ) ⊆ U n , ∀ n ∈ N . Therefore, ℱ n = { U n } is a finite subset of A n , for all n ∈ N and ⋃ n ∈ N ℱ n is an open cover of X . Therefore, ( X , τ ) is a Menger space.

In the same space, V = { { r } : r ∈ X } is a g -open cover of X and { ℰ n = V : n ∈ N } is a sequence of g -open cover of X . Now if we consider any finite subset ℱ n of ℰ n , for all n ∈ N . Then ⋃ n ∈ N F n will never be able to cover of X . Since countable union of finite set is countable. But X is an uncountable set, i.e., X is not a GO-Menger space.

Example 3.5

A Lindelöf space may not be a GO-Lindelöf space.

Let ℬ = { ( 1 2 n , 1 2 n − 1 ] : n ∈ N } ⋃ { ∅ } is a base for a suitable topology on X = ( 0 , 1 ] and τ be the topology generated by ℬ . Then ( X , τ ) is a Lindelöf space.

Now consider a cover V = { { p } : p ∈ ( 0 , 1 ] } , which forms a g -open cover. Since { p } does not contain any closed set other than ∅ . Therefore, each { p } is g -open set but not open. Here, V is an infinite collection of uncountable disjoint g -open sets, so there does not exist any countable subcover of V . Therefore, ( X , τ ) is not a GO-Lindelöf space.

Example 3.6

A GO-Menger space may not be a GO-compact space.

Let X = N equipped with the discrete topology τ δ and { A n : n ∈ N } be an arbitrary sequence of g -open covers of X . For each n ∈ N , there exists a A n ∈ A n such that n ∈ A n . So for each n ∈ N , if we choose U n = { A n } . Then U n ⊆ A n is a finite subset for each n ∈ N . Also ⋃ n ∈ N U n forms a g -open cover of X . So the space is GO-Menger.

Open Problem 3.7

Does there exists a space which is GO-Lindelöf but not GO-Menger?

Definition 7

Let G ( X ) and G ( Y ) denote the collection of all g -open sets of X and collection of all g -open sets of Y , respectively. A map f : ( X , τ ) → ( Y , σ ) is called

1. g -continuous if f − 1 ( A ) ∈ G ( X ) for all A ∈ σ [12]

2. g c -irresolute if f − 1 ( A ) ∈ G ( X ) for all A ∈ G ( Y ) [12]

Theorem 4.1

g-continuous image of a GO-Menger space is a Menger space.

Proof

Let f : X → Y be a g -continuous map from a GO-Menger space ( X , τ ) onto a topological space ( Y , σ ) . Let { A n : n ∈ N } be an arbitrary sequence of open covers of Y .

Then { ℬ n = f − 1 ( A n ) = { f − 1 ( A ) : A ∈ A n } : n ∈ N } is a sequence of g -open cover of X . Now X is a GO-Menger space, therefore for every n ∈ N , there exists a finite subset C n ⊆ ℬ n = f − 1 ( A n ) such that ⋃ n ∈ N C n is an g -open covers of X . Since f is onto so f ( ⋃ n ∈ N C n ) = ⋃ n ∈ N f ( C n ) = Y . Moreover, { f ( C n ) : n ∈ N } is a sequence such that f ( C n ) is a finite subset of A n for each n ∈ N .

Hence, Y is a Menger space.□

Theorem 4.2

gc-irresolute image of a GO-Menger space is GO-Menger space.

Proof

Let f : X → Y be a gc-irresolute map from a GO-Menger space ( X , τ ) onto a topological space ( Y , σ ) . Let { U n : n ∈ N } be an arbitrary sequence of g -open covers of Y .

Then { V n = f − 1 ( U n ) = { f − 1 ( U ) : U ∈ U n } : n ∈ N } is a sequence of g -open cover of X . Now X is a GO-Menger space, therefore for every n ∈ N , there exists a finite subset W n ⊆ V n = f − 1 ( U n ) such that ⋃ n ∈ N W n is an g -open covers of X . Since f is onto so f ( ⋃ n ∈ N W n ) = ⋃ n ∈ N f ( W n ) = Y . Moreover, { f ( W n ) : n ∈ N } is a sequence such that f ( W n ) is a finite subset of U n for each n ∈ N .

Hence, Y is a GO-Menger space.□

Definition 8

In a space X , a set A ⊆ X will be called a g -clopen set if A is both g -open and g -closed.

Theorem 4.3

Every g-clopen subspace of a GO-Menger space is GO-Menger space.

Proof

Let A be a g -clopen subspace of X and { U n : n ∈ N } be a sequence of g -open covers of A . As A is a g -open set and g -open sets in a g -open subspace are g -open in the whole space, so each U n is a collection of g -open sets in X . On the other hand, since A is also g -closed, we conclude that each U n ⋃ { X ⧹ A } = G n is a g -open cover of X . But X is a GO-Menger space. So there exists a finite subset W n ⊆ G n , for all n ∈ N such that ⋃ n ∈ N W n is an g -open cover of X .

Now V n = W n ⧹ { X ⧹ A } is a finite subset of U n for each n ∈ N , such that ⋃ n ∈ N V n is an g -open cover of A . Hence, A is GO-Menger space.□

Theorem 5.1

Following four statements are equivalent in a topological space ( X , τ ) .

1. X is GO-compact.

2. For a family G of g -open sets, if X can not be covered by any finite subfamily of G , then G can not cover X.

3. For a family ℱ of g-closed sets, if ℱ satisfies finite intersection property, then ⋂ { F : F ∈ ℱ } ≠ ∅ .

4. For a family ℱ of subsets of X , if ℱ has finite intersection property, then ⋂ { g − c l ( F ) : F ∈ ℱ } ≠ ∅ .

Proof

(1) ⇔ (2) The definition of GO-compact space says that every g -open cover of X has a finite subcover. So statements ( 1 ) and ( 2 ) are contrapositive to each other. Hence, ( 1 ) ⇔ ( 2 ) .

( 1 ) ⇒ ( 3 ) Let statement ( 1 ) is true and suppose ℱ is a family of g -closed sets with finite intersection property and ⋂ { F : F ∈ ℱ } = ∅ .

Therefore, G = { X ∕ F : F ∈ ℱ } is a family of g -open sets of X .

Also, X ∕ { ⋂ { F : F ∈ ℱ } } = X ∕ ∅ = X

⇒ ⋃ { X ∕ F : F ∈ ℱ } = X

Therefore, G is a g -open cover of X . By statement ( 1 ) , G has a finite subcover G ′ , so ⋃ G ′ = X and G ′ is finite.

⇒ X ∕ ⋃ G ′ = ∅ and G ′ is finite.

⇒ ⋂ { X ∕ G : G ∈ G ′ } = ∅ and G ′ is finite.

⇒ { X ∕ G : G ∈ G } is a finite subsets of ℱ and ⋂ { X ∕ G : G ∈ G } = ∅ .

i.e., ℱ does not satisfy the finite intersection property, which is a contradiction.

Therefore ⋂ { F : F ∈ ℱ } ≠ ∅ .

( 3 ) ⇒ ( 1 ) Let condition (3) holds, and G be a g -open cover of X .

Then ⋃ G = X

⇒ X ∕ ( ⋃ G ) = ∅

⇒ ⋂ { X ∕ G : G ∈ G } = ∅ .

Therefore, ℱ = { F = X ∕ G : G ∈ G } is a family of g -closed sets with ⋂ ℱ = ∅ . Therefore, by contrapositivity of statement (3), there exists a finite subset ℱ ′ of ℱ , such that ⋂ ℱ ′ = ∅ .

⇒ ⋂ { F : F ∈ ℱ ′ } = ∅ , where ℱ ′ is finite.

⇒ X ∕ ⋂ { F : F ∈ ℱ ′ } = X

⇒ ⋃ { X ∕ F : F ∈ ℱ ′ } = X

⇒ ⋃ { G = X ∕ F : F ∈ ℱ ′ } = X .

Therefore, { G = X ∕ F : F ∈ ℱ ′ } is a finite subcover of G . Hence, X is GO-compact.

( 3 ) ⇔ ( 4 ) Let statement (3) holds and ℱ is a family of subsets of X with finite intersection property. Then { g - c l ( A ) : A ∈ ℱ } is a family of g -closed subsets of X and by statement (2) ⋂ { g - c l ( A ) : A ∈ ℱ } ≠ ∅ .

( 4 ) ⇔ ( 3 ) Follows directly as g - c l ( A ) = A for every g -closed set A .□

Definition 9

A point a is said to be a g-cluster point of a net S = { a α : α ∈ ∑ } if for every g -open set G containing “a” and α 0 ∈ ∑ , and there exists a α ∈ ∑ and α ≥ α 0 , such that a α ∈ G .

Theorem 5.2

The set of all g-cluster points of an arbitrary net in a topological space X is g-closed.

Proof

Let us consider a net { a α : α ∈ ∑ } in X and suppose g - c p { a α } α ∈ ∑ = A . Let x ∈ X ⧹ A . Therefore, there exists an g -open set U x containing “x” and α 0 ∈ ∑ such that x β ∉ U x , ∀ β ∈ ∑ and β ≥ α 0 ; therefore, U x is a g -open set such that U x ⊂ X ∕ A .

⇒ x ∈ U x = g - int ( U x ) ⊆ g - int ( X ∕ A ) = X ∕ g - c l ( A ) .

By contrapositivity, g - c l ( A ) ⊂ A .

Therefore, A = g - c l ( A ) . Hence, A is g -closed.□

Theorem 5.3

In a topological space ( X , τ ) each net { a α : α ∈ ∑ } has at least one g-cluster point if and only if the space is a GO-compact space.

Proof

Let in a topological space ( X , τ ) each net has at least one g -cluster point, but it is not a GO-compact space. Therefore, there exists a g -open cover { U α : α ∈ Λ } of X , which does not have any finite subcover. Let P ( Λ ) be the family of all finite subsets of Λ . Clearly, ( P ( Λ ) , ⊆ ) is a directed set. For each I ∈ P ( Λ ) , we can choose x I ∈ X ⧹ ⋃ { U K : K ∈ I } . Now, we think of the net { x I : I ∈ P ( Λ ) } .

But g - c l { x I } I ∈ P ( Λ ) is non-empty. Let p ∈ g - c l { x I } I ∈ P ( Λ ) . Let α 0 ∈ Λ such that p ∈ U α 0 and take I 0 = { U α 0 } therefore I 0 ∈ P ( Λ ) . By the definition of g-cluster point for each I 0 ∈ P ( Λ ) , then there exists I 0 * ∈ P ( Λ ) with I 0 ⊆ I 0 * and x I 0 * ∈ U α 0 .

But, x I 0 * ∈ X ⧹ ⋃ { U k : k ∈ I 0 * } ⊆ X ∕ U α 0 .

Which is a contradiction. Hence, ( X , τ ) is GO-compact space.

Conversely, Let X be a GO-compact space, but in X , there exists a net { a α : α ∈ ∑ } , which does not have any g-cluster points. i.e. g − c l { a α } α ∈ ∑ = ∅ . Therefore, for every x ∈ X , ∃ x ∈ U x ∈ G O ( X ) and α x ∈ ∑ such that a β ∉ U x ∀ β ≥ α x and β ∈ ∑ . Then the collection { U x : x ∈ X } is a g -open cover of X , and it has finite subcover say { U k = U x k : k = 1 , 2 , 3 , … , n } .

Now, we take α ∈ ∑ such that α ≥ α x k { k = 1 , 2 , … , n } for every β ≥ α , we have a β ∉ U x k ∀ { k = 1 , 2 , … , n } ⇒ a β ∉ ⋃ k = 1 n U x k = X , which is a contradiction. Hence, every net in X at least one g-cluster point.□

Definition 10

In a topological space ( X , τ ) , a filter base ℱ will be called g-convergent to a point x ∈ X if for every g -open neighborhood U of x , we can find a B ∈ ℱ such that B ⊂ U . Symbolically, we will write ℱ → g -con x .

Definition 11

In a topological space ( X , τ ) , filter ℱ will be said to g-accumulate at x ∈ X , if U ∩ B ≠ ∅ for every g -open neighborhood U of x and every B ∈ ℱ . Symbolically, we will write ℱ → g -accu x .

Remark

In a topological space ( X , τ ) for a filter base ℱ and a point x ∈ X , then

Theorem 6.1

In a topological space ( X , τ ) , for a filter base ℱ and ℱ → g -accu x where x ∈ X , if and only if x ∈ ⋂ { g - c l ( B ) : B ∈ ℱ } .

Proof

Let the filter base ℱ be such that ℱ → g -accu x .

⇔ U ∩ B ≠ ∅ ∀ x ∈ U ∈ G ( X ) and B ∈ ℱ .

⇔ x ∈ g - c l ( B ) ∀ B ∈ ℱ .

⇔ x ∈ ⋂ { g - c l ( B ) : B ∈ ℱ } .□

Proposition 3

In a topological space ( X , τ ) , if ℱ is a maximal filterbase such that ℱ → g -accu a , where a ∈ X then ℱ → g -con a .

Proof

Let ℱ is a maximal filter base and ℱ → g -accu a , where a ∈ X but ℱ ⟶ g̸ -con a .

Therefore, there exists a g -open set U α containing x such that B ⊈ U α , ∀ B ∈ ℱ . So, B ∩ ( X ∕ U α ) ≠ ∅ , ∀ B ∈ ℱ . Then ℱ ∪ { U α ∩ B : B ∈ ℱ } become a filter-base for X such that ℱ ⊂ ℱ ∪ { U α ∩ B : B ∈ ℱ } , which is a contradiction to the fact that ℱ is a maximal filter-base. Hence, the proposition is proved.□

Theorem 6.2

In a topological space ( X , τ ) , next few statements are equivalent:

1. X is GO-compact.

2. For a maximal filter-base ℱ 0 , ℱ 0 → g -con a , for some a ∈ X .

3. For a filter-base ℱ , ℱ → g -accu a , for some a ∈ X .

Proof

( 1 ) ⇒ ( 2 ) Let ( X , τ ) be a GO-compact space and ℱ 0 be a maximal filterbase with ℱ 0 not being g-convergent to any point of X . So by the proposition [6.4], ℱ 0 does not g-accumulate to any point of X , i.e., for every a ∈ X , we can find a g -open set U a containing a and F a ∈ ℱ 0 with U a ∩ F a = ∅ . Thus, { U a : a ∈ X } is a g -open cover of X . But X is GO-compact. So there exists a finite g -open subcover { U a i : i = 1 , 2 , 3 , … , k } . So, X = ⋃ i = 1 k U a i . Also, F a i ⊆ X ∕ U a i .

But ℱ 0 being a filterbase contains F 0 ∈ ℱ such that F 0 ⊆ ⋂ i = 1 k F a i ⊆ ⋂ i = 1 k X ∕ U a i = X ∕ ⋃ i = 1 k U a i = X ∕ X = ∅ .

⇒ F 0 = ∅ , which is a contradiction to the fact that ℱ is a filter-base. Hence, ℱ 0 convergence to some point of X .

( 2 ) ⇒ ( 3 ) Let statement ( 2 ) holds and ℱ is a filter-base on X . Obviously, there exists a maximal filterbase ℱ 0 with ℱ ⊆ ℱ 0 . By ( 2 ) ℱ 0 → g -con a , for some a ∈ X and B ∈ ℱ be arbitrary. Now for every g -open set U containing a , ∃ B U ∈ ℱ 0 such that B U ⊆ U . So, ∅ ≠ B U ∩ B ⊂ U . Hence, U ∩ B ≠ ∅ , i.e., ℱ → g -accu a .

( 3 ) ⇒ ( 1 ) Let { F α : α ∈ Λ } be a family of g -closed sets such that ⋂ α ∈ Λ F α = ∅ . P ( Λ ) be the family of all finite subsets of Λ . Suppose ∩ { F α : α ∈ J } ≠ ∅ , ∀ J ∈ P ( Λ ) . Therefore, ℱ = { ∩ { F α : α ∈ J } : J ∈ P ( Λ ) } is a filterbase on X . But ℱ → g -accu a for some a ∈ X . Now, { X ∕ F α : α ∈ Λ } is a g -open cover of X . So, there exists α 0 ∈ Λ such that a ∈ X ∕ F α 0 . Therefore, X ∕ F α 0 is a g -open set containing a and F α 0 ∈ ℱ . Moreover, ( X ∕ F α 0 ) ∩ F α 0 = ∅ , which is a contradiction to the fact that ℱ → g -accu a . So ∩ { F α : α ∈ J } = ∅ .

Thus, by contrapositivity of theorem 5.1, X is a GO-compact space.□