A variation of the class of statistical γ covers

Prasenjit Bal; Debjani Rakshit

doi:10.1515/taa-2023-0101

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A variation of the class of statistical γ covers

Prasenjit Bal and Debjani Rakshit

Published/Copyright: August 29, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

In this article, we introduce s-s- γ cover using the notion of star operator, which is an extension of the previous results on s- γ covers. We also define s-s-dense set to address the issue that a statistical dense subset of an s-s- γ cover is not an s-s- γ cover. Furthermore, we study some results on s- γ covers in subspace topology.

Keywords: asymptotic density; open cover; γ cover; statistical-γ cover; star operator

MSC 2010: 40A35; 54A05; 54D20; 54E15

1 Introduction

The notion of natural density or asymptotic density is defined as:

δ ( A ) = lim n → ∞ 1 n ∣ { k < n : k ∈ A } ∣ ,

where A ⊆ N (the set of all natural numbers). In this respect, it should be mentioned that Fast [16] used the concept of asymptotic density to expand the concept of usual convergence to statistical convergence (also Schoenberg [26]). After Fridy’s [17] and Connor’s [11] work on statistical convergence, a lot of research has been carried out on this convergence and it is topological repercussions (see [13,23,25]).

In particular, Maio and Kočinac [22] have lately introduced the idea of statistical convergence in topological spaces as well as in uniform spaces. They also constructed a new class of open covers that arose from the interplay of an existing open cover with asymptotic density. Recently, Das [12] generalized this approach by using ideals instead of statistical density.

In [14], van Douwen introduced the star operator as follows: S t ( A , U ) represents the set ⋃ { U ∈ U : A ∩ U ≠ ∅ } if A ⊆ X , and U represents a collection of subsets of X . S t ( x , U ) = ⋃ { U ∈ U : x ∈ U } for a point x ∈ X . Later on, in [18–21], Kočinac extended the concept of selection principles to star selection principles by incorporating the star operator. Several topological properties related to this operator have been studied in many recent research studies [2–10,24,27–29].

Inspired by [22], we use the star operator to create and study a new class of γ covers in topological spaces. No particular separation axiom is assumed in this article otherwise stated.

2 Preliminaries

In this section, we have mentioned some pre-requisite definitions and results for ready references.

Definition 2.1

[15] If A ⊆ X in a topological space ( X , τ ) , then τ A = { U ∩ A : U ∈ τ } forms a topology on A . This topological space ( A , τ A ) is called a subspace of ( X , τ ) .

A collection U of subsets of X in a topological space ( X , τ ) is called a cover if ⋃ U = X . If every element of U is an open set, then U is called an open cover [15].

Definition 2.2

[1] In a topological space, a family of pairwise disjoint open sets is called a cellular open family.

Definition 2.3

[22] A countable cover U = { U n : n ∈ N } of a topological space ( X , τ ) is said to be a statistical γ cover (in short s- γ cover) if for each x ∈ X , the set { n ∈ N : x ∉ U n } has natural density zero, i.e., δ ( { n ∈ N : x ∉ U n } ) = 0 .

Definition 2.4

[22] Let U = { U n : n ∈ N } be a cover of a topological space ( X , τ ) . A subset V = { U m 1 , U m 2 , U m 3 , … } of U is called s-dense in U if the set M = { m 1 , m 2 , m 3 , … } has natural density 1.

Theorem 2.5

[22] A s-dense subset of a s- γ cover of a topological space is also an s- γ cover for that topological space.

Theorem 2.6

[22] An open cover U = { U n : n ∈ N } of a topological space ( X , τ ) is a s- γ cover if and only if for each finite F ⊆ X , the set { n ∈ N : F ⊈ U n } has asymptotic density zero.

3 On statistical γ covers

Example 3.1

Sub-cover of an s- γ cover may not be an s- γ cover.

Let X = { p , q } , τ = { ∅ , { p } , X } . Then, ( X , τ ) is a topological space. Consider the cover U = { U n : n ∈ N } , where

U n = { p } if n ∈ { n 2 : n ∈ N } , X if n ∉ { n 2 : n ∈ N } .

For every x ∈ X , δ ( { n ∈ N : x ∉ U n } ) = 0 . ∴ U is an s- γ cover of ( X , τ ) . Consider the sub-cover V of U such that V = { V n : n ∈ N } , where

V n = U 2 if n = 2 , U n 2 otherwise.

Here, δ ( { n ∈ N : x ∉ V n } ) = 1 . Hence, the result.

Theorem 3.2

Let U and V be two s- γ covers of a topological space ( X , τ ) . Then, U ⊔ V = { U i ∪ V i : U i ∈ U , V i ∈ V , and i ∈ N } is also an s- γ cover.

Proof

Let U and V be two s- γ covers of the topological space ( X , τ ) .

So for all x ∈ X ,

δ ( { n ∈ N : x ∉ U n } ) = 0 ⇒ δ ( { n ∈ N : x ∈ U n } ) = 1

and

δ ( { n ∈ N : x ∉ V n } ) = 0 ⇒ δ ( { n ∈ N : x ∈ V n } ) = 1 .

Obviously, { n ∈ N : x ∈ U n } ⊆ { n ∈ N : x ∈ U n ∪ V n } . Therefore, δ ( { n ∈ N : x ∈ U n ∪ V n } ) = 1 . Thus, δ ( { n ∈ N : x ∉ U n ∪ V n } ) = 0 . Hence, U ⊔ V is an s- γ cover.□

Theorem 3.3

Let ( X , τ ) be a topological space and A ⊆ X . If U = { U n : n ∈ N } is an s- γ cover of X, then U A = { U n ∩ A : U n ∈ U a n d n ∈ N } forms an s- γ cover for ( A , τ A ) ( τ A is the sub-space topology).

Proof

Let x ∈ A be arbitrary.

∴ x ∈ X , and also, U = { U n : n ∈ N } is an s- γ cover of ( X , τ ) .

Therefore, δ ( { n ∈ N : x ∉ U n } ) = 0 .

⇒ δ ( { n ∈ N : x ∈ U n } ) = 1 .

⇒ δ ( { n ∈ N : x ∈ U n ∩ A } ) = 1 [ ∵ x ∈ A ].

⇒ δ ( { n ∈ N : x ∉ U n ∩ A } ) = 0 .

Since x is an arbitrary element of A , δ ( { n ∈ N : x ∉ U n ∩ A } ) = 0 ∀ x ∈ A . U A is an s- γ cover for ( A , τ A ) .□

Theorem 3.4

Let U = { U n : n ∈ N } be an open cover of a topological space ( X , τ ) , for i = 1 , 2 , 3 , … n such that X = ⋃ i = 1 n X i . If U X i = { X i ∩ U n : n ∈ N } are s- γ covers of ( X i , τ X i ) for i = 1 , 2 , 3 , … n , then U is an s- γ cover for ( X , τ ) .

Proof

Let U X i ’s be s- γ covers of ( X i , τ X i ) . Therefore, δ ( { n ∈ N : x ∉ X i ∩ U n } ) = 0 .

Let p ∈ X = ⋃ i = 1 n X i be arbitrary.

So there exists at least one j ∈ { 1 , 2 , 3 , … n } such that p ∈ X j .

But δ ( { n ∈ N : p ∉ U n } ) ≤ δ ( { n ∈ N : p ∉ X j ∩ U n } ) = 0 [ ∵ X j ∩ U n ⊆ U n ].

i.e., δ ( { n ∈ N : x ∉ U n } ) = 0 for all x ∈ X .

Hence, U is an s- γ cover of ( X , τ ) .□

According to Maio and Kočinac (Lemma 3.1 of [22]), an open cover U = { U n : n ∈ N } of a topological space ( X , τ ) is an s- γ cover if and only if for each finite F ⊆ X , the set { n ∈ N : F ⊈ U n } has asymptotic density zero. We propose a strong condition to the result of Maio and Kočinac by stating that the set F need not to be finite, it is enough if F \ U p is finite for each p ∈ { n ∈ N : F ⊈ U n } .

Theorem 3.5

An open cover U = { U n : n ∈ N } of a topological space ( X , τ ) is a s- γ cover if and only if for each F ⊆ X such that F \ U p is a finite set for each p ∈ { n ∈ N : F ⊈ U n } implies that δ ( { n ∈ N : F ⊈ U n } ) = 0 .

Proof

Let U = { U n : n ∈ N } be an s- γ cover and F ⊆ X be such that F \ U p is finite for each p ∈ { n ∈ N : F ⊈ U n } .

Let k ∈ { n ∈ N : F ⊈ U n } be arbitrary.

∴ F ⊈ U k .

⇒ there exists a x ∈ F \ U k .

⇒ x ∈ F and x ∉ U k .

⇒ k ∈ { n ∈ N : x ∉ U n } ∀ x ∈ F \ U k .

∴ { n ∈ N : F ⊈ U n } ⊆ { n ∈ N : x ∉ U n } ∀ x ∈ F \ U k .

Thus, ∀ x ∈ F \ U k , δ ( { n ∈ N : F ⊈ U n } ) ≤ δ ( { n ∈ N : x ∉ U n } ) = 0 [ ∵ U is an s- γ cover of ( X , τ ) ].

Hence, δ ( { n ∈ N : F ⊈ U n } ) = 0 .

Conversely, for each x ∈ X , { x } ⊆ X and { x } \ U p is finite for each p ∈ { n ∈ N : { x } ⊈ U n } implies that δ ( { n ∈ N : { x } ⊈ U n } ) = 0 .

i.e., δ ( { n ∈ N : x ∉ U n } ) = 0 for each x ∈ X .

Hence, U is an s- γ cover.□

4 On statistical star γ covers

We incorporate the star operator in the definition of s- γ cover to bring a variation in the class of s- γ covers.

Definition 4.1

A countable cover U = { U n : n ∈ N } of a topological space ( X , τ ) is said to be a statistical star γ cover (in short s-s- γ cover) if for each x ∈ X , the set { n ∈ N : x ∉ S t ( U n , U ) } has natural density zero, i.e., δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 .

First, we tried to discuss the interdependences of s- γ cover and s-s- γ covers. Then, some basic topological properties are investigated.

Proposition 4.2

In a topological space ( X , τ ) , every s- γ cover is an s-s- γ cover.

Proof

Let U = { U n : n ∈ N } be an s- γ cover of a topological space ( X , τ ) . Then, for every x ∈ X , δ ( { n ∈ N : x ∉ U n } ) = 0 .

Let p ∈ { n ∈ N : x ∉ S t ( U n , U ) } for some x ∈ X . Therefore, x ∉ S t ( U p , U ) . Also, U p ⊆ S t ( U p , U ) . ∴ x ∉ U p . So, p ∈ { n ∈ N : x ∉ U n } .

Thus, { n ∈ N : x ∉ S t ( U n , U ) } ⊆ { n ∈ N : x ∉ U n } for each x ∈ X .

⇒ δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) ≤ δ ( { n ∈ N : x ∉ U n } ) = 0 for each x ∈ X .

⇒ δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 for each x ∈ X .

∴ U is an s-s- γ cover. Hence, the theorem.□

Example 4.3

The converse of the aforementioned proposition (Proposition 4.2) may not be true, i.e., there exists an s-s- γ cover of a topological space, which is not s- γ cover.

Let X = ( − 1 , 1 ) and τ = { ( − r , r ) : r ∈ [ 0 , 1 ] } be a topology on X . Consider the countable open cover U = { U n = ( − 1 n , 1 n ) : n ∈ N } . So, for every n ∈ N , S t ( U n , U ) = X .

Thus, for every x ∈ X , δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 . Therefore, U is an s-s- γ cover. But δ ( { n ∈ N : 1 2 ∉ U n } ) = 1 . Therefore, U is not an s- γ cover.

The relationship between s- γ cover and s-s- γ cover is shown in Figure 1.

$Figure 1 Relationship between s- γ \gamma cover and s-s- γ \gamma cover.$

Figure 1

Relationship between s- γ cover and s-s- γ cover.

Proposition 4.4

A cellular s-s- γ cover of a topological is an s- γ cover of that space.

Proof

Let U = { U n : n ∈ N } be a cellular s-s- γ cover of a topological space ( X , τ ) , i.e., U i ∩ U j = ∅ if i ≠ j . Then, for every n ∈ N , S t ( U n , U ) = U n .

Thus, for every x ∈ X , δ ( { n ∈ N : x ∉ U n } ) = δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 . So, U is an s- γ cover of ( X , τ ) .□

Theorem 4.5

An open cover U = { U n : n ∈ N } of a topological space ( X , τ ) is an s-s- γ cover if and only if for each F ⊆ X such that F \ S t ( U p , U ) is finite for each p ∈ { n ∈ N : F ⊈ S t ( U n , U ) } implies that the natural density of set { n ∈ N : F ⊈ S t ( U n , U ) } is zero.

Proof

Let U = { U n : n ∈ N } be an s-s- γ cover of a topological space ( X , τ ) and F ⊆ X such that F \ S t ( U p , U ) is finite for each p ∈ { n ∈ N : F ⊈ S t ( U n , U ) } .

Let k ∈ { n ∈ N : F ⊈ S t ( U n , U ) } be arbitrary.

⇒ F ⊈ S t ( U k , U ) .

⇒ there exists a x ∈ F \ S t ( U k , U ) .

⇒ x ∈ F and x ∉ S t ( U k , U ) .

⇒ k ∈ { n ∈ N : x ∉ S t ( U n , U ) } ∀ x ∈ F \ S t ( U k , U ) .

∴ { n ∈ N : F ⊈ S t ( U n , U ) } ⊆ { n ∈ N : x ∉ S t ( U n , U ) } . ∀ x ∈ F \ S t ( U k , U ) .

So, δ ( { n ∈ N : F ⊈ S t ( U n , U ) } ) ≤ δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 . [ ∵ U is an s-s- γ cover].

⇒ δ ( { n ∈ N : F ⊈ S t ( U n , U ) } ) = 0 .

Conversely, let for each F ⊆ X such that F \ S t ( U p , U ) is finite for each p ∈ { n ∈ N : F ⊈ S t ( U n , U ) } implies that δ ( { n ∈ N : F ⊈ S t ( U n , U ) } ) = 0 .

Now, for arbitrary x ∈ X , { x } ⊆ X such that { x } \ S t ( U p , U ) is finite for each p ∈ N .

⇒ δ ( { n ∈ N : { x } ⊈ S t ( U n , U ) } ) = 0 , ∀ x ∈ X .

⇒ δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 , ∀ x ∈ X .

∴ U is an s-s- γ cover.□

Theorem 4.6

Let U and V be two s-s- γ covers of a topological space ( X , τ ) . Then, U ⊔ V = { U i ∪ V i : U i ∈ U , V i ∈ V a n d i ∈ N } is also an s-s- γ cover of ( X , τ ) .

Proof

Let U and V be two s-s- γ covers of a topological space ( X , τ ) .

Then, for every x ∈ X ,

δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 and δ ( { n ∈ N : x ∉ S t ( V n , V ) } ) = 0 .

Let p ∈ { n ∈ N : x ∉ S t ( U n ∪ V n , U ⊔ V ) } for some x ∈ X .

⇒ x ∉ S t ( U p ∪ V p , U ⊔ V ) .

⇒ x ∉ U p ∪ V p . [ ∵ U p ∪ V p ⊆ S t ( U p ∪ V p , U ⊔ V ) ].

⇒ x ∉ U p and x ∉ V p .

⇒ p ∈ { n ∈ N : x ∉ U n } .

So, { n ∈ N : x ∉ S t ( U n ∪ V n , U ⊔ V ) } ⊆ { n ∈ N : x ∉ U n } .

⇒ δ ( { n ∈ N : x ∉ S t ( U n ∪ V n , U ⊔ V ) } ) ≤ δ ( { n ∈ N : x ∉ U n } ) = 0 .

∴ ( U ⊔ V ) is an s-s- γ cover.□

Example 4.7

U = { U n : n ∈ N } be an s-s- γ cover for a topological space ( X , τ ) and A ⊆ X . Then, U A = { A ∩ U n : U n ∈ U and n ∈ N } may not be an s-s- γ cover for ( A , τ A ) ( τ A is the subspace topology).

Let ( X , τ ) be a topological space with the cellular open base ℬ = { ∅ , B 1 , B 2 , B 3 } .

∴ U = { U n : n ∈ N } , where

U n = B 1 ∪ B 2 if n ∈ N \ 2 N , B 2 ∪ B 3 if n ∈ 2 N .

is a cover for ( X , τ ) . Moreover, for every n ∈ N , S t ( U n , U ) = X . ∴ δ ( { n ∈ N : x ∉ U n } ) = 0 ∀ x ∈ X .

∴ U is an s-s- γ cover for ( X , τ ) .

Consider the subset A = B 1 ∪ B 3 ⊆ X , then U A = { V n = A ∩ U n : U n ∈ U and n ∈ N } , where

V n = B 1 if n ∈ N \ 2 N , B 2 if n ∈ 2 N .

⇒ S t ( V n , V ) = B 1 if n ∈ N \ 2 N , B 2 if n ∈ 2 N .

Moreover, for every x ∈ A , δ ( { n ∈ N : x ∉ S t ( V n , U A ) } ) = 1 2 ≠ 0

∴ U A is not an s-s- γ cover for ( A , τ A ) . Hence, the result.

The aforementioned result explains the main difference in between the properties of s- γ cover and s-s- γ covers.

Theorem 4.8

Let U = { U n : n ∈ N } be an open cover of a topological space ( X , τ ) and ( X i , τ X i ) are the subspaces of ( X , τ ) for i = 1 , 2 , 3 , … n such that X = ⋃ i = 1 n X i . If U X i = { X i ∩ U n : n ∈ N } are s-s- γ covers of ( X i , τ X i ) , for i = 1 , 2 , 3 , … n , then U is also an s-s- γ cover for ( X , τ ) .

Proof

Let p ∈ X = ⋃ i = 1 n X i be arbitrary.

So there exists at least one j ∈ { 1 , 2 , 3 , … n } such that p ∈ X j .

Since U X j is an s-s- γ cover of ( X j , τ X j ) , ∴ δ ( { n ∈ N : p ∉ S t ( X j ∩ U n , U X j ) } ) = 0 .

⇒ δ ( { n ∈ N : p ∈ S t ( X j ∩ U n , U X j ) } ) = 1 .

Now, let q ∈ { n ∈ N : p ∈ S t ( X j ∩ U n , U X j ) } . So p ∈ S t ( ( X j ∩ U q ) , U X j ) .

But S t ( X j ∩ U q , U X j ) ⊆ S t ( U q , U ) . ∴ q ∈ { n ∈ N : p ∈ S t ( U n , U ) } .

⇒ δ ( { n ∈ N : p ∈ S t ( U n , U ) } ) ≥ δ ( { n ∈ N : p ∈ S t ( X j ∩ U n , U X j ) } ) = 1 .

⇒ δ ( { n ∈ N : p ∈ S t ( U n , U ) } ) = 1 .

⇒ δ ( { n ∈ N : p ∉ S t ( U n , U ) } ) = 0 for all p ∈ X .□

5 Nature of various types of γ covers under variation of dense subsets

It is obvious that infinite subset of γ cover of a topological space is also a γ cover for that topological space. But it may not true for an s- γ cover. We also know that an s-dense subset of an s- γ cover of a topological space is also an s- γ cover for that topological space. Now, the question arises whether s-dense subset of an s-s- γ cover is an s-s- γ cover.

Example 5.1

Let ( X , τ ) be a topological space, where X = [ 0 , a ) and τ is the topology on X induced by the upper limit topology, i.e., ℬ = { [ 0 , α ) : α ∈ [ 0 , a ] } is the base for the topology τ .

Let U = { U n : n ∈ N } , where

U n = [ 0 , a ) if n = k 2 and k ∈ N , 0 , a 2 otherwise .

S t ( U n , U ) = X if n = k 2 and k ∈ N , X otherwise .

∴ δ ( { n ∈ N : x ∉ S t ( U n , U ) } ) = 0 ; thus, U is an s-s- γ cover for the topological space ( X , τ ) .

Consider the subset V = { V 1 = U 2 , V 2 = U 3 , V 3 = U 5 , V 4 = U 6 , V 5 = U 7 , V 6 = U 8 , V 7 = U 10 , … } of U .

Here, V is an s-dense subset of U , since δ ( { 2 , 3 , 5 , 6 , 7 , 8 , 10 , … } ) = 1 . But ⋃ U = 0 , a 3 ≠ X . ∴ V is not a cover at all. Thus, s-dense subset of an s-s- γ cover for a topological space may not be a cover at all.

Since s-dense subset of an s-s- γ cover may not be an s-s- γ cover (may not be a cover either), so what condition makes an s-dense subset of an s-s- γ cover to becomes an s-s- γ cover? One obvious answer is that an s-dense subset of a cellular s-s- γ cover becomes an s-s- γ cover. Since a cellular s-s- γ cover is an s- γ cover, s-dense subset of an s- γ cover is s- γ cover and every s- γ cover is an s-s- γ cover. But we do not want to put a restriction on the nature of s-s- γ cover; rather, we want to modify the restriction on the nature of the subset of the cover. So, we introduce the following definition.

Definition 5.2

Let U = { U n : n ∈ N } be a cover of a topological space ( X , τ ) and V = { U m 1 , U m 2 , U m 3 , … } be a subset of U such that m 1 < m 2 < m 3 < … . Then, V is said to be s-s-dense in U if δ ( { m i ∈ N : U m i ∈ V and S t ( U m i , V ) = X } ) = 1 .

First, we verify how different is this s-s-dense subset of a countably infinite cover from the s-dense subset of that cover.

Theorem 5.3

Every s-s-dense subset of a countably infinite cover of a topological space is an s-dense subset of that cover for that topological space.

Proof

Let V = { U m 1 , U m 2 , U m 3 , … } be an s-s-dense subset of a countably infinite cover U = { U n : n ∈ N } in a topological space ( X , τ ) .

Now, { m i ∈ N : U m i ∈ V } ⊇ { m i ∈ N : U m i ∈ V and S t ( U m i , V ) = X } .

So, δ ( { m i ∈ N : U m i ∈ V } ) ≥ δ ( { m i ∈ N : U m i ∈ V and S t ( U m i , V ) = X } ) = 1 .

i.e., δ ( { m i ∈ N : U m i ∈ V } ) = 1 .

Therefore, V is an s-dense subset of U in ( X , τ ) .□

Example 5.4

But converse of Theorem 5.3 may not be true, i.e., statistically dense subset of an open cover of a topological space may not be an s-s dense subset for that open cover.

In Example 5.1, V is an s-dense subset of the cover U n = { U n : n ∈ N } in the topological space ( X , τ ) .

Now, for every U m i ∈ V , S t ( U m i , V ) = [ 0 , a 2 ) ≠ X .

∴ δ ( { m i ∈ N : U m i ∈ V and S t ( U m i , V ) = X } ) = 0 ,

i.e., V is not an s-s dense subset of that open cover U . Hence, s-dense subset of an open cover of a topological space may not be an s-s dense subset for that open cover.

The relationship between s dense subset and s-s dense subset is shown in Figure 2.

Figure 2

Relationship between s dense subset and s-s dense subset.

Lemma 5.5

If V = { U n p : p ∈ N } is a subsequence of the sequence U = { U n : n ∈ N } of open sets in a topological space ( X , τ ) , then δ ( { p ∈ N : x ∈ S t ( U n p , V ) } ) ≥ δ ( { n p ∈ N : x ∈ S t ( U n p , V ) } ) for all x ∈ X .

Proof

Let x ∈ X be arbitrary, A = { p ∈ N : x ∈ S t ( U n p , V ) } and B = { n p ∈ N : x ∈ S t ( U n p , V ) } . But p ≤ n p ∀ p ∈ N .

∴ ∣ { k ∈ A : k ≤ n } ∣ ≥ { k ∈ B : k ≤ n } ∀ n ∈ N .

⇒ lim n → ∞ ∣ { k ∈ A : k ≤ n } ∣ n ≥ lim n → ∞ ∣ { k ∈ B : k ≤ n } ∣ n .

⇒ δ ( A ) ≥ δ ( B ) .

□ ⇒ δ ( { p ∈ N : x ∈ S t ( U n p , V ) } ) ≥ δ ( { n p ∈ N : x ∈ S t ( U n p , V ) } ) .

Theorem 5.6

Every s-s dense subset of a countable cover in a topological space is an s-s- γ cover of that space.

Proof

Let U = { U n : n ∈ N } be a countable cover of a topological space ( X , τ ) .

V = { U n k : k ∈ N } be an s-s dense subset of U .

∴ δ ( { n k : S t ( U n k , V ) = X } ) = 1 .

Now, { n k : x ∈ S t ( U n k , V ) } ⊇ { n k : S t ( U n k , V ) = X } ∀ x ∈ X .

δ ( { n k : x ∈ S t ( U n k , V ) } ) = 1 ∀ x ∈ X .

Now, by Lemma 5.5,

δ ( { k ∈ N : x ∈ S t ( U n k , V ) } ) ≥ δ ( { n k ∈ N : x ∈ S t ( U n k , V ) } ) = 1 ∀ x ∈ X .

⇒ δ ( { k ∈ N : x ∈ S t ( U n k , V ) } ) = 1 ∀ x ∈ X .

⇒ δ ( { k ∈ N : x ∉ S t ( U n k , V ) } ) = 0 ∀ x ∈ X .

Hence, V is an s-s- γ cover of ( X , τ ) .□

Corollary 5.7

Every s-s-dense subset of an s-s- γ cover of a topological space is also an s-s- γ cover for that space.

Proof

The corollary is a direct consequence of Theorem 5.6 and the fact that every s-s- γ cover is a countable cover.□

6 Conclusion

The relationship between various types of γ covers under the variation of dense subsets is shown in Figure 3, and this figure concludes our study.

$Figure 3 Relationship between various types of γ \gamma covers under the variation of dense subsets.$

Figure 3

Relationship between various types of γ covers under the variation of dense subsets.

Acknowledgement

The authors are thankful to the referees for their valuable suggestions that had significantly improved the content and representation of this article.

Funding information: The authors have no financial or proprietary interests in any material discussed in this article.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: In this article, no dataset has been generated or analyzed. So, data sharing is not applicable here.

References

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Received: 2022-07-19

Revised: 2023-05-25

Accepted: 2023-07-31

Published Online: 2023-08-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/taa-2023-0101

Keywords for this article

asymptotic density; open cover; γ cover; statistical-γ cover; star operator

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