Center group actions and related concepts

Eman Yahea Habeeb; Sattar Hameed Hamzah

doi:10.1515/eng-2022-0512

Article Open Access

Center group actions and related concepts

Eman Yahea Habeeb and Sattar Hameed Hamzah

Published/Copyright: March 2, 2024

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From the journal Open Engineering Volume 14 Issue 1

Abstract

In order to examine the characteristics of center group spaces, center orbits, center stabilizers, and center kernels, the major purpose of this work is to introduce the definition of center group actions. We review the definition and proposition of the center limit set, as well as the center thin sets (briefly C -thin) and the center Cartan Cp ( G ) -spaces (briefly C -Cartan Cp ( G ) -spaces).

Keywords: 𝒞-topological spaces; 𝒞-topological group; 𝒞-convergent; 𝒞-group action; Cp(G)-spaces; 𝒞-thin set

1 Introduction

Many other types of “fuzzy sets,” “intuitionist fuzzy sets,” “vague sets,” “soft sets,” [1] and “fuzzy soft sets” have been developed by mathematicians as tools for tackling ambiguity. Proximity spaces was initially proposed by Efremovic. To offer a subjective topology on the hidden set, Leader and Lodato [2] used more explicit proverbs than those found in the surrounding area of Efremovic. Image analysis [3] and facial recognition [4] are two examples of challenges that benefit from human observation, and proximity is a key factor to keep in mind while working with these types of situations. Several results, such as in the study by Gupta and Patnaik [5], have been produced in support of the fixed point hypothesis, with cyclic compression and the best proximity point being two of the most relevant topics. Recently, A. Kandil et al. [6] developed a new method of proximity structures [7] that is grounded in the ideas of ideal and soft sets. Using the advantages of nearby locations, center sets were first introduced in the study by Abdulsada et al. [8]. One of the most important concepts in geometrical topology is the concept of group actions.

The purpose of this study is to introduce the definition of center group actions and examine the characteristics of center group spaces, center orbits, center stabilizers, and center kernels, as well as the center thin sets (briefly C-thin) and the center Cartan Cp ( G ) -spaces (briefly C -Cartan Cp ( G ) -spaces).

1.1 Basic definitions and notations

Definition 1.1. [2,9]. Let X be a nonempty set and δ ⊆ P(X) × P(X) a binary relation, then (X, δ) is proximity space iff for each A ′, A ″, A ‴ ⊆ X,

A ′δ A ″ iff A ″δ A ′,
A ′δ( A ″ ∪ A ‴) iff A ′δ A ″ or A ′δ A ‴,
X δ ̅ ∅ ,
{x}δ{x} for each x ∈ X,
A ′ δ ̅ A ″ implies there is an E ⊆ X such that A ′ δ ̅ E and (X–E) δ ̅ A ″,

where A ′ δ ̅ A ″ means it is not true that A ′ δ A ″.

Definition 1.2. [4] Let (X, δ) be a proximity space and A ⊆ X. A center set C A = { 〈 A , A ′ 〉 ∶ A δ A ^′}.

Definition 1.3. [4]. Let (X, δ) be a proximity space and {x}, B ⊆ X such that {x}δB.

Then, x _B = {〈{x}, B〉} is called the a center point in X, and we denote the set of all center points in X by Cp ( X ) .

Definition 1.4. [10] Let (X, δ) be a proximity space and τ cent ( X ) ⊆ P C (X), then τ cent ( X ) is said to be a C -toplogy if

C ∅ , C X ∈ τ cent ( X ) ,
{ C A i : i ∈ I } ∈ τ cent ( X ) ⇒ ⋎ C { C A i : i ∈ I } ∈ τ cent ( X ) ,
C A 1 , C A 2 ∈ τ cent ( X ) ⇒ C A 1 ⋏ C C A 2 ∈ τ cent ( X ) .

The triplet (X, δ X , and τ cent ( X ) ) is called a C -topological space, and the members of τ cent ( X ) are said to be C -open. τ cent ( X ) is called indiscrete C -toplogy if τ cent ( X ) = { C X , C ∅ } and called discrete C -toplogy if τ cent ( X ) ⊆ P C (X).

Definition 1.5. [10] Let (X, δ X , τ cent ( X ) ) be a C -topological space. A center set C B over (X, δ X ) is said to be C -closed set, if there exists C -open C A such that cop. ( C A ) = C B . Since cop. ( C A ) ⊆ C X for each A ⊆ X, then we denote C -closed set by C XA .

Definition 1.6. [10] Let (X, δ X , τ cent ( X ) ) be a C -topological space and C B be a center set. Then, the C -closure of C B , denoted by C B ̅ C = ⋏ C { C XA : C XA C -closed and C B ≼ C C XA } .

Definition 1.7. [10] Let (X, δ X , τ cent ( X ) ) be a C -topological space, C A be a center set, and x B be a center point. Then, C A is said to be a C -neighborhood of x B , if there exists a C -open set C A' such that x B ∈ C A' ≼ C C A .

Proposition 1.8

[10] Let (X, δ X , τ cent ( X ) ) be a C -topological space. Then, each center point x B has a C -neighborhood. If C A and C A' are C -neighborhoods of some x B , then C A ⋏ C C A' is also a C -neighborhood of x B . If C A is C -neighborhood of x B and C A ≼ C C A ′ , then C A' is also C -neighborhood of x B .

Definition 1.9. [11] Let ( X , δ X ) and ( Y , δ Y ) be two proximity spaces and A ⊆ X , B ⊆ Y , and let C A and C B be two center sets. The center product of these center sets is defined by:

C A X C C B = { 〈 A × B , C × D 〉 : 〈 A , C 〉 ∈ C A and 〈 B , D 〉 ∈ C B }

Definition 1.10. [11] The function cent ( f ) : ( X , δ X , τ cent ( X ) ) → ( Y , δ Y , τ cent ( Y ) ) is said to be center continuous function if ( cent ( f ) ) ⁻¹ ( C A ) is C -open set in X for every C -open set C A in Y.

Theorem 1.11

[12] Let ( G , μ , τ ) be a topological group. Then, there exists a proximity relation δ such that ( G , δ ) is a proximity space.

Definition 1.12. [12] Let ( G , μ , τ ) be a topological group and ( G , δ ) be the proximity space, which is defined in (Theorem 1.11) and let μ and v be a proximity map, then the fourfold ( G , μ , τ cent ( G ) , δ ) is center topological group if:

The center function cent ( μ ) : Cp ( G ) X c Cp ( G ) → Cp ( G ) is center continuous.
The center inverse function cent ( v ) : Cp ( G ) → Cp ( G ) is center continuous.

Definition 1.13. [13] A center net is a function S of directed set D to the set of all center points Cp ( X ) that is S : D → Cp ( X ) , and the center point S ( d ) is denoted by χ B d = χ d B .

Theorem 1.14

[13] Let (X, δ X , τ cent ( X ) ) and (Y, δ Y , τ cent ( Y ) ) be center topological spaces. And let f :(X, δ X ) → (Y, δ Y ) be a proximity mapping a center function cent ( f ) : (X, δ X , τ cent ( X ) ) → (Y, δ Y , τ cent ( Y ) ) is continuous if and only if whenever ( χ d B )_d∈D is a center net in (X, δ X , τ cent ( X ) ) such that χ d B → c x B o , then cent ( f ) ( χ d B ) → c cent ( f ) ( x B o ).

Theorem 1.15

[13] Let (X, δ X , τ cent ( X ) ) be a center topological space and C A ≼ C C X .

Then, x B o ∈ C A ̅ C if and only if there is a center net ( χ d B ) d ∈ D in C A such that χ d B → c x B o .

Theorem 1.16

[13] Let ( χ d B ) _d∈D be a center net in a center topological space (X, δ X , τ cent ( X ) ), and for each d _o ∈ D, C A do = { χ d B : d ≥ d _o}, x B o ∈ C X is the center cluster point of ( χ d B )^{d ∈ D} if and only if x B o ∈ C A d o ̅ C for each d _o ∈ D.

2 Center compact space and center subnet

Definition 2.1. Let H = { C A i : i ∈ I } be the collection of non-empty center sets defined on ( X , δ ) is called is C -cover of the center set C X . If C X = ⋎ C i ∈ I C A i and this collection is called finite C -cover if I is finite set.

Also, it is called C -cover of the non-empty center set C A if C A ≼ C ⋎ C i ∈ I C A i . And the mention said to the family H is C -open cover, if each member is C -open. In other words, if C A i is C -open set for all i , then the { C A i : i ∈ I } is said to be C -open cover.

Definition 2.2. Let ( X , δ X , τ cent ( X ) ) be a C -topological space, the universal set C X (a non-empty center set C A ) is called C -compact space if each C -open cover { C A i : i ∈ I } of C X ( C A ) has finite ∆ subset of I , i.e., if C X = ⋎ C i ∈ I C A i ( C A ≼ C ⋎ C i ∈ I C A i ) , then there exist i = 1,…., n such that C X = ⋎ n C i ∈ I C A i ( C A ≼ C ⋎ n C i = 1 C A i ) .

Definition 2.3. A collection F C of center sets is said to have the center finite intersection property ( C FIP ) iff the center intersection of each finite subcollection of F C is center nonempty.

Theorem 2.4

A C -topological space ( X , δ X , τ cent ( X ) ) is C -compact iff every collection of C -closed center subset with the C FIP .

Proof: It is clear.

Definition 2.5. A center net ( t β A ) β ∈ E in ( X , δ X , τ cent ( X ) ) is called a center subnet of a center net ( x α B ) α ∈ D iff there is a mapping g : E → D such that t A = x B o g , i.e., for each i ∈ E , t i A = x g ( i ) B ; for each α ∈ D , there exists some m ∈ E , such that if m ≥ p ∈ E , g ( p ) ≤ α .

Definition 2.6. [14] Let ( A , ≥ ) and ( B , ≥ * ) be two directed sets. Then, a mapping ω : B → A is said to be isotone iff x ≥ y ⟹ ω ( x ) ≥ * ω ( y ) for each x , y ∈ B .

Definition 2.7. [14] Let ( A , ≥ ) be a directed set and let B ⊂ A . If for every a ∈ A , there exist an element b ∈ B such that b ≥ a , then B is called the cofinal subset of A .

Theorem 2.8

Let ω be an isotone map of directed set ( B , ≥ * ) into directed set ( A , ≥ ) such that ω ( B ) is cofinal in A. Let ( χ a H ) a ∈ A be a center net. Then, χ H o ω is the center subnet of ( χ a H ) a ∈ A .

Proof

It is clear.

Theorem 2.9

Let ( X , δ X , τ cent ( X ) ) be a C -topological space and let ( χ d B ) d ∈ D be a center net in X . Let H C be the collection of center subnet of X satisfying the following two conditions:

( χ d B ) d ∈ D is center frequently in each member of H C .
If C S , C T ∈ H C , then there exists C U ∈ H C such that C U ≼ C C S ⋏ C C T . Then, there exists a center subnet of ( χ d B ) d ∈ D , which is eventually in each member of H C .

Proof

It is evident from (ii) that H C directed set by the center inclusion relation ≼ C . Now, consider F C of the center Cartesian product D × H C defined by F C = { ( d , C U ) : d ∈ D , C U ∈ H C and χ d B ∈ C U } . We define a binary relation ≫ in F C as follows:

Let ( d , C S ) and ( b , C T ) be any two members of F C . Then, ( d , C S ) ≫ ( b , C T ) iff d ≥ b and C S ≼ C C T . We now show that F C is directed by ≫ .

Let ( d , C S ) and ( b , C T ) be any two members of F C . Then, by (ii), there exists C U in H C such that C U ≼ C C S ⋏ C C T so that C U ≼ C C S and C U ≼ C C T . Again by (i) and by the definition of ≫ , there exist an element a in D such that a ≥ d and a ≥ b and χ a B ∈ C U . Then, ( a , C U ) ∈ F C and is such that ( a , C U ) ≫ ( d , C S ) and ( a , C U ) ≫ ( b , C T ) . Define a mapping ω : F C → D : ω ( ( d , C S ) ) = d , ∀ ( d , C S ) ∈ F C . Since ( d , C S ) ≫ ( b , C T ) ⟹ d ≥ b it follows that ( d , C S ) ≫ ( b , C T ) → ω ( d , C S ) ≥ ω ( b , C T ) . Hence, ω is an isotone mapping. Furthermore, ω ( F C ) is cofinal in D . [For let d be any element of D . Then, d ≥ d . But ω ( d , C S ) = d for all C S ∈ H C . Hence, ω ( d , C S ) ≥ d . Thus, for every element, d ∈ D . There exists an element ω ( a , C S ) of ω ( F C ) such that ω ( a , C S ) ≥ d ]. It follows by Theorem 2.8. that the mapping φ B 1 = χ B o ω : F C → Cp ( X ) is a center subnet of χ d B .

We now show that this center subnet is eventually in each member of H C . Let C U o be any member of H C . By (i), there exists a member d o of D such that χ d o B ∈ C U o . Hence, by definition of F C , b o = ( d o , C U o ) is a member of F C . Now, let b = ( a , C U ) be any element of F C such that b ≫ b o . Then, a ≥ d o and C U ≼ C C U o . We then have φ B 1 ( b ) = ( χ B o ω ) ( b ) = χ B ( ω ( b ) ) = χ B ( ω ( a , C U ) ) = χ a B ∈ C U ≼ C C U o .

Thus, there exists an element b 1 ∈ F C such that for every b ∈ F C , b ≥ b 1 → ω ( b ) ∈ C U o . Hence, ω is eventually in C U o .

Theorem 2.10

Let ( X , δ X , τ cent ( X ) ) be a C -topological space. A center point x B o is a center cluster point of a center net ( χ d B ) d ∈ D iff there exists a center subnet ( y α A ) α ∈ β , which center converges to x B o .

Proof

Assume that ( χ d B ) d ∈ D has a center subnet ( y α A ) α ∈ β , which center converges to x B o . To prove that x B o is a center cluster point of a center net ( χ d B ) d ∈ D . Let C N be a center neighborhood of x B o , and let a o be any element of D . Since ( y α A ) α ∈ β is a center subnet of ( χ d B ) d ∈ D , there exists a mapping ω : β → D such that:

y A = χ B o ω ,
For each d ∈ D , there exists an element c in β such that ω ( c ) ≥ d for each c ≥ * b in β . Hence, by (ii) corresponding to a o ∈ D , there exists an element b o ∈ β such that ω ( c ) ≥ a o for every c ≥ * b o since ( y α A ) α ∈ β center converges to x B o . Now, let ω ( p ) = q . Then, q ∈ D and q ≥ a o . Also, χ B ( q ) = χ B ( ω ( p ) ) = ( χ B o ω ) ( p ) = y A ( p ) ∈ C N . Thus, we have shown that for each element a o in D , there exists an element q ≥ a o in D such that χ B ( q ) ∈ C N . Hence, ( χ d B ) d ∈ D is center frequently in C N .

It follows that x B o is a center cluster point of ( χ d B ) d ∈ D .

Conversely, let x B o be a center cluster point of ( χ d B ) d ∈ D , and let N C ( x B o ) be the collection of all center neighborhoods of x B o . If C L and C M are any two center members of N C ( x B o ) , then C L ≼ C C M is also a center member of N C ( x B o ) . Also, since is a center cluster point of ( χ d B ) d ∈ D . ( χ d B ) d ∈ D is center frequently in each member of N C ( x B o ) . Hence, by Theorem 2.9, there exists a subnet ( y α A ) α ∈ β of ( χ d B ) d ∈ D , which is center eventually in each member of N C ( x B o ) . This implies that ( y α A ) α ∈ β center converges to x B o .

Theorem 2.11

C -topological space ( X , δ X , τ cent ( X ) ) is C -compact iff each center net in X has a center cluster point.

Proof

Let every center net in X be a center cluster point, and let F C be a collection of C -closed subset in X with C FIP . Let K C = { C A : C A be the center intersection of a finite subcollection of F C } . Since the center intersection of every two center members of K C is a member of K C , it is evident that K C is directed by the center inclusion relation ≼ C . Since each C A is non-empty center set, by the axiom of choice, we may choose a center point x B ( C A ) in C A . Now, consider the mapping f : K C → Cp ( X ) : f ( C A ) = x B ( C A ) ∀ C A ∈ K C . Then, f is a center net in X . By hypothesis, f must have a center cluster point, say x B o . Let C E be any member of K C . Then, for every C A ≥ C E (i.e., C A ≼ C C E ) in K C , we have f ( C A ) = x B ( C A ) ∈ C A ≼ C C E . Hence, f is center eventually in the C -closed set C E . It follows from Theorem 1.16 that x B o ∈ C E . We have x B o ∈ ∩ K C ⊂ ∩ F C (since F C ⊂ K C ). Hence, ⋏ C F C ≠ C ∅ , and consequently, X is C -compact by Theorem 2.4.

Conversely, let ( X , δ X , τ cent ( X ) ) be C -compact and let ( χ d B ) d ∈ D be a center net in X . For each d in D , let C M d = { χ h B : h ≥ d in D } . Since D is directed by ≥ , the collection { C M d : d ∈ D } has the C FIP . Hence, { C M d ̅ C : d ∈ D } also has the C FIP : it follows from Theorem 2.4 that ⋏ C { C M d ̅ C : d ∈ D } ≠ C ∅ . Let p Y ∈ ⋏ C { C M d ̅ C : d ∈ D } . Then, p Y ∈ C M d ̅ C ∀ d ∈ D . Hence, by Theorem 1.16, p Y is a center cluster point of ( χ d B ) d ∈ D .

Corollary 2.12

A C -topological space ( X , δ X , τ cent ( X ) ) is C -compact iff each center net in X has a center subnet which has center convergence to some center point in X .

Proof

It is an immediate consequence of Theorems 2.10 and 2.11.

3 Center group actions

Definition 3.1. [15] Let G be a topological group and X be a topological space. A left action of G on X is a continuous function φ : G × X → X such that:

φ ( e , x ) = x , for all x ∈ X , where e is the identity element in G .
φ ( μ ( g 1 , g 2 ) , x ) = φ ( g 1 , φ ( g 2 , x ) ) , for all x ∈ X , and μ is the law of multiplication of G . The space X together with action is called group space (for brief G -space) (or more precisely left G -space). In a similar way, one can define a right G -space. We denoted it by φ ( g , x ) = gx .

Now, we introduce the following definition.

Definition 3.2. Let ( G , μ , τ cent ( G ) , δ ) be a center topological group and ( X , δ X , τ cent ( X ) ) be a center topological space. A left center action of Cp ( G ) on Cp ( X ) is a center continuous map cent( φ ): Cp ( G ) X c Cp ( X ) → Cp ( X ) such that:

cent( φ ) ( e B o , x B ) = x B , for all x B ∈ Cp ( X ) , where e B o is the center identity in Cp ( G ) .
cent ( φ ) ( cent ( μ ) ( g 1 C 1 , g 2 C 2 ) , x B ) = cent ( φ ) ( g 1 C 1 , cent ( φ ) ( g 2 C 2 , x B ) ) for all x B ∈ Cp ( X ) and g 1 C 1 , g 2 C 2 ∈ Cp ( G ) , and cent ( μ ) is the law of center multiplication of Cp ( G ) .

We call the triple ( Cp ( G ) , Cp ( X ) , cent ( φ ) ) a center topological transformation group (center group space), and we also express this same thing by simple saying that Cp ( X ) is Cp ( G ) -space, more precisely (left Cp ( G ) -space). In a similar way, one can define a right Cp ( G ) -space. We denoted it by cent( φ ) ( g C , x B ) = g C x B .

Example 3.3. Let G be a center topological group, then Cp ( G ) is Cp ( G ) -space by center multiplication

cent ( φ ) = cent ( μ ) : Cp ( G ) X c Cp ( G ) → Cp ( G ) . ( g 1 C 1 , g 2 C 2 ) → g 1 C 1 . C g 2 C 2 .

Example 3.4. Let G be a center topological group, then Cp ( G ) is Cp ( G ) -space by center conjugation.

cent ( φ ) : Cp ( G ) X c Cp ( G ) → Cp ( G ) . ( g 1 C 1 , g 2 C 2 ) → g 1 C 1 . C g 2 C 2 . C g 1 − 1 C 1 − 1 , cent ( φ ) is center continuous since cent ( φ ) = cent ( r g 1 − 1 ) o cent ( μ ) and:

(i) cent ( φ ) ( e B o , g C ) = e B o . C g C . C e − 1 B o − 1 = g C for all g C ∈ Cp ( G ) .

( ii ) cent ( φ ) ( g 1 C 1 , cent ( φ ) ( g 2 C 2 , g 3 C 3 ) )

= cent ( φ ) ( g 1 C 1 , g 2 C 2 . C g 3 C 3 . C g 2 − 1 C 2 − 1 ) = g 1 C 1 . C ( g 2 C 2 . C g 3 C 3 . C g 2 − 1 C 2 − 1 ) . C g 1 − 1 C 1 − 1 = ( g 1 C 1 . C g 2 C 2 ) . C g 3 C 3 . C ( g 1 C 1 . C g 2 C 2 ) − 1

= cent ( φ ) ( g 1 C 1 . C g 2 C 2 , g 3 C 3 ), for all g 1 C 1 , g 2 C 2 , g 3 C 3 ∈ Cp ( G ) .

Definition 3.5. Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then:

The center orbit of x B is defined to be the center set C G x B = { cent ( φ ) ( g C , x B ) : g C ∈ Cp ( G ) } , the set of all center orbits denoted by Cp ( X ) / Cp ( G ) and called it center orbit space.
The center stabilizer of x B ∈ Cp ( X ) is defined to be the center set.

C S x B = { g C ∈ Cp ( G ) : cent ( φ ) ( g C , x B ) = x B } .
The center kernel of the center action is defined to be the center set.

C -ker ( cent ( φ ) ) = { g C ∈ Cp ( G ) : cent ( φ ) ( g C , x B ) = x B , ∀ x B ∈ Cp ( X ) } .

Proposition 3.6

Let Cp ( X ) be a Cp ( G ) -space, then:

C S x B is a center subgroup of G .
C -ker ( cent ( φ ) ) = ⋏ C x B ∈ Cp ( X ) C S x B .
C -ker ( cent ( φ ) ) is a center normal subgroup of G .

Proof

Let g 1 C 1 , g 2 C 2 ∈ C S x B then cent ( φ ) ( g 1 C 1 , x B ) = cent ( φ ) ( g 2 C 2 , x B ) = x B .
cent ( φ ) ( g 1 C 1 . C g 2 C 2 , x B ) = cent ( φ ) ( cent ( μ ) ( g 1 C 1 , g 2 C 2 ) , x B ) = cent ( φ ) ( g 1 C 1 , cent ( φ ) ( g 2 C 2 , x B ) ) = cent ( φ ) ( g 1 C 1 , x B ) = x B .

Hence, g 1 C 1 . C g 2 C 2 ∈ C S x B . Now, let g C ∈ C S x B then cent ( φ ) ( g C , x B ) = x B .
cent ( φ ) ( g − 1 C − 1 , x B ) = cent ( φ ) ( g − 1 C − 1 , cent ( φ ) ( g C , x B ) )

= cent ( φ ) ( g − 1 C − 1 . C g C , x B ) = cent ( φ ) ( e B o , x B ) = x B .

Hence, cent ( φ ) ( g − 1 C − 1 , x B ) = x B . Therefore, g − 1 C − 1 ∈ C S x B .

So C S x B is a center subgroup of G .
Let g C ∈ C -ker ( cent ( φ ) ) iff cent ( φ ) ( ( g C , x B ) = x B for all x B ∈ Cp ( X ) . Iff g C ∈ C S x B for all x B ∈ Cp ( X ) .

Iff g C ∈ ⋏ C x B ∈ Cp ( X ) C S x B . Then, C -ker ( cent ( φ ) ) = ⋏ C x B ∈ Cp ( X ) C S x B .
From (ii), C -ker ( cent ( φ ) ) is a center subgroup of G .

Let g C ∈ C -ker ( cent ( φ ) ) , then cent ( φ ) ( ( g C , x B ) = x B for all x B ∈ Cp ( X ) and cent ( φ ) ( h C 1 . C g C . C h − 1 C 1 − 1 , x B ) = cent ( φ ) ( h C 1 , cent ( φ )

( g C . C h − 1 C 1 − 1 , x B ) = cent ( φ ) ( h C 1 , cent ( φ ) ( g C , cent ( φ ) ( h − 1 C 1 − 1 , x B ) )

= cent ( φ ) ( ( h C 1 , cent ( φ ) ( h − 1 C 1 − 1 , x B ) ) = cent ( φ ) ( h C 1 . C h − 1 C 1 − 1 , x B )

= cent ( φ ) ( e B o , x B ) = x B . Hence, cent ( φ ) ( h C 1 . C g C . C h − 1 C 1 − 1 , x B )

= x B for all x B ∈ Cp ( X ) .

Hence, h C 1 . C g C . C h − 1 C 1 − 1 ∈ C -ker ( cent ( φ ) ) ; therefore, h C 1 . C C -ker ( cent ( φ ) ) . C h − 1 C 1 − 1 ≼ C C -ker ( cent ( φ ) ) for all x B ∈ Cp ( X ) . Since C -ker ( cent ( φ ) ) ≼ C h C 1 . C C -ker ( cent ( φ ) ) . C h − 1 C 1 − 1 . Thus, h C 1 . C C -ker ( cent ( φ ) ) . C h − 1 C 1 − 1 = C -ker ( cent ( φ ) ) . Therefore, C -ker ( cent ( φ ) ) is a center normal subgroup of G .

Definition 3.7. Let Cp ( X ) be Cp ( G ) -space and x B ∈ Cp ( X ) . A center action of Cp ( G ) on Cp ( X ) with C -ker ( cent ( φ ) ) is said to be:

C -Transitive if C G x B = Cp ( X ) for all x B ∈ Cp ( X ) .
C -Effective if C -ker ( cent ( φ ) ) = e B o .
C -Free if C S x B = e B o for all x B ∈ Cp ( X ) .
C -Trivial if C -ker ( cent ( φ ) ) = Cp ( G ) .

Theorem 3.8

Let Cp ( X ) be Cp ( G ) -space, then:

The center function cent ( φ g C ) : Cp ( X ) → Cp ( X ) defined by cent ( φ g C ) ( x B ) = cent ( φ ) ( g C , x B ) , ∀ x B ∈ Cp ( X ) is a center homeomorphism ∀ g C ∈ Cp ( G ) .
The center function cent ( φ x B ) : Cp ( G ) → Cp ( X ) defined by cent ( φ x B ) = cent ( φ ) ( g C , x B ) , ∀ g C ∈ Cp ( G ) is a center continuous ∀ x B ∈ Cp ( X ) .

Proof

Clear.
Clear.

Theorem 3.9

Let Cp ( X ) be a center Hausdorff Cp ( G ) -space and G is a center compact. Then, the center action cent ( φ ) : Cp ( G ) X c Cp ( X ) → Cp ( X ) is a center closed function.

Proof

Let C A ≼ C C G X c C X be a center closed set in C G X c C X .

Let y B 1 ∈ cent ( φ ) ( C A ) ̅ C , then there exists a center net ( y d B ) d ∈ D ∈ cent ( φ ) ( C A ) such that y d B → c y B o (Theorem 1.15). This implies that there exist ( g d A , x d B ) d ∈ D ∈ C A such that cent ( φ ) ( g d A , x d B ) = y d B .

Since G is a center compact, then the center net ( g d A ) d ∈ D has center convergent subnet ( g nd A ) such that g nd A → c g C and y nd B → y B 1 . Now,

x nd B = cent ( φ ) ( e B o , x nd B ) = cent ( φ ) ( g − 1 nd A − 1 . C g nd A , x nd B )

= cent ( φ ) ( g − 1 nd A − 1 , cent ( φ ) ( g nd A , x nd B ) = cent ( φ ) ( g − 1 nd A − 1 , y nd B ) .

And then, x nd B → c cent ( φ ) ( g − 1 C − 1 , y B 1 ) . Therefore, ( g nd A , x nd B ) → c ( g C , cent ( φ ) ( g − 1 C − 1 , y B 1 ) , and since C A is center closed, then ( g C , cent ( φ ) ( g − 1 C − 1 , y B 1 ) ∈ C A . Thus, cent ( φ ) ( ( g C , cent ( φ ) ( g − 1 C − 1 , y B 1 ) ) = cent ( φ ) ( g C . C g − 1 C − 1 , y B 1 ) = cent ( φ ) ( e B o , y B 1 ) = y B 1 ∈ cent ( φ ) ( C A ) . Therefore, cent ( φ ) ( C A ) = cent ( φ ) ( C A ) ̅ C , and then, cent ( φ ) ( C A ) is center closed.

4 Center Cartan Cp ( G ) -space

Definition 4.1. [16] Let X be an G -space, and a subset A of X is said to be thin relative to a subset B of X if the set (( A , B )) = { g ∈ G : gA ∩ B ≠ ∅ } has neighborhood whose closure is compact in G . If A is thin relative to itself, then it is called thin.

Definition 4.2. Let Cp ( X ) be a Cp ( G ) -space, and a center subset C A of cent ( X ) is said to be center thin relative to a center subset C B of X if the set (( C A , C B )) = { g C ∈ Cp ( G ) : g C C A ⋏ C C B ≠ C ∅ } has a center neighborhood whose center closure is a center compact in G . If C A is a C -thin relative to itself, then it is called C -thin.

Theorem 4.3

Let Cp ( X ) be a Cp ( G ) -space and C K 1 and C K 2 be a center compact subset of X . Then:

(( C K 1 , C K 2 )) is a center closed subset.
(( C K 1 , C K 2 )) is a center compact when C K 1 and C K 2 are relatively C -thin.

Proof

Let g C . Then, by Theorem 1.15, there is a center net ( g d A ) d ∈ D in (( C K 1 , C K 2 )) such that g d A → c g C . Since g d A ∈ (( C K 1 , C K 2 )), then there is a center net ( k 1 d C 1 ) d ∈ D in C K 1 , which is a center compact, such that g d A k 1 d C 1 ∈ C K 2 . Since C K 2 is a center compact, then there exists a center subnet ( g dm A k 1 dm C 1 ) of ( g d A k 1 d C 1 ) such that g dm A k 1 dm C 1 → c k o 2 B , where k o 2 B ∈ C K 2 , but ( k 1 dm C 1 ) in C K 1 and C K 1 is center compact; thus, there is a center point k o 1 B 1 ∈ C K 2 and a center subnet of k 1 dm C 1 say itself such that k 1 dm C 1 → c k o 1 B 1 , then by Theorem 1.14, g dm A k 1 dm C 1 → c g C k o 1 B 1 , which means that g C ∈ (( C K 1 , C K 2 )); therefore, (( C K 1 , C K 2 )) is a center closed.
Let C K 1 and C K 2 be the center compact subsets of center G -space X such that C K 1 and C K 2 are center relatively thin, then (( C K 1 , C K 2 )) has a center neighborhood whose center closure is a center compact, since C K 1 and C K 2 are center compact by (i), (( C K 1 , C K 2 )) is a center closed, and thus, (( C K 1 , C K 2 )) is a center compact.

Recall that a G -space X is called Cartan G -space if every point in X has a thin neighborhood [17].

Definition 4.4. A Cp ( G ) -space Cp ( X ) is said to be a center cartan Cp ( G ) -space (briefly C -Cartan Cp ( G ) -spaces) if every center point in Cp ( X ) has a C -thin center neighborhood.

Example 4.5. A Cp ( G ) -space Cp ( X ) is a C -Cartan Cp ( G ) -space if G is a center compact.

Theorem 4.6

Let Cp ( X ) be a C -Cartan Cp ( G ) -space, then each C G x B is center closed, and C S x B is a center compact.

Proof

Let y B o ∈ C G x B ̅ C . Then, there is a center net ( y d B ) d ∈ D in C G x B such that y d B → c y B o . Since X is a C -Cartan center G-space, then y B o has C -thin center neighborhood C U . Since y d B ∈ C G x B , then there exists a center net ( g d A ) d ∈ D in G such that y d B = g d A x B for each d ∈ D . Fixed d o , a center net ( g d A . C g − 1 d o A − 1 ) ( g d o A x C ) = g d A x B So g d A . C g − 1 d o A − 1 ∈ (( C U , C U )) and g d A . C g − 1 d o A − 1 has a center cluster point g A .

Then, by Corollary 2.12, there exists a center subnet of g d A . C g − 1 d o A − 1 , say itself, such that g d A . C g − 1 d o A − 1 → c g A , then g d A → c g d A . C g d o A x B and y B o = g d A . C g d o A x C , so y B o ∈ C G x B . Thus, C G x B is a center closed.

Now, let x B o ∈ Cp ( X ) , then there exists a C -thin center neighborhood C V of x B o . Then, (( C V , C V )) has a center neighborhood whose center closure is a center compact since the center stabilizer C S x B o group of G is a center closed and C S x B o ∈ (( C V , C V )). Hence, C S x B o is a center compact.

Dydo in the study by Al-Srrai [16] developed the concept of the sets Λ ( x ) and J ( x ) in any space and used only J ( x ) as a characterization of Cartan G -space as follows:

For any point x in a G -space.

J ( x ) = { y ∈ X : there is a net ( g d ) d ∈ D in G and there is a net ( x d ) d ∈ D in X with g d → ∞ and x d → x such that g d x d → y }.

Λ ( x ) = { y ∈ X : there is a net ( g d ) d ∈ D in G with g d → ∞ such that g d x d → y }. Also, it is clear that the set Λ ( x ) is a subset of J ( x ) .

Now, we introduce the following definition and prove some results.

Definition 4.7. Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then:

C Λ ( x B ) = { y C : there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ such that g d A x d E → c y C } is called the center limit set of x B .
C J ( x B ) = { y C : there is a center net ( g d A ) d ∈ D in G and there is a center net ( x d E ) d ∈ D in X with g d A → c ∞ and x d E → c x B such that g d A x d E → c y C } is called the center first prolongation limit set.

It is clear that the set C Λ ( x B ) is a center subset of C J ( x B ) .

Definition 4.8. Let Cp ( X ) be a Cp ( G ) -space and C A ≼ C C X . Then, C A is called the center invariant under G if C G C A = C A .

Theorem 4.9

Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then:

C Λ ( x B ) and C J ( x B ) are the center invariant sets under Cp ( G ) .
The center orbit C G x B is center closed if and only if C Λ ( x B ) ≼ C C G x B .
If x B ∉ C Λ ( x B ) , then the center stabilizer subgroup C S x B of G is a center compact.
C G x B ̅ C = C G x B ⋎ C C Λ ( x B ) .
g C C Λ ( x B ) = C Λ ( g C x B ) = C Λ ( x B ) for each g C ∈ Cp ( G ) .

Proof

Let y C ∈ C Λ ( x B ) and g A o ∈ Cp ( G ) . Then, there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ and g d A x d E → c y C , and it is clear that ( g A o g d A ) d ∈ D is a center net in G with g A o g d A → c ∞ . Since the center action is center continuous, then g A o g d A x B → c g A o y C , which implies that g A o y C ∈ C Λ ( x B ) , and hence, C Λ ( x B ) is center invariant.

The proof of C J ( x B ) is similar.
Let C G x B be a center closed and let y C ∈ C Λ ( x B ) , then there is a center net ( g d A ) d ∈ D in G such that g d A → c ∞ and g d A x d E → c y C . Since g d A x d E ∈ C G x B and ( g d A ) d ∈ D is a center net in C G x B , then by Theorem 1.15, y C ∈ C G x B ̅ C = C G x B . Therefore, a center net in C Λ ( x B ) ≼ C C G x B .

Conversely:

Let y C ∈ C G x B ̅ C , then by Theorem 1.15, there is a center net ( y d F ) d ∈ D is a center net in C G x B such that y d F → c y C , then ∀ d ∈ D , there is g d A ∈ G such that y d F = g d A x B .

Then, ( g d A ) d ∈ D is a center net in G and g d A x B → c y C . Now either g d A → c g A o or g d A → c ∞ . If g d A → c g A o , then g d A x B → c g d A x B = y C , which implies that y C ∈ C G x B . If g d A → c ∞ , then y C ∈ C Λ ( x B ) ≼ C C G x B ; thus, C G x B is a center closed.
Let x B ∉ C Λ ( x B ) and suppose that C S x B is not center compact. Then, there is a center net ( g d A ) d ∈ D in G such that g d A → c ∞ . Since g d A ∈ C S x B , then g d A x B = x B , i.e., g d A x B → c x B . Then, x B ∈ C Λ ( x B ) , which is a contradiction; thus, C S x B is a center compact.
If C G x B is a center closed, then C G x B = C G x B ̅ C from (ii), C G x B = C G x B ⋎ C C Λ ( x B ) .

Let y C ∉ C G x B and y C ∈ C Λ ( x B ) , then there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ such that g d A x d E → c y C since ( h H g d A ) d ∈ D is a center net in G with h H g d A → c ∞ . So, h H g d A x d E → c h H y C , which implies that h H y C ∈ C Λ ( x B ) , but h H y C ∈ C G x B and then C Λ ( x B ) ≼ C C G x B ; therefore by (ii), C G x B is center closed; hence, C G x B ̅ C = C G x B ⋎ C C Λ ( x B ) .
Clear.

Theorem 4.10

Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then:

(i) y C ∈ C J ( x B ) if and only if x B ∈ C J ( y C ) .

(ii) g A o C J ( x B ) = C J ( g A o x B ) = C J ( x B ) for each g A o ∈ Cp ( G ) .

Proof

(i) Let y C ∈ C J ( x B ) , then there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ , and there is a center net ( x d E ) d ∈ D in X with x d E → c x B such that g d A x d E → c y C . Put γ d H = g d A x d E → c y C . Then, g − 1 d A − 1 → c y C , and g − 1 d A − 1 γ d H = g − 1 d A − 1 g d A x d E = x d E → c x B . Thus, x B ∈ C J ( y C ) .

The converse is similar.

Clear.

Theorem 4.11

Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then:

If x B ∈ C J ( x B ) , then for each y C ∈ C G x B , y C ∈ C J ( y C )
If y C ∈ C Λ ( x B ) for some y C ∈ Cp ( X ) , then y C ∈ C J ( y C ) .
If x B ∉ C J ( y C ) for each x B ∈ Cp ( X ) , then C Λ ( x B ) = C ∅ .

Proof

(i) Let x B ∈ C J ( x B ) and y C ∈ C G x B . Since C J ( x B ) is a center invariant, then x B ∈ C J ( x B ) for each y C ∈ C G x B : by Theorem 4.10. (i), x B ∈ C J ( x B ) . But C J ( x B ) is a center invariant; thus, y C ∈ C J ( y C ) .

(ii) Let y C ∈ C J ( x B ) , then there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ , such that g d A x d E → c y C , and put γ d H = g d A x d E → c y C . Then, it is clear that g − 1 d A − 1 → c ∞ and g − 1 d A − 1 ( γ d H ) = g − 1 d A − 1 . C g d A x d E = x d E → c x B ; thus, x B ∈ C J ( y C ) , which is center closed and center invariant, then we have g d A x d E ∈ C J ( y C ) . Since g d A x d E ∈ y C , then y C ∈ C J ( y C ) .

Let x B ∉ C J ( y C ) for each x B ∈ Cp ( X ) . To prove C Λ ( x B ) = C ∅ . If y C ∈ C J ( x B ) , then from (ii) y C ∈ C J ( y C ) , it is a contradiction; thus, C Λ ( x B ) = C ∅ for each x B ∈ Cp ( X ) .

Theorem 4.12

Let Cp ( X ) be a Cp ( G ) -space. If Cp ( X ) is C -Cartan Cp ( G ) -space, then x B ∉ C J ( x B ) for each x B ∈ Cp ( X ) .

Proof

If Cp ( X ) is a C -Cartan Cp ( G ) -space, let x B ∈ C J ( x B ) , then there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ and there is a center net ( x d E ) d ∈ D in X with x d E → c x B such that g d A x d E → c x B . Since x B ∈ Cp ( X ) and X is a C -Cartan Cp ( G ) -space, then x B has a center open neighborhood C U such that (( C U , C U )) is a center relative thin. Then, (( C U , C U )) is a center relative compact. Thus, there is d ∈ D , and x d E and g d A x d E are in C U so that g d A is in (( C U , C U )). Then, ( g d A ) d ∈ D contains a convergent subnet, which is a contradiction.

Theorem 4.13

Let Cp ( X ) be a C -Cartan Cp ( G ) -space, then C Λ ( x B ) = C ∅ for each x B ∈ Cp ( X ) .

Proof

Suppose that there is a center point y C ∈ Cp ( X ) such that y C ∈ C Λ ( x B ) ; then, there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ such that g d A x d E → c y C . Let C U y C be a C -thin center neighborhood of y C . Then, there is d o ∈ D such that g d A x d E ∈ C U y C for each d ≥ d o , we get center net g d A . C g − 1 d 1 A 1 − 1 . C g d 1 A 1 x d E = g d A x d E ∈ C U y C ; thus, g d A . C g − 1 d 1 A 1 − 1 ∈ (( C U y C , C U y C )), which has center compact closure. Hence, the center net g d A . C g − 1 d 1 A 1 − 1 has center convergent subnet, say itself, i.e, there is g o A o ∈ Cp ( G ) such that g d A . C g − 1 d 1 A 1 − 1 → c g o A o , then g d A → c g o A o . C g d 1 A 1 , which is a contradiction; therefore, y C ∉ C Λ ( x B ) . Since y C is arbitrary, then C Λ ( x B ) = C ∅ .

Theorem 4.14

Let Cp ( X ) be a C -Cartan Cp ( G ) -space and x B ∈ Cp ( X ) . If x B ∈ C J ( x B ) , then x B has no C -thin center neighborhood.

Proof

Let x B ∈ C J ( x B ) and suppose that x B has C -thin center neighborhood, then there is a center neighborhood C U of x B such that the center set (( C U , C U )) has center compact closure. By hypothesis x B ∈ C J ( x B ) , there is a center net ( g d A ) d ∈ D in G with g d A → c ∞ and there is a center net ( x d E ) d ∈ D in X with x d E → c x B such that g d A x d E → c x B ; since C U is a center neighborhood of x B , there is d o ∈ D such that g d A x d E ∈ C U for all d ≥ d o , which is center compact closure, so the center net ( g d A ) d ∈ D has center convergent subnet. This is a contradiction.

Definition 4.15. Let Cp ( X ) be a Cp ( G ) -space and x B ∈ Cp ( X ) . Then, the center point x B is said to be the fixed center point if C G x B = x B .

Theorem 4.16

Let Cp ( X ) be a C -Cartan Cp ( G ) -space, and then, there is no fixed center point.

Proof

Let Cp ( X ) be a C -Cartan Cp ( G ) -space, x B ∈ Cp ( X ) and x B be fixed center point. Then, for each center net ( g d A ) d ∈ D in G with g d A → c ∞ , g d A x B = x B → c x B , i.e., x B ∈ C Λ ( x B ) , which is a contradiction to C Λ ( x B ) = C ∅ . Therefore, x B is not fixed center point.

Theorem 4.17

Let Cp ( X ) be a Cp ( G ) -space if C Λ ( x B ) = C ∅ , for each x B ∈ Cp ( X ) . Then, the center orbit C G x B is not center compact.

Proof

Suppose that C G x B is a center compact. Since C Λ ( x B ) = C ∅ , then there is net ( g d A ) d ∈ D in G with g d A → c ∞ ; since ( g d A x B ) d ∈ D is a center net in C G x B , then g d A x B → c y C , for some y C ∈ Cp ( X ) . Hence, y C ∈ C Λ ( x B ) , which is a contradiction with C Λ ( x B ) = C ∅ for each x B ∈ Cp ( X ) .

5 Conclusion

One of the most important concepts in geometrical topology is the concept of group actions. In this study, we introduce the concept of center group space and review the definition and proposition of the center limit set, as well as the center thin sets and the center Cartan Cp ( G ) -spaces.

Conflict of interest: Authors state no conflict of interest.
Data availability statement: The most datasets generated and/or analysed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2023-05-29

Revised: 2023-08-02

Accepted: 2023-08-08

Published Online: 2024-03-02

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

Articles in the same Issue

https://doi.org/10.1515/eng-2022-0512

Keywords for this article

𝒞-topological spaces; 𝒞-topological group; 𝒞-convergent; 𝒞-group action; Cp(G)-spaces; 𝒞-thin set

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