I -convergence classes D. N. Georgiou 1 , S. D. Iliadis 2 , A. C. Megaritis 3 and G. A. Prinos 1 1 Department of Mathematics, University of Patras, Greece 2 Department of General Topology and Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia 3 Technological Educational Institute of Western Greece, Greece 1 / 31
Abstract Let X be a non-empty set. We consider the class C consisting of triads ( s , x , I) , where s = ( s d ) d ∈ D is a net in X , x ∈ X and I is an ideal of D . We shall find several properties of C such that there exists a topology τ for X satisfying the following equivalence: (( s d ) d ∈ D , x , I) ∈ C , where I is a proper D -admissible, if and only if ( s d ) d ∈ D I -converges to x relative to the topology τ . 2 / 31
Outline Preliminaries 1 3 / 31
Outline Preliminaries 1 Basic propositions 2 3 / 31
Outline Preliminaries 1 Basic propositions 2 Main theorem 3 3 / 31
Outline Preliminaries 1 Basic propositions 2 Main theorem 3 Problems 4 3 / 31
Outline Preliminaries 1 Basic propositions 2 Main theorem 3 Problems 4 Bibliography 5 3 / 31
Preliminaries In this section, we recall some of the basic concepts related to the con- vergence of nets in topological spaces and we refer to [10] for more details. Ideals Let D be a non-empty set. A family I of subsets of D is called ideal if I has the following properties: ∅ ∈ I . 1 If A ∈ I and B ⊆ A , then B ∈ I . 2 If A , B ∈ I , then A ∪ B ∈ I . 3 The ideal I is called proper if D ∉ I . 4 / 31
Preliminaries Directed set A partially ordered set D is called directed if every two elements of D have an upper bound in D . If ( D , ⩽ D ) and ( E , ⩽ E ) are directed sets, then the Cartesian product D × E is directed by ⩽ , where ( d 1 , e 1 ) ⩽ ( d 2 , e 2 ) if and only if d 1 ⩽ D d 2 and e 1 ⩽ E e 2 . Also, if ( E d , ⩽ d ) is a directed set for each d in a set D , then the product E d = { f ∶ D → ⋃ E d ∶ f ( d ) ∈ E d for all d ∈ D } ∏ d ∈ D d ∈ D is directed by ⩽ , where f ⩽ g if and only if f ( d ) ⩽ d g ( d ) , for all d ∈ D . 5 / 31
Preliminaries Net A net in a set X is an arbitrary function s from a non-empty directed set D to X . If s ( d ) = s d , for all d ∈ D , then the net s will be denoted by the symbol ( s d ) d ∈ D . Semisubnet A net ( t λ ) λ ∈ Λ in X is said to be a semisubnet of the net ( s d ) d ∈ D in X if there exists a function ϕ ∶ Λ → D such that t = s ○ ϕ . We write ( t λ ) ϕ λ ∈ Λ to indicate the fact that ϕ is the function mentioned above. 6 / 31
Preliminaries Subnet A net ( t λ ) λ ∈ Λ in X is said to be a subnet of the net ( s d ) d ∈ D in X if there exists a function ϕ ∶ Λ → D with the following properties: t = s ○ ϕ , or equivalently, t λ = s ϕ ( λ ) for every λ ∈ Λ . 1 For every d ∈ D there exists λ 0 ∈ Λ such that ϕ ( λ ) ⩾ d whenever 2 λ ⩾ λ 0 . Remark Suppose that ( t λ ) ϕ λ ∈ Λ is a subnet of the net ( s d ) d ∈ D in X . For every ideal I of the directed set D , we consider the family { A ⊆ Λ ∶ ϕ ( A ) ∈ I } . This family is an ideal of Λ which will be denoted by I Λ ( ϕ ) . 7 / 31
Preliminaries Convergence of a net We say that a net ( s d ) d ∈ D converges to a point x ∈ X if for every open neighbourhood U of x there exists a d 0 ∈ D such that x ∈ U for all d ⩾ d 0 . d ∈ D s d = x . In this case we write lim 8 / 31
Preliminaries I -convergence of a net ([14]) Let X be a topological space and I an ideal of a directed set D . We say that a net ( s d ) d ∈ D I -converges to a point x ∈ X if for every open neighbourhood U of x , { d ∈ D ∶ s d ∉ U } ∈ I . In this case we write I − lim d ∈ D s d = x and we say that x is the I -limit of the net ( x d ) d ∈ D . If X is a Hausdorff space, then a proper I -convergent net has a unique I -limit ([14]). 9 / 31
Preliminaries Natural (Asymptotic) density ([8], [17]) If A ⊆ N , then A ( n ) will denote the set { k ∈ A ∶ k ≤ n } and ∣ A ( n )∣ will stand for the cardinality of A ( n ) . The natural density of A is defined by ∣ A ( n )∣ d ( A ) = lim , n n →∞ if the limit exists. 10 / 31
Preliminaries In what follows ( X ,ρ ) is a fixed metric space and I denotes a proper ideal of subsets of N . I -convergence of a sequence in a metric space ([12]) A sequence ( x n ) n ∈ N of elements of X is said to be I -convergent to x ∈ X if and only if for each ǫ > 0 the set A ǫ = { n ∈ N ∶ ρ ( x n , x ) ≥ ǫ } ∈ I . Example Take for I the class I f of all finite subsets of N . Then I f is a proper ideal and I f -convergence coincides with the usual convergence with respect to the metric ρ in X . Example Denote by I d the class of all subsets A of N with d ( A ) = 0. Then I d is a proper ideal and I d -convergence coincides with the statistical conver- gence. 11 / 31
Preliminaries Let D be a directed set. For all d ∈ D we set M d = { d ′ ∈ D ∶ d ′ ≥ d } . D -admissible ideal ([14]) An ideal I of D is called D-admissible , if D ∖ M d ∈ I , for all d ∈ D . Proposition ([14]) Let X be a topological space, x ∈ X , and D a directed set. Then, I 0 ( D ) = { A ⊆ D ∶ A ⊆ D ∖ M d for some d ∈ D } is a proper ideal of D . Moreover, a net ( s d ) d ∈ D converges to a point x of a space X if and only if ( s d ) d ∈ D I 0 ( D ) -converges to x . 12 / 31
Preliminaries Proposition ([14, Theorem 3]) Let X be a topological space and A ⊆ X . If the net ( s d ) d ∈ D in A I - converges to the point x ∈ X , where I is a proper ideal of D , then x ∈ Cl X ( A ) . 13 / 31
Basic propositions In what follows X is a topological space, x ∈ X , ( s d ) d ∈ D is a net of X , and I is an ideal of D . Proposition If ( s d ) d ∈ D is a net such that s d = x for every d ∈ D , then I − lim d ∈ D s d = x . Proposition d ∈ D s d = x , then for every subnet ( t λ ) λ ∈ Λ of the net ( s d ) d ∈ D we If I 0 ( D ) − lim λ ∈ Λ t λ = x . have I 0 ( Λ ) − lim Proposition d ∈ D s d = x , then for every semisubnet ( t λ ) ϕ If I − lim λ ∈ Λ of the net ( s d ) d ∈ D we λ ∈ Λ t λ = x . have I Λ ( ϕ ) − lim 14 / 31
Basic propositions Proposition If I − lim d ∈ D s d = x , where I is a proper ideal of D , then there exists a semisubnet ( t λ ) λ ∈ Λ of the net ( s d ) d ∈ D such that I 0 ( Λ ) − lim λ ∈ Λ t λ = x . Proposition Let D be a directed set and I a D -admissible ideal of D . If ( s d ) d ∈ D does not I -converge to x , then there exists a subnet ( t λ ) ϕ λ ∈ Λ of the net ( s d ) d ∈ D such that: Λ ⊆ D . 1 ϕ ( λ ) = λ , for every λ ∈ Λ . 2 No semisubnet ( r k ) f k ∈ K of ( t λ ) ϕ λ ∈ Λ I K -converges to x , for every 3 proper ideal I K of K . I Λ ( ϕ ) is a proper and Λ -admissible ideal of Λ . 4 15 / 31
Basic propositions Proposition We suppose the following: D is a directed set. 1 I D is a proper ideal of D . 2 E d is a directed set for each d ∈ D . 3 I E d is a proper ideal of E d for each d ∈ D . 4 I D × I ∏ d ∈ D E d is the family of all subsets of D × ∏ d ∈ D E d for which: 5 A ∈ I D × I ∏ d ∈ D E d if and only if there exists A D ∈ I D such that { f ( d ) ∶ ( d , f ) ∈ A } ∈ I E d , for each d ∈ D ∖ A D . Then, the family I D × I ∏ d ∈ D E d is a proper ideal of D × ∏ d ∈ D E d . 16 / 31
Basic propositions Proposition We suppose the following: D is a directed set. 1 I D is a proper ideal of D . 2 E d is a directed set for each d ∈ D . 3 I E d is a proper ideal of E d for each d ∈ D . 4 ( s ( d , e )) e ∈ E d is a net from E d to a topological space X for each 5 d ∈ D . I D − lim d ∈ D ( I E d − lim s ( d , e )) = x . 6 e ∈ E d Then, the net r ∶ D × ∏ d ∈ D E d → X , where r ( d , f ) = s ( d , f ( d )) , for every ( d , f ) ∈ D × ∏ d ∈ D E d , I D × I ∏ d ∈ D E d -converges to x . 17 / 31
Main theorem I -convergence classes Let X be a non-empty set and let C be a class consisting of triads ( s , x , I ) , where s = ( s d ) d ∈ D is a net in X , x ∈ X , and I is an ideal of D . We say that the net s I -converges ( C ) to x if ( s , x , I ) ∈ C . We write I − lim d ∈ D s d ≡ x ( C ) . 18 / 31
Main theorem I -convergence classes Let X be a non-empty set and let C be a class consisting of triads ( s , x , I ) , where s = ( s d ) d ∈ D is a net in X , x ∈ X and I is an ideal of D . We say that C is a I -convergence class for X if it satisfies the follow- ing conditions: (C1) If ( s d ) d ∈ D is a net such that s d = x for every d ∈ D and I is an ideal of D , then I − lim d ∈ D s d ≡ x ( C ) . (C2) If I 0 ( D ) − lim d ∈ D s d ≡ x ( C ) , then for every subnet ( t λ ) λ ∈ Λ of the net ( s d ) d ∈ D we have I 0 ( Λ ) − lim λ ∈ Λ t λ ≡ x ( C ) . (C3) If I − lim d ∈ D s d ≡ x ( C ) , where I is an ideal of D , then for every semisubnet ( t λ ) λ ∈ Λ of the net ( s d ) d ∈ D we have I Λ ( ϕ ) − lim λ ∈ Λ t λ ≡ x ( C ) . 19 / 31
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