On Filter Convergence of Nets in Uniform Spaces

Home , Filter (mathematics)

Filomat 33:2 (2019), 353–357 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1902353S University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Ekrem Savas¸a, Ulas¸Yamancıb

aUs¸ak University, Department of Mathematics, Us¸ak, Turkey bS¨uleymanDemirel University, Faculty of Arts and Sciences, Department of Statistics, Isparta, Turkey

Abstract. In this paper, we introduce -convergent and st-fundamental nets in uniform spaces and study F F some their properties.

1. Introduction and Notations

The concept of statistical convergence was introduced by Fast [7] and Schonberg [21], and its topological properties were discussed by Fridy [8], Salat [18] and Maddox [15]. Fridy [8] also introduced the concept of statistically fundamental sequence and showed its equivalence to statistical convergence with respect to numerical sequences. This problem on the uniform space was raised in [16]. The authors [16] showed that if the sequence xn n N is statistically convergent in a uniform space, then it is statistically fundamental. { } ∈ Recently, Bilalov and Nazarova [3] gave the concept of st-fundamental sequences in uniform spaces and obtain some results related with this concept. F Kostyrko et al. [12] introduced the notion of -convergence of sequences in a metric space and discussed some properties of such convergence. RecallI that -convergence is a generalization of statistical convergence. Some problems about the ideals or filters canI be found in [4, 5, 13, 14]. We now recall some concepts of ideal and filter [3, 12, 17]. A family of sets 2N is said to be an ideal if (i) ; (ii) A, B imply A B ; (iii) A , B A imply B . I ⊂ ∅ ∈ I ∈ I ∪ ∈ I ∈ I ⊂ ∈ I A family of sets 2N is said to be a filter if (i) < ; (ii) A, B imply A B ; (iii) A , A B imply B . F ⊂ ∅ F ∈ F ∩ ∈ F ∈ F ⊂ ∈ F If filter satisfy the following axioms: F (iv) if A1 A2 ... and An for all n N, then there exists nm m N N; n1 < n2 < ... such that ⊃ ⊃ ∈ F ∈ { } ∈ ⊂ ∞ (αm, αm ] A , ∪m=1 +1 ∩ (m) ∈ F (v) Fc (N F) for any finite subset F N, \ ∈ F ⊂ then filter is said to be a monotone closed filter and a right filter, respectively [2, 3]. F An ideal is said to be non-trivial if , and , N. 2N is a non-trivial ideal if and only if = ( ) = IN A : A is a filter. A non-trivialI ∅ idealI is saidI ⊂ to be admissible if n : n N . Filter convergenceF F I { was\ introduced∈ I} in [1] and described in detailsI in the paper [9]. ConvergenceI ⊃ {{ } with∈ respect} to

2010 Mathematics Subject Classification. Primary 40A35; Secondary 54A20, 54E15 Keywords. Ideal, filter, nets, -convergence, st-convergence Received: 02 June 2017; Revised:F 11 OctoberF 2017; Accepted: 17 October 2017 Communicated by Ljubisaˇ D.R. Kocinacˇ Email addresses: [email protected] (Ekrem Savas¸), [email protected] (Ulas¸Yamancı) E. Savas¸, U. Yamancı / Filomat 33:2 (2019), 353–357 354 set of filters was studied in the paper [11]. More information about filters and convergence with respect to filters can be found in [1, 12, 17, 19, 20]. Now we recall the definition of uniformity on a set X [6, 10]. Λ = (x, x) : x X is said to be a diagonal or the identity relation. If U X X is a relation, then the inverse{ of this∈ relation} U 1 is defined as the set of all pairs x, y such that⊂ y,×x U, that is, U 1 = − − x, y X X : y, x U . Let U, V X X be some relation. The composition U ∈V of the relations U and∈V is× defined as∈ the set of all pairs⊂ ×(x, z), we get x, y V and y, z U for some◦ y X, that is, U V = (x, z) : y X, x, y V and y, z U . Let K X∈be some set and∈ U X X be∈ a relation. Assume◦ U [K] = ∃y ∈X : x K∈ = x, y U∈. For K = x⊂suppose U [K] = U [x]. Uniformity⊂ × on the set X ∈ ∃ ∈X X ⇒ ∈ { } is a non-empty family Ω 2 × which satisfies the following axioms: ⊂ (a) Λ U, U Ω; ⊂ ∀ ∈ 1 (b) U Ω imply U− Ω; ∈ ∈ (c) U Ω imply V Ω such that V V U; ∈ ∃ ∈ ◦ ⊂ (d) U, V Ω imply U V Ω; ∈ ∩ ∈ (e) U Ω and U V X X imply V Ω. ∈ ⊂ ⊂ × ∈ (X, Ω) is said to be a uniform space. Subfamily ∆ Ω of the uniformity Ω is said to be its base if any element of the family Ω contains an element of the family⊂ ∆. Let (X, Ω) be a uniform space. The topology τ, associated with a uniformity Ω, is the family of all sets K X such that for each x K there exists a U Ω such that U [x] K. ⊂ ∈ ∈ ⊂ The uniform space (X, Ω) is called Hausdorff if U ΩU = Λ. Let (X, Ω) be a uniform space and xn n N ∩ ∈ { } ∈ be some sequence. xn n N is called fundamental if U Ω, there exists a n0 N such that (xn, xm) U for all n, m n . { } ∈ ∀ ∈ ∈ ∈ ≥ 0 Throughout the paper (D, ) will denote a directed set and a non-trivial proper ideal of D.A net is ≥ I a mapping from D to X and will be denoted by sα : α D . Let Dα = β D : β α for α D. Then the { ∈ } ∈ ≥ c ∈ collection 0 = A D : A Dα for some α D forms a filter in D. Let 0 = A D : A 0 . Then 0 is a non-trivialF ideal{ of⊂D. A nontrivial⊃ ideal of∈ D}will be said to be D-admissibleI { if⊂D for∈ F all} α D.I A net I α ∈ F ∈ sα : α D in a topological space (X, τ) is called -convergent to s X if α D : sα U for any open set{ U containing∈ } s. F ∈ { ∈ ∈ } ∈ F

2. Main Results

In this section, we introduce -convergent and st-fundamental nets in uniform spaces and study some of their properties. F F Now we introduce our main deﬁnitions.

Deﬁnition 2.1. Let (X, Ω) be a uniform space and s : α D be a net in X. The net s : α D is said to { α ∈ } { α ∈ } be -convergent to s (in short, -lim sα = s) if for every U Ω, α D : (sα, s) U . In other words, for UF Ω, α D : s U [s] F. ∈ { ∈ ∈ } ∈ F ∀ ∈ { ∈ α ∈ } ∈ F Deﬁnition 2.2. Let (X, Ω) be a uniform space and s : α D be a net in X. The net s : α D is said to be { α ∈ } {α ∈ } st-fundamental in X if for every U Ω, there exist a α D such that α D : s U s . F ∈ 0 ∈ ∈ α ∈ α0 ∈ F

Lemma 2.3. Let (X, Ω) be a Hausdorﬀ uniform space and sα : α D be a net in X. If there exists -lim sα, then it is unique. { ∈ } F

Proof. Let (X, Ω) be a Hausdorﬀ uniform space. Accordingly, s = U ΩU [s]. Let sα : α D be a net in X. We prove that if there exists -lim s , then it is unique. Supposed{ } ∩ ∈ to contrary,{ that∈ is, } -lim s has F α F α two values t , t . Then it is obvious that there exists a Uk Ω such that t < U [t ] and t < U [t ]. If 1 2 ∈ 1 2 2 2 1 1 U = U1 U2, then U Ω. Furthermore, t1 < U [t2] and t2 < U [t1]. Since U Ω, there exists a V Ω such ∩ ∈ 1 ∈ ∈ that V V U and V = V− . It is clear that t < V [t ] and t < V [t ]. Suppose that ◦ ⊂ 1 2 2 1

A1 = α D : s V [t1] { ∈ α ∈ } E. Savas¸, U. Yamancı / Filomat 33:2 (2019), 353–357 355 and

A2 = α D : s V [t2] . { ∈ α ∈ } If A , A , then A A . On the other hand, A A = . If A A , , then there exists a 1 2 1 2 1 2 1 2 α D such∈ F that s A∩ A∈ F. Moreover, s , t V and∩ s , t ∅ ∈V F. From the∩ symmetry∅ of V, we have 0 α0 1 2 α0 1 α0 2 t ,∈s V. Consequently,∈ ∩(t , t ) V V U. This∈ is a contradiction,∈ that is, -lim s is unique. 2 α0 ∈ 1 2 ∈ ◦ ⊂ F α Theorem 2.4. Let (X, Ω) be a Hausdorff uniform space and s : α D be a net in X which is -convergent. Then { α ∈ } F s : α D is st-fundamental. { α ∈ } F Proof. Let (X, Ω) be a uniform space, sα : α D be a net in X and -lim sα = s. Now we prove that { ∈ } F 1 s : α D is st-fundamental. Let U Ω. Then there exists a V Ω such that V V U and V = V− . { α ∈ } F ∈ ∈ ◦ ⊂ Take α0 α D : s V [s] . It is obvious that ∈ { ∈ α ∈ } α D : s V [s] . { ∈ α ∈ } ∈ F If s V [s], then s , s V V U. As a result, α ∈ α α0 ∈ ◦ ⊂ α D : s V [s] α D : s U s { ∈ α ∈ } ⊂ ∈ α ∈ α0 and so α D : s U s . ∈ α ∈ α0 ∈ F Hence, the theorem is proved. Theorem 2.5. Let (X, Ω) be a Hausdorff, complete uniform space with a countable base and s : α D be a net in { α ∈ } X. If the net s : α D is st-fundamental, then there exists s X such that -lim s = s. { α ∈ } F ∈ F α Proof. Let (X, Ω) be a complete uniform space. We suppose that (X, Ω) has a countable base and it is Hausdorff. Then, there exists Uα Ω such that α DUα = Λ and Uα U for all α D. Without loss of ∈ ∩ ∈ 1 ⊂ ∈ (α+1) (α+1) (α) (α) (α) − generality, we suppose that U U U and U = U . Let s : α D be st-fundamental in ◦ ⊂ { α ∈ } F n (i) o X. Hence, by definition there exists αi D such that Ai , where Ai = α D : sα U sαi for i = 1, 2. It ∈ (1) ∈ F(2) ∈ ∈ is obvious that A(1) = A1 A2 . Let B1 = U sα1 U sα2 . Clearly, sα B1 for all α A(1). Likewise, ∩ ∈ F n (3) ∩o ∈ ∈ there exists α3 D such that A3 = α D : sα U sα3 . Suppose that A(2) = A(1) A3. It is obvious ∈ ∈ ∈ ∈ F ∩ that A . Put B = B U(3) s . As a result, B , and so s B for all α A . Continuing in the (2) ∈ F 2 1 ∩ α3 2 ∅ α ∈ 2 ∈ (2) same way, we get the net of open non-empty sets Bα α D X such that { } ∈ ⊂ (α+1) B B ..., Bn U sk for all α D, 1 ⊃ 2 ⊃ ⊂ α+1 ∈ such as A(i) such that A(i) = k D : sk Bi for all i D. Take esα Bα for all α D. Now we prove that ∈ F { ∈ ∈ } ∈ ∈ ∈ esα : α D is a fundamental net. Let U Ω be an arbitrary element. Then, it is clear that there exists α0 D ∈ (α0) ∈ ∈ such that U U for α α0. Let α α0 be arbitrary. We obtain esα+p Bα+p Bα for all p D. Since, we ⊂ ≥(α+1) ≥ (α+1∈) ⊂ (α+1) ∈ have Bα such that Bα+p U skα+1 , it is obvious that esα, skα+1 U and esα+p U skα+1 . Moreover, (α+1) (α+⊂1) (α) ∈ ∈ esα,esα+p U U U for all p D. As a result, esα,esα+p U for all α α0 and p D. Since U is ∈ ◦ ⊂ ∈ ∈ ≥ ∈ arbitrary, the net esα : α D is fundamental in (X, Ω) and let limesα = s. Now prove that -lim sα = s. Take ∈ (α) (α+1) F U Ω. Then, there exists a α D such that U U for all α α . Since B U sk , we have ∈ 0 ∈ ⊂ ≥ 0 α ⊂ α+1 n (α+1) o A α D : s U sk (α) ⊂ ∈ α ∈ α+1 ∈ F

(α0+1) for all α D. Let α1 D such that esk U [s] for all k α1. Without loss of generality, we suppose that ∈ ∈ ∈ h i ≥ (α1+1) (α1+1) (α1+1) (α1+1) α1 α0 +1. As a result,esα1 Bα1 U skα +1 . We put sk, skα +1 U . Then sk,esα1 U U ≥ ∈ ⊂ 1 1 ∈ ∈ ◦ ⊂ (α1) (α1+1) (α1) U . Since, esα1 , skα +1 U U , then it is obvious that 1 ∈ ⊂ (α1) (α1) (α1 1) (α0) sk, sk U U U − U U. α1+1 ∈ ◦ ⊂ ∈ ⊂ E. Savas¸, U. Yamancı / Filomat 33:2 (2019), 353–357 356

This implies that α D : s B α D : s U [s] . ∈ α ∈ α0 ⊂ { ∈ α ∈ } Therefore, A = α D : s B . (α0) ∈ α ∈ α0 ∈ F From the previous inclusion it follows that α D : s U [s] . { ∈ α ∈ } ∈ F Since U was arbitrary, we have -lim s = s. F α Theorem 2.6. Let (X, Ω) be a uniform space with a countable base and let sα : α D be an st-fundamental net in X. Then: { ∈ } F i) if is monotone closed filter and -lim sα = s, then there exists tα α D X such that lim tα = s and F F { } ∈ ⊂ α D : sα = tα ; { ∈ii) if is a right} ∈ F filter and lim t = s and α D : s = t , then -lim s = s. F α { ∈ α α} ∈ F F α Proof. i) Suppose that the net s : α D is st-fundamental, is monotone closed filter and the space { α ∈ } Fn o F (X, Ω) has a countable base. Consider the net A(α) , constructed in the proof of Theorem 2.5. We get α D ∈ A A ... and A for α D. (1) ⊃ (2) ⊃ (α) ∈ F ∈ Then by condition (iv) of filter we get αm : α1 < α2 < ... such that { } ∞ (αm, αm ] A . ∪m=1 +1 ∩ (m) ∈ F Suppose that

n c o D = k D : k (αm, αm ] A , m D [1, α ] . 0 ∈ ∈ +1 ∩ (m) ∈ ∪ 1 Define ( s, k D0 tk = ∈ , sk, k < D0 where s = -lim sα. Now we prove that lim tk = s. Let U Ω be an arbitrary element. If k D0, then it is F ∈ ∈c obvious that tk U [s]. If k < D , then there exists a m D such that αm < k αm and k < A . Moreover, ∈ 0 ∈ ≤ +1 (m) (α0 1) if k A , then sk Bm. Let α D be a number such that U − U. Let k be sufficiently large m α . We ∈ (m) ∈ 0 ∈ h i ⊂ ≥ 0 (α0) (α0+1) (α0+1) (α0) get sk U [s] and so sk U sk +1 and sk +1 U [s]. Hence, (tk, s) U U, since, in this case ∈ ∈ α0 α0 ∈ ∈ ⊂ sk = tk. Since U is arbitrary, lim tk = s. Now we prove that Ae = k D : sk = tk . It is clear that { ∈ } ∈ F ∞ (αm, αm ] A Ae. ∪m=1 +1 ∩ (m) ⊂ Hence, ∞ (αm, αm ] A and we obtain Ae from the condition (iii) of filter. Therefore, if ∪m=1 +1 ∩ (m) ∈ F ∈ F -lim s = s, then there exists an Ae such that lim t = s and s = t for all α Ae. F α ∈ F α α α ∈ ii) Suppose that lim tα = s, Ae = α D : sα = tα and is a right filter. Let U Ω be arbitrary. Then there exists α D such that t U{[s]∈for all α α} ∈. WeF get F ∈ 0 ∈ α ∈ ≥ 0 α D : α α0 Ae α D : s U [s] . { ∈ ≥ } ∩ ⊂ { ∈ α ∈ } It is obvious that α D : α α0 Ae . { ∈ ≥ } ∩ ∈ F Then we have α D : s U [s] from the condition (iii) of filter. { ∈ α ∈ } ∈ F E. Savas¸, U. Yamancı / Filomat 33:2 (2019), 353–357 357

The following results are immediate consequences of Theorems 2.5 and 2.6.

Corollary 2.7. Let (X, Ω) be a uniform space with a countable base, sα : α D be a net in X and be a monotone closed and a right ﬁlter. Then the followings are equivalent: { ∈ } F i) -lim s = s, F α ii) sα : α D is st-fundamental, iii){lim t ∈= s} andFα D : s = t . α { ∈ α α} ∈ F

Corollary 2.8. Let (X, Ω) be a uniform space with a countable base, s : α D be an st-fundamental net in X, { α ∈ } F and be a right ﬁlter. If -lim s = s, then there exists a αk : α1 < α2 < ... such that lim s = s. F F α { } ∈ F αk

Acknowledgement

The authors are most grateful to anonymous referees for careful reading of the manuscript and valuable suggestions.

References

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