On G -sequential connectedness

📝 Abstract

Recently, Cakalli has introduced a concept of $G $-sequential connectedness in the sense that a non-empty subset $A$ of a Hausdorff topological group $X$ is $G $-sequentially connected if there are no non-empty, disjoint $G $-sequentially closed subsets $U$ and $V$ meeting $A$ such that $A\subseteq U\bigcup V $. In this paper we investigate further properties of $G $-sequential connectedness and prove some interesting theorems.

💡 Analysis

Recently, Cakalli has introduced a concept of $G $-sequential connectedness in the sense that a non-empty subset $A$ of a Hausdorff topological group $X$ is $G $-sequentially connected if there are no non-empty, disjoint $G $-sequentially closed subsets $U$ and $V$ meeting $A$ such that $A\subseteq U\bigcup V $. In this paper we investigate further properties of $G $-sequential connectedness and prove some interesting theorems.

📄 Content

arXiv:1201.1796v1 [math.GN] 22 Dec 2011 ON G-SEQUENTIAL CONNECTEDNESS H¨USEY˙IN C¸AKALLI*, AND OSMAN MUCUK** *DEPARTMENT OF MATHEMATICS, MALTEPE UNIVERSITY, ISTANBUL, TURKEY ** DEPARTMENT OF MATHEMATICS, ERCIYES UNIVERSITY, KAYSERI, TURKEY Abstract. Recently, Cakalli has introduced a concept of G-sequential connectedness in the sense that a non-empty subset A of a Hausdorfftopological group X is G-sequentially connected if there are no non-empty, disjoint G-sequentially closed subsets U and V meeting A such that A ⊆U S V . In this paper we investigate further properties of G-sequential connectedness and prove some interesting theorems.

1. Introduction The concept of connectedness and any concept related to connectedness play a very important role not only in pure mathematics but also in other branches of science involving mathematics especially in geographic information systems, population modeling and motion planning in robotics. In [1], Connor and Grosse-Erdmann have investigated the impact of changing the definition of the convergence of sequences on the structure of sequential continuity of real functions. Cakalli [2] extended this concept to topological group setting and has introduced the concepts of G-sequential compactness and G-sequential continuity; and has investigated some results in this generalized setting (see also [3]). One is often relieved to find that the standard closed set definition of connectedness for topological spaces can be replaced by a sequential definition of connectedness. That many of the properties of connectedness of sets can be easily derived using sequential arguments has also been, no doubt, a source of relief to the interested mathematics instructor. Date: November 5, 2018. 2000 Mathematics Subject Classification. Primary: 40J05 ; Secondary: 54A05, 22A05. Key words and phrases. Sequences, series, summability, sequential closure, G-sequential continuity, G-sequential connectedness. 1 2H¨USEY˙IN C¸ AKALLI*, AND OSMAN MUCUK** *DEPARTMENT OF MATHEMATICS, MALTEPE UNIVERSITY, ISTANBUL, TURKEY ** Recently, Cakalli [4] has defined G-sequential connectedness of a topological group and investi- gated some results in this generalized setting. The purpose of this paper is to develop some further properties of G-sequential connectedness in metrizable topological groups, and present some interesting results.
2. Preliminaries Before giving some results on G-sequential connectedness we remark some background as follows. Throughout this paper, N denotes the set of all positive integers and X denotes a Hausdorff topological group written additively satisfying the first axiom of countability. We use boldface letters x, y, z, … for sequences x = (xn), y = (yn), z = (zn), … of terms of X. s(X) and c(X) respectively denote the set of all X-valued sequences and the set of all X-valued convergent sequences of points in X. Following the idea given in a 1946 American Mathematical Monthly problem [5], a number of authors Posner [6], Iwinski [7], Srinivasan [8], Antoni [9], Antoni and Salat [10], Spigel and Krupnik [11] have studied A-continuity defined by a regular summability matrix A. Some authors ¨Ozt¨urk [12], Sava¸s [13], Sava¸s and Das [14], Borsik and Salat [15] have studied A-continuity for methods of almost convergence or for related methods. See also [16] for an introduction to summability matrices. A sequence (xk) of points in X is called to be statistically convergent to an element ℓof X if for each neighborhood U of 0 lim n→∞ 1 n|{k ≤n : xk −ℓ/∈U}| = 0, and this is denoted by st −limn→∞xn = ℓ. Statistical limit is an additive function on the group of statistically convergent sequences of points in X (See [17] for the real case and [18], [19], [20] for the topological group setting and see [21], and [22] for the most general case, i.e., topological space setting). A sequence (xk) of points in X is called lacunary statistically convergent to an element ℓof X if lim r→∞ 1 hr |{k ∈Ir : xk −ℓ/∈U}| = 0, for every neighborhood U of 0 where Ir = (kr−1, kr] and k0 = 0, hr : kr −kr−1 →∞as r →∞and θ = (kr) is an increasing sequence of positive integers. For a constant lacunary sequence θ = (kr), the lacunary statistically convergent sequences in a topological group form a subgroup of the group ON G-SEQUENTIAL CONNECTEDNESS 3 of all X-valued sequences and lacunary statistical limit is an additive function on this space (see [18] for topological group setting and see [23] and [24] for the real case). Throughout this paper, we assume that lim infr kr kr−1 > 1. By a method of sequential convergence, or briefly a method, we mean an additive function G defined on a subgroup cG(X) of s(X) into X [2]. A sequence x = (xn) is said to be G-convergent to ℓif x ∈cG(X) and G(x) = ℓ. In particular, lim denotes the limit function lim x = limn xn on the group c(X). A method G is called regular if every convergent sequence x = (xn) is G-convergent with G(x) = lim x. Clearly if f is G-sequentially continuous on X, then

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