TAILIEUCHUNG - The relation between rough Wijsman convergence and asymptotic cones
In this context, we often study sufficiently small neighborhoods of a point in a topological space, and investigate the behavior of a sequence approaching that point. In this paper, we explore the effect of the asymptotic cone of the limit set of a sequence that is rough Wijsman convergent. | Turk J Math (2016) 40: 1349 – 1355 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The relation between rough Wijsman convergence and asymptotic cones ∗ ¨ ¨ Oznur OLMEZ , Salih AYTAR Department of Mathematics, S¨ uleyman Demirel University, Isparta, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we explore the effect of the asymptotic cone of the limit set of a sequence that is rough Wijsman convergent. Key words: Rough convergence, Wijsman convergence, asymptotic cone 1. Introduction As is well known, one of the fundamental concepts in mathematical analysis is the concept of the limit of a sequence. In this context, we often study sufficiently small neighborhoods of a point in a topological space, and investigate the behavior of a sequence approaching that point. In recent years, many authors have explored sequences diverging to infinity, by using much larger neighborhoods of a point. This idea is a useful way in both differential geometry and geometric group theory. In addition, it has led to the emergence of asymptotic cones. The notion of an asymptotic cone was first introduced by Steinitz [10]. He also gave the asymptotic properties of unbounded convex sets. Later, asymptotic cones were called ”horizon cones” by some authors. In 1940, the theory of asymptotic cones was developed by Stoker [11]. In 1953, Fenchel [4] introduced the concept of the convergence of a sequence of rays by using the distance between two rays. Moreover, he proved that such a distance is a metric on the space of rays. In 1966, an alternative definition of the asymptotic cone was given by Wijsman [13] via normalized sequences. As for the notion of rough convergence of a sequence, it was first introduced by Phu [7] in a finitedimensional normed space as follows. Let r be a nonnegative real number. A sequence {xn } is said to be r r
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