CONDITIONS ON UNIQUENESS OF LIMIT POINT AND COMPLETENESS IN CONE POLYGONAL METRIC SPACES Sie Evan Setiawan 1* , Mahmud Yunus 2 1,2 Institut Teknologi Sepuluh Nopember, Surabaya Email : 1 evan.setiawan15@mhs.matematika.its.ac.id, 2 yunusm@matematika.its.ac.id * Corresponding author Abstract. This paper discusses cone polygonal metric spaces. We analyze some characteristics derived from convergence and Cauchyness of sequences. Our result consists of some conditions on uniqueness of limit point and completeness in cone polygonal metric spaces. Keywords: Cone metric spaces, cone polygonal metric spaces, convergence and Cauchyness, completeness, nested closed-ball property I. INTRODUCTION The concept of metric spaces is a major subject in mathematical analysis. There had been introductions of generalized metric spaces within various forms. An interesting generalization was found in cone metric spaces [1]. Cone metric space is obtained by substituting the codomain of metric function with a partially ordered Banach space. The order in Banach spaces is defined by a subset called cone, similar to how ordering in the set of real numbers is defined by the set of positive real numbers. The concept of cone metric spaces was generalized further by replacing the property of triangular inequality with similar inequality which contains four or more elements. The results are cone rectangular metric spaces [2], cone pentagonal metric spaces [3], cone hexagonal metric spaces [4], cone heptagonal metric spaces [5], and so on. However, these spaces share similar characteristics. Therefore, they can be studied as one group, namely cone polygonal metric spaces. Some basic properties of metric spaces are not present in cone polygonal metric spaces. For example, a convergent sequence can have multiple limit points and is not necessarily Cauchy [6]. This motivates us to further analyze correlation between convergent sequences and Cauchy sequences. We discovered that simultaneous presence of convergence and Cauchyness would lead to uniqueness of limit point. We also analyze conditions that indicate completeness of a given space. We found sequential compactness to be a sufficient condition for completeness. Other than that, we establish a necessary and sufficient condition by expanding a nested closed-ball theorem [7] from cone metric spaces to cone polygonal metric spaces. Our result could be applied in other theories that use iterative sequences. For example, our sufficient conditions on completeness could be directly linked to fixed point theorems [1, 2, 8, JOURNAL OF FUNDAMENTAL MATHEMATICS AND APPLICATIONS (JFMA) VOL. 4 NO. 1 (JUN 2021) Available online at www.jfma.math.fsm.undip.ac.id 133 p-ISSN: 2621-6019 e-ISSN: 2621-6035 https://doi.org/10.14710/jfma.v4i1.10653