Thai Journal of Mathematics Volume 16 (2018) Number 1 : 229–242 http://thaijmath.in.cmu.ac.th ISSN 1686-0209 Some Notes on Cone Metric Spaces M. Tavakoli † , A.P. Farajzadeh †, 1 , T. Abdeljawad ‡ and S. Suantai § † Department of Mathematics, Razi University, Kermanshah 67149, Iran e-mail : mahtab.tavakoli1414@gmail.com (M. Tavakoli) farajzadehali@gmail.com (A.P. Farajzadeh) ‡ Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia e-mail : tabdeljawad@psu.edu.sa (T. Abdeljawad) § Department of Mathematics, Faculty of Science, Chiang Mai University Chiang Mai 50200, Thailand e-mail : suthep.s@cmu.ac.th (S. Suantai) Abstract : Recently, several articles have been written on the cone metric spaces. Despite the fact that any cone metric space is equivalent to a usual metric space, we aim in this paper to deal with some of the published articles on cone metric spaces by repairing some gaps, providing new proofs and extending their results to topological vector spaces. Several authors have worked with a class of special cones which known as strongly minhedral cones where the strongly minihedrality condi- tion (that is, each nonempty bounded above subset has a least upper bound) is very restrictive. Another goal of this article is to eliminate or mitigate this condition. Furthermore, we present some examples in order to show that the imagination of many authors that the behavior of the ordering induced by a strongly minihedral cone is just as the behavior of the usual ordering on the real line, that has caused an error in their proofs, is not correct. We establish a relationship between strong minihedrality and total orderness. Finally, a fixed point theorem for a contractive mapping, which generalizes the corresponding result given in [1], is investigated. One can consider the results of this paper as a generalization and correction of some recent papers that have been written in this area. Keywords : cone metric space; first countable; strongly minihedral cone; totally ordered; sequentially compact; contractive mapping. 1 Corresponding author. Copyright c  2018 by the Mathematical Association of Thailand. All rights reserved.