On the Para - Compactness of Cone Metric Spaces 1

In 2007, Long-Guang and Xian[3] replaced introduced cone metric spaces. They replaced the set of real numbers by an ordered Banach space in the definition of metric and generalized the notion of metric space. Recently, Ayse Sönemaz [5] proved a cone metric space with a normal cone, of course it has to be strongly minihedral, is paracompact. In this paper we omit the strongly minihedral of cone. It is well-known that any metric space is paracompact [4]. This result plays a crucial role in this note. We assign to any cone metric a metric that their topology are homeomorphic and as we know to be paracompact is a topological property. Hence we obtain our result. To set up our results in the next section we recall some definitions and facts.