Normed Ordered and E-Metric Spaces

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space E, and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of R, and we discuss their extension to E. We introduce the notion of E-metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of convergence and Cauchy theory. In particular, we get a fixed point theorem of a contractive mapping in E-metric spaces. Finally, by extending the notion of a contractive sequence in a real-valued metric space, we show that in E-metric spaces, a contractive sequence is Cauchy.