On the Completion of Fuzzy Number Space with Respect to Endograph Metric

The endograph metric plays an important role in fuzzy number theory. The endograph metric on the fuzzy number space E¹ is known to be separable but not complete. This paper deals with the completion of E¹ with respect to the endograph metric. It is shown that the space of all non-compact fuzzy number space F*(R) is the completion of E¹ with respect to the endograph metric. It is proved that the endograph metric is approximative with respect to order on fuzzy number spaces F*(R), also, the endograph metric is computable. Finally some analytic theorems are given with respect to the endograph metric.