Enumeration degrees and enumerability of familes

We study the enumerability of families relative to the enumeration degrees. It is shown that if a family of finite sets is e-reducible to every non-zero e-degree, then the family is computably enumerable (c.e). On the another hand, we will find a non-c.e. family which is e-reducible to all non-zero e-degree. This allows to construct a model, whose (extended) degree spectrum coincides with the non-zero e-degrees