Domain and range characterizations of a nonempty computably enumerable set
= Domain and range characterizations of a nonempty computably enumerable set
For a nonempty set $A$ of words, the following are equivalent: $A$ is <computably enumerable set>[computably enumerable], $A$ is the domain of a <partial computable function>, $A$ is the range of a partial computable function, and $A$ is the range of a <total computable function>. The nonempty hypothesis is needed only for the last formulation, since the range of a total function cannot be empty.