Is there a name for complement of recursively enumerable set? If there is a recursive function
$$
f(x) = 
\begin{cases}
1 &amp; \text{if $x \notin A$}\\
\text{undefined} &amp; \text{if $x \in A$}\\
\end{cases}
$$

Is there a special name for the class of sets like $A$?
The complement of a recursively enumerable (RE) language is called co-recursively enumerable (co-RE).

As RE and co-RE form the first level of the arithmetic hierarchy, they are also known by aliases $\Sigma_1^0$ and $\Pi_1^0$.