# Family of Computably Enumerable Sets

1) Let $$S$$ be nonempty countable set (possibly finite). Any surjective map $$\nu\colon \omega \twoheadrightarrow S$$ from the set $$\omega$$ of natural numbers onto the set $$S$$ is called an enumeration.
2) An enumeration $$\nu$$ is single-valued if $$\nu$$ is a bijective, i.e. $$\nu$$(x) /= $$\nu$$(y) for any x /= y