What is the relation between countability and recursive enumeration?

Question

Does recursive enumeration implies countability? Does countability implies recursive enumeration?

I believe the first implication holds but not sure about the second. A good example would suffice.

Answer 1

If a language is recursively enumerable then it is countable. However, there are languages that are countable but not recursively enumerable. For example, consider the $A_{TM}$, the language of the Halting problem. This language is recursively enumerable while its complement $\Sigma^*-A_{TM}$ is not recursively enumerable, otherwise $A_{TM}$ would be decidable. However, $\Sigma^*-A_{TM}$ is clearly countable since it is a subset of a countable set $\Sigma^*$.

Hence one simple relation between countability and enumeration is that every r.e. set is countable, but the opposite is not always true.