Is every subset of a RE language also RE, in general?
No, it is not true that every subset of a recursively enumerable (RE) language is also RE. In fact, it make sense to say that a subset of a RE language is not RE more often than not.

For an example, let us consider the RE language, $\Sigma^*$ that contains all words. It is the language of the Turing machine which always halts and accepts in its first step. Note that every language is a subset of $\Sigma^*$. Every language that is not RE is an example.



Here are a few exercises for you to check.

Exercise 1. Is every subset of a regular language regular?

Exercise 2. Is every subset of a context-free language context-free?

Exercise 3. Is every subset of a recursive language recursive?