Definition of a computable (or recursive) set I am new to computability theory, but I understand the usual definition of a "computable set" S when S is a subset of the natural numbers.  Is there a notion of "computable set" that doesn't involve just subsets of the real numbers?  In particular, I am interested in the set $\Sigma^{\omega}$ for some finite alphabet $\Sigma$ (this is the set of right-infinite words over $\Sigma$).  Is there a notion of a computable subset of $\Sigma^{\omega}$?   Off the top of my head, it seems that a possible definition would be to say that a subset $X$ of $\Sigma^{\omega}$ is computable if each sequence in $X$ represents a computable function $\mathbb{N} \rightarrow \Sigma$, but that doesn't seem correct.  Any help is appreciated.

"Computable" subsets of $2^\omega$ are the clopen sets under the product topology (generated by basic (cl)open sets of the form $[\sigma] := \{X \in 2^{\omega} \colon \sigma \prec X \}$, where $\prec$ denotes "is a prefix of"), which is why clopen sets are called $\Delta_0$ in the Lightface Borel hierarchy. (Recall that $\Delta_0$ in the arithmetical hierarchy means computable.)
Prateek's answer is essentially proving that any clopen subset of Cantor space $2^\omega$ is a finite union of basic opens. I think a better word than "computable" for these sets is "decidable", since given any infinite sequence and clopen set you'll eventually know if that sequence is the clopen set or not (by Prateek's answer).
