# Terminology for set of infinite strings with a certain prefix

Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology.

Given a finite string $s \in \mathcal{A}^*$, let $C(s)$ be the set of all strings in $\mathcal{A}^\omega$ that have $s$ as a prefix.

Is there a standard name for such sets $C(s)$ in $C$? If we think of $C$ as a Cantor set, then $C(s)$ is one the canonical "Cantor subsets".

Yes, $C(s)$ is an example of a cylinder set.
More specifically, $C(s)$ is called a basic open cylinder (since other cylinder sets are unions of such sets). See e.g. Andre Nies' monograph Computability and Randomness; the notation $[s]$ is used for $C(s)$.
• In Kechris' book the notation $N_s$ is used, but no name for it is assigned there. This applies to any countable power. – Pedro Sánchez Terraf Jun 20 '15 at 0:29
• If one uses an arbitrary countable power $A^X$ and call cylinder any subset of the form $\{f\in A^X:f|_F=g\}$ (for finite $F\subset X$ and $g\in A^F$), one gets something more general, since here $F$ is assumed to be an initial segment of $\omega$. I'm not sure which definition of cylinder is more common (of course I rather like the latter one). – YCor Jun 20 '15 at 15:58