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".



Terminology for set of infinite strings with a certain prefix
Yes, $C(s)$ is an example of a [cylinder set](https://en.wikipedia.org/wiki/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 $C(s)$.


