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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".

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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 $C(s)$.

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    $\begingroup$ In Kechris' book the notation $N_s $ is used, but no name for it is assigned there. This applies to any countable power. $\endgroup$
    – Pedro Sánchez Terraf
    Commented Jun 20, 2015 at 0:29
  • $\begingroup$ Indeed. Also, since a cylinder is shift-invariant, this is a very common object in symbolic dynamics. $\endgroup$
    – Nikita Sidorov
    Commented Jun 20, 2015 at 15:17
  • $\begingroup$ 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). $\endgroup$
    – YCor
    Commented Jun 20, 2015 at 15:58

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