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I'm interested in subsets $X$ of the Cantor space ($2^\omega$) or the Baire space ($\omega^\omega$) that are closed under prepending an arbitrary finite prefix: $$ (x_1, x_2, \dots) \in X \implies (s_1, \dots, s_k, x_1, x_2, \dots) \in X $$ for any $(s_1, \dots, s_k)$. In particular I'm interested on what Borel and Wadge hierarchies look like when limited to such subsets. I haven't been able to find much online.

Questions:

  1. Do such subsets X have a name? What keyword should I be searching for?

  2. What's good place to start exploring this topic (a paper or a book)?

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  • $\begingroup$ As for question 1, I've heard them called "tail sets." $\endgroup$
    – Will Brian
    Commented Feb 17, 2022 at 13:39
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    $\begingroup$ @WillBrian I've heard "tail set" used for the concept that has $\iff$ instead of $\implies$ in the OP's concept. $\endgroup$
    – Andreas Blass
    Commented Feb 17, 2022 at 16:22
  • $\begingroup$ @AndreasBlass: You're right -- my mistake. $\endgroup$
    – Will Brian
    Commented Feb 17, 2022 at 16:25

1 Answer 1

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I am not sure of a specific, good source for you, but it looks like you are interested in sets that are closed under the 'inverse' of the shift map.

The shift $S:A^\omega\to A^\omega$ is $S(x_1,x_2,x_3,\ldots)=(x_2,x_3,\ldots)$. This isn't injective so it doesn't have an inverse, but the reverse process is adding something (anything) from $A$ to the beginning of a sequence.

Hopefully this gives you some more material to help with your search. Although searching for the inverse shift map might send you to articles concerning two-way infinite sequences, which is not exactly what you are looking for.

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