I'm interested in sets $X$ in 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 sets. I haven't been able to find much online.
1. Do such sets 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)?