# Borel / Wadge hierarchies on sets closed under prepending a finite prefix

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.

Questions:

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)?