Given two subsets $A,B$ of the Cantor cube $2^\omega$ we write $A\le_W B$ (resp. $A\le_H B$) if there is a continuous (and injective) function $f:2^\omega\to 2^\omega$ such that $f^{-1}(B)=A$. The relation $\le_W $ is the well-known (and well-studied) Wadge reducibility relation. The relation $\le_H$ is called *the Hurewicz reducibility* to make associations with some known Hurewicz-type tests (proved by Hurewicz, Louveau, Saint Raymond and others).

**Problem 1.** *What is known about the relation of Hurewicz reducibility between Borel subsets of the Cantor cube?
Is it equivalent to the Wadge reducibility for sufficiently complex Borel subsets of $2^\omega$?*

More precisely: *Assume that $A,B\subset 2^\omega$ are two Borel subsets of complexity (say) $A,B\notin \Sigma^0_3\cap \Pi^0_3$. Does $A\le_W B$ imply $A\le_HB$?*

By a result of Louveau and Saint Raymond (1987) this implication is true if $A,B$ belong to the class $\Gamma\setminus \check\Gamma=\{C\in\Gamma:2^\omega\setminus C\notin\Gamma\}$ for some class $\Gamma$ of Borel subsets of $2^\omega$ which is closed under intersections with $G_\delta$-sets and unions with $F_\sigma$-sets in $2^\omega$. So, for any $\xi\ge 3$ and any subsets $A,B\in\Pi^0_\xi\setminus \Sigma^0_\xi$ we get $A\le_H B$ and $B\le_H A$.

Let us say that two topological spaces $X,Y$ are *Hurewicz equivalent* if each of them admits a closed topological embedding into the other.
It is clear that two subsets $A,B\subset 2^\omega$ are Hurewicz equivalent if $A\le_H B$ and $B\le _H A$. The converse is not true: the Cantor cube contains two homeomorphic dense $G_\delta$-sets $A,B$ such that $A\not\le_H B$.

**Problem 2.** Assume that $A,B\subset 2^\omega$ are two Borel subsets of sifficiently hight Borel complexity (say $A,B\notin \Sigma^0_3\cap \Pi^0_3$). Assume that $A,B$ are Hurewicz equivalent. Is $A\le_H B$?

**Problem 3.** Classify Borel subsets of $2^\omega$ up to Hurewicz equivalence. Is it true that for each Borel subset $B\subset 2^\omega$ of sufficiently hight Borel complexity (say $B\notin\Pi^0_2\cup \Sigma^0_2$) the Wadge class $W(A)=\{B\subset 2^\omega:B\le_W A,\;A\le_W B\}$ of $A$ contains only finitely many spaces (better, a unique space) that are pairwise Hurewicz non-equivalent?