the set of countable subsets of $2^ω$

Question

Let $X$ denote the two-element set {$0, 1$} and let $A$ be the set of countable subsets of $X^ω$. Show that $X^ω$ and $A$ have the same cardinality. (This is from Topology by James Munkres) How can I prove this without using the Axiom of Choice? It this trivial that there exists an injective function from $X^ω$ to $A$ but I have no idea how to prove the opposite direction.