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Let [tex]X[/tex] be a countable set. Then [tex]X^{n}[/tex] is countable for each [tex]n \in N[/tex].

Now it should also be true that [tex]\bigcup^{\infty}_{n=1} X^{n}[/tex] is countable. How is this different from [tex]X^{\omega}[/tex], which is uncountable?