The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\hookrightarrow X$.

It is known that without Axiom of Choice the injectivity cannot be replaced by the surjectivity in this result.

Nonetheless, without AC, we have the following simple

Fact.If for two sets $X,Y$ there are surjective functions $X\twoheadrightarrow Y$ and $Y\twoheadrightarrow X$, then there exists a bijective function $\mathcal P(X)\leftrightarrow \mathcal P(Y)$ between the corresponding power-sets.

Can the power-sets in this result be replaced by some simpler sets built over $X$, for example, $X^\omega$ or $X\times\omega$?

More precisely:

Question.Assume that for two sets $X,Y$ there exist surjective functions $X\twoheadrightarrow Y$ and $Y\twoheadrightarrow X$.Is it true (in ZF) that the sets

(i) $X^\omega$ and $Y^\omega$ have the same cardinality?

(ii) $X\times\omega$ and $Y\times\omega$ have the same cardinality?

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