Is there a condensation (continuous bijective mapping) from $D^{\aleph\_0}$ onto a metrizable compact space ?


$D$ - discrete space of cardinality $\aleph\_1$.


CH implies it is a positive answer. In general, I don’t know the answer.



Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?
If $|D| < \aleph\_\omega$, then there is a condensation from $D^\omega$ onto $\omega^\omega$ (the Baire space) if and only if there is a partition of $\omega^\omega$ into exactly $|D|$ Borel sets. 


As far as I know, this theorem was first proved by me and Arnie Miller in 



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> "Partitions of $2^\omega$ and completely ultrametrizable spaces," *Topology and its Applications* **184** (2015), pp. 61-71.
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See Theorem 3.9. I still don't know what happens for $|D| \geq \aleph\_\omega$.


It is a theorem of Hausdorff that $\omega^\omega$ can be partitioned into $\aleph\_1$ Borel sets, regardless of whether $\mathsf{CH}$ holds or not. This, together with the theorem quoted above, provides a positive answer to your question.


It is consistent with any allowed value of $\mathfrak{c}$ that there is a partition of $\omega^\omega$ into $\kappa$ Borel sets for every $\kappa \leq \mathfrak{c}$. (This is Theorem 3.11 in the linked paper.) It is also consistent with any allowed value of $\mathfrak{c}$ that the only partitions of $\omega^\omega$ into Borel sets have size $\leq \aleph\_0$, $\aleph\_1$, and $\mathfrak{c}$ (see Corollary 3.16 in the linked paper), or size $\leq \aleph\_0$, $\aleph\_1$, $\kappa$, and $\mathfrak{c}$ for any particular $\kappa$ between $\aleph\_1$ and $\mathfrak{c}$ (see Proposition 3.17 and the comments following). Generally, though, it's not known how to get some prescribed list of sizes and not others.


