TITLE: Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?
QUESTION [9 upvotes]: 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.

REPLY [11 votes]: 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 

"Partitions of $2^\omega$ and completely ultrametrizable spaces," Topology and its Applications 184 (2015), pp. 61-71.

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.