How strong is Sierpiński theorem about continua?

This is actually quite a deep question. See this mathoverflow question plus its comments and answers.

In short: it's complicated. I do see some consistency options there in the absence of CH (or MA). Read up on Cichon's diagram and cardinal invariants of the continuum (i.e. a lot of books/papers..) when looking through this thread.

In this paper by Miller it's shown that consistenly we can have non-CH and $2^\omega$ partitioned into $\aleph_1$ many closed subsets (thm 4 and 5). Now apply theorem 3. of that paper to get such a partition of $[0,1]$, say.

I hope this helps.

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