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Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}$. Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ a connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$

We know $$\aleph_0<\kappa \leq |\mathbb R|;$$

the first inequality is due to Sierpinski (Theorem 6.1.27 in Engelking's Topology), and the second is true because every connected space has a partition into $\mathbb R$-many singletons.

Is it necessarily true that $\kappa=|\mathbb R|$? Or is this axiom-dependent?

EDIT: What about just for the space $X=[0,1]$? Is the minimum cardinality of a compact partition of $X$ necessarily $|\mathbb R|?$

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It is indeed axiom independent. In the Solovay random model (adding $\aleph_2$ random reals over a model of ZFC+GCH) you have $2^\omega=\aleph_2$ and there exists a $\aleph_1$ partition of $[0,1]$ into nowhere dense closed sets (hence compact). The idea is to define such a collection in the ground model and use the key fact that no Cohen real is added in random forcing to capture each new real using some nowhere dense closed set. In this way, no new (random) real escapes from the partition we fixed to begin with in the gound model. I thought it was folklore but actually found that Jacques Stern has an article about this.

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  • $\begingroup$ Is that the "$\neg$AC$+$every subset of reals is measurable" model? $\endgroup$ – D.S. Lipham Apr 5 '18 at 1:39
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    $\begingroup$ it is not. It is a model of ZFC. The forcing poset is measurable subsets of $2^{\omega_2}$ with the product measure, ordered by taking subsets modulo null sets. $\endgroup$ – Jing Zhang Apr 5 '18 at 2:12

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