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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 compact sets. 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?

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 $|\mathbb R|?$

Let's assume all spaces are metrizable.

It is a well-known theorem of Sierpinski that a connected compact space cannot be the union of a countable number of disjoint compact subsets.

For each connected compactum $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into compact sets.

  Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$

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

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

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 compact sets. 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?

Let's assume all spaces are metrizable.

It is a well-known theorem of Sierpinski that a connected compact space cannot be the union of a countable number of disjoint compact subsets.

For each connected compactum $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into compact sets.

Let $\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ connected compactum }X)(\mathcal K\in \mathscr K(X))\}$.

Obviously $\kappa \leq \mathfrak c=|\mathbb R|$.

Is it necessarily true that $\kappa=\mathfrak c$? Or is this axiom-dependent?

Let's assume all spaces are metrizable.

It is a well-known theorem of Sierpinski that a connected compact space cannot be the union of a countable number of disjoint compact subsets.

For each connected compactum $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into compact sets.

Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$

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

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