This question already has an answer here:

Consider how we might partition the unit interval in the reals into disjoint closed sets $$[0,1]=\bigsqcup_i C_i.$$ Of course, we could partition the unit interval into singletons, which would make a partition of size continuum. And it turns out not to be possible to find such a partition with only countably many closed sets.

If the continuum hypothesis fails, perhaps there is a nontrivial partition into fewer than continuum many disjoint closed sets? Yes, indeed, it turns out to be relatively consistent with ZF that this is possible.

So let us define the *closed partition number* $\kappa$, which is
the size of the smallest nontrivial partition of the unit interval
$[0,1]$ in the reals into closed sets.

This number appeared in my answer to a MathOverflow question concerning the continuum hypothesis and cofinite topologies being contractible, where I proved that the cofinite topology on a nontrivial set $X$ is contractible or equivalently path connected if and only if $X$ has size at least the closed partition number.

In that answer, I had noted that we have some information about the closed partition number.

- It is uncountable and at most continuum. $$\omega_1\leq \kappa\leq \frak{c}$$
- $\text{cov}(\mathcal{M})\leq\kappa$.
- $\frak{d}\leq\kappa$.
- It is relatively consistent that $\kappa=\omega_1<\frak{c}$.

My question is whether we might have more complete information about this cardinal.

**Question.** How does the closed partition number relate to the
other better-known cardinal characteristics? Is it already known by
another name?