# How big is the smallest nontrivial partition of the unit interval into closed disjoint closed sets? [duplicate]

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.

• 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 $\kappa=\omega_1<\frak{c}$.