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?


marked as duplicate by Joel David Hamkins set-theory Apr 11 '18 at 0:06

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.