This question was motivated by an answer to this question of Dominic van der Zypen.

It relates to the following classical theorem of Sierpiński.

**Theorem (Sierpiński, 1921).** For any countable partition of the unit interval $[0,1]$ into closed subsets exactly one set of the partition is non-empty.

Motivated by this Sierpiński Theorem we can ask about the smallest infinite cardinality $\acute{\mathfrak n}$ of a partition of the unit interval into closed non-empty subsets. It is clear that $\acute{\mathfrak n}\le\mathfrak c$. The Sierpinski Theorem guarantees that $\omega_1\le\acute{\mathfrak n}$. So, $\acute{\mathfrak n}$ is a typical small uncountable cardinal living in the segment $[\omega_1,\mathfrak c]$.

**Problem 1.** Is $\acute{\mathfrak n}$ equal to some other known small uncountable cardinal?

**Problem 2.** Is $\acute{\mathfrak n}$ equal to $\mathfrak c$ under Martin's Axiom?

We can also can consider the measure modification of the cardinal $\acute{\mathfrak n}$.

Namely, let $\acute{\mathfrak m}$ be the smallest cardinality of a cover of $[0,1]$ by pairwise disjoint closed subsets of Lebesgue measure zero.

The definitions imply that $\acute{\mathfrak n}\cdot\mathrm{cov}({\mathcal N})\le\acute{\mathfrak m}\le\mathfrak c$.

According to Theorem 4 of Miller, the strict inequality $\acute{\mathfrak m}<\mathfrak c$ is consistent. So, $\acute{\mathfrak m}$ is a non-trivial small uncountable cardinal.

**Problem 3.** Is it consistent that $\acute{\mathfrak n}<\acute{\mathfrak m}$?

**Problem 4.** Is $\acute{\mathfrak m}$ equal to some known small uncountable cardinal?

**Added after analyzing comments to these problems:** As was observed by @Ashutosh, the answer to Problem 2 is affirmative. In his paper Miller writes that this was done by Both (1968, unpublished) and Weiss (1972, unpublished). The MA equality $\acute{\mathfrak n}=\mathfrak c$ can be also derived from the ZFC inequality $$\mathfrak d\le\acute{\mathfrak n},$$ which can be proved as follows: given a partition $\mathcal P$ of $[0,1]$ into pairwise disjoint closed sets with $|\mathcal P|=\acute{\mathfrak n}$, we can choose a countable subfamily $\mathcal P'\subset\mathcal P$ such that the space $X=[0,1]\setminus\bigcup\mathcal P'$ is nowhere locally compact and hence is homeomorphic to $\omega^\omega$. Then $\mathcal P\setminus\mathcal P'$ is a cover of $X\cong\omega^\omega$ by compact subsets, which implies that $\acute{\mathfrak n}=|\mathcal P\setminus\mathcal P|'\ge\mathfrak d$ by the definition of the cardinal $\mathfrak d$.

Miller proved the consistency of the strict inequality $\acute{n}<\mathfrak c$. Looking at the diagram of small uncountable cardinals in Vaughan, I found only three small uncountable cardinals above $\mathfrak d$: $\mathfrak i$, $cof(\mathcal M)$ and $cof(\mathcal L)$.

**Problem 5.** Is $\acute{\mathfrak n}$ equal to one of the cardinals $\mathfrak d$, $\mathfrak i$, $cof(\mathcal M)$ or $cof(\mathcal L)$ in ZFC?

**Summing up the progress made sofar.** The cardinals $\acute{\mathfrak n}$ and $\acute{\mathfrak m}$ satisfy the following ZFC-inequalities:

$$\mathfrak d\le \acute{\mathfrak n}\le\acute{\mathfrak m}=\acute{\mathfrak n}\cdot\mathrm{cov}_{\sqcup}(\mathcal N)\le\mathfrak c.$$

Here by $\mathrm{cov}_{\sqcup}(\mathcal N)$ we denote the smallest cardinality of a disjoint cover of $[0,1]$ by Borel Lebesgue null sets.

It is clear that $\mathrm{cov}(\mathcal N)\le\mathrm{cov}_\sqcup(\mathcal N)$ and $\mathrm{cov}(\mathcal N)=\aleph_1$ implies $\mathrm{cov}_\sqcup(\mathcal N)=\aleph_1$.

So, $\mathrm{cov}(\mathcal N)=\aleph_1$ implies the equality $\acute{\mathfrak n}=\acute{\mathfrak m}$.

Below I collect two consistency results observed by

**Will Brian**: $\acute{\mathfrak m}<\mathrm{non}(\mathcal N)$ is consistent;

**Ashutosh**: $\acute{\mathfrak n}=\aleph_1$ in the random real model (in this model $\acute{\mathfrak m}=\mathrm{cov}(\mathcal N)=\mathrm{non}(\mathcal M)=\mathfrak i=\mathfrak r=\mathfrak c$), so $\acute{\mathfrak n}<\acute{\mathfrak m}$ is consistent.

The above results suggest the following questions:

**Problem 6.** Is $\mathfrak d<\acute{\mathfrak n}$ consistent?

**Problem 7.** Is $\acute{\mathfrak n}$ upper bounded by some known small uncountable cardinal different from $\mathfrak c$? For example, is $\acute{\mathfrak n}\le \mathrm{cof}(\mathcal N)$ true in ZFC?

**Problem 8.** Is $\acute{\mathfrak m}=\acute{\mathfrak n}\cdot\mathrm{cov}(\mathcal N)$?

**Problem 9.** Is $\mathrm{cov}_\sqcup(\mathcal N)=\mathrm{cov}(\mathcal N)$?