A continuum is a compact connected metrizable topological space.


Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint union of more than one and at most $\kappa$ many closed subsets. In particular a space is connected iff it is $2$-connected iff it is $n$-connected for all $2\leq n<\aleph\_0$.


It is a standard result that continua are $\aleph\_0$-connected and, by writing a continuum as a union of singletons, it is clear that continua are $\mathfrak c$-disconnected.


Given a continuum $X$ let $\mathrm{disc}(X)$ denote the smallest $\kappa$ such that $X$ is $\kappa$-disconnected.


Is it consistent with $\mathsf{ZFC}$ to have a continuum $X$ with $\aleph\_0<\mathrm{disc}(X)<\mathfrak c$?


Is it a theorem of $\mathsf{ZFC}$ that if $X$ and $Y$ are nontrivial continua, then $\mathrm{disc}(X)=\mathrm{disc}(Y)$?


Assuming a positive answer to the previous two questions, is $\mathrm{disc}(X)$ equal to some standard cardinal invariant of the continuum?



How “disconnected” can a continuum be?
The answer to all three of your questions is yes.


The cardinal $\mathrm{disc}([0,1])$ is discussed in [this](https://mathoverflow.net/questions/285780/are-the-sierpi%C5%84ski-cardinal-acute-mathfrak-n-and-its-measure-modification?rq=1) MO question of Taras Banakh. He calls this cardinal the *Sierpiński cardinal* and denotes it $\acute{\mathfrak n}$. The comments and answers there give a good bit of information about this cardinal, including that it is consistent to have $\acute{\mathfrak n} < \mathfrak{c}$. This consistency result was first proved by Stern, who showed that it is true in the random real model.


The Sierpiński cardinal also goes by another name in the literature: $\mathfrak{a}\_T$. This is because it can be characterized as the smallest size of an infinite maximal family of almost disjoint subtrees of the infinite binary tree $2^{<\omega}$.


Arnie Miller shows in Theorem 3 of [this](https://people.math.wisc.edu/%7Eawmille1/res/cov.pdf) paper that $\mathfrak{a}\_T = \aleph\_1$ if and only if every uncountable Polish space can be partitioned into $\aleph\_1$ closed sets. As every continuum is Polish, this shows that $\mathrm{disc}(X) = \mathfrak{a}\_T = \aleph\_1$ for every continuum $X$, provided that $\mathfrak{a}\_T = \aleph\_1$. This almost answers your second question.


Miller's argument does not readily generalize to cardinals $>\!\aleph\_1$. Recently I wrote a paper about partitions of Polish spaces into closed sets (and Borel sets, more generally), and I found a a way to extend Miller's result to cardinals $>\!\aleph\_1$ (via a different approach): see Theorem 2.4 [here](https://arxiv.org/pdf/2112.00535.pdf). So it is true that $\mathrm{disc}(X) = \mathfrak{a}\_T$ for any continuum $X$; and more generally, $\mathfrak{a}\_T$ is equal to the smallest size of any partition of a Polish space $X$ into uncountably many closed sets (or compact sets, or $F\_\sigma$'s).


